electromagnetism-a collection of problems 20101026

EEM015 – ElectromagnetismA collection of problems

Thomas Rylander

Department of Signals and systems,

Chalmers University of Technology,

SE-412 96, Göteborg, Sweden

October 25, 2010

Preface

This collection of problems are intended for the course “EEM015 – Electro-magnetism” at Chalmers University of Technology. The problems mainlydivide into two groups: (i) qualitative questions; and (ii) quantitative ques-tions. The qualitative questions are suitable for discussion among the un-dergraduate students during Supplementary Instruction (SI) meetings, andthese questions are labeled “S-x.y” where x is a number that refers to thechapter and y is an index for the questions within that chapter. The quan-titative questions are suitable for Tutorial Sessions (TS) lead by a teachingassistant, and these questions are labeled “T-x.y” where x is a number thatrefers to the chapter and y is an index for the questions within that chapter.Questions that are not discussed or solved during SI meetings or tutorialsessions, the undergraduate students are expected to work out on their ownas home work assignments.Many of the problems and questions intended for the tutorial sessionsare collected from a previous collection of example problems used for thecourse “EEM015 – Electromagnetism” at Chalmers University of Technol-ogy. The previous collection of example problems was authored and as-sembled by many teachers and the persons known to the author are HansDesaix, Eva Palmberg and Hans-Georg Gustafsson. The author is indebtedto them. Also, the author would like to thank teaching assistant Per Ja-cobsson for valuable input and suggestions for improvements. Moreover,the undergraduate students are gratefully acknowledged for their feedbackand suggestions on the material and, in particular, the author wishes tomention the SI-meeting leaders David Carlsson and Jonathan Lock.

Thomas Rylander

Göteborg, SwedenOctober, 2008

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Contents

1 Introduction 6

2 Vector analysis 9

2.1 Review questions . . . . . . . . . . . . . . . . . . . . . . . . . 92.1.1 Scalars and vectors . . . . . . . . . . . . . . . . . . . . 92.1.2 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.1.3 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . 102.1.4 Coordinate systems . . . . . . . . . . . . . . . . . . . . 112.1.5 Integration of fields . . . . . . . . . . . . . . . . . . . . 122.1.6 Differentiation of fields . . . . . . . . . . . . . . . . . 132.1.7 The usage of BETA . . . . . . . . . . . . . . . . . . . . 14

2.2 Example problems . . . . . . . . . . . . . . . . . . . . . . . . 142.2.1 A scalar field . . . . . . . . . . . . . . . . . . . . . . . 152.2.2 A vector field #1 . . . . . . . . . . . . . . . . . . . . . 152.2.3 A vector field #2 . . . . . . . . . . . . . . . . . . . . . 16

3 Electrostatics 183.1 Example problems – vacuum and charge distributions . . . . 18

3.1.1 Force between two point charges . . . . . . . . . . . . 183.1.2 Electric field given two point charges . . . . . . . . . 193.1.3 Uniform line charge . . . . . . . . . . . . . . . . . . . 203.1.4 Two parallel line charges of finite length . . . . . . . . 213.1.5 Circular line charge . . . . . . . . . . . . . . . . . . . . 223.1.6 Line charge of finite length . . . . . . . . . . . . . . . 223.1.7 Electric flux integral . . . . . . . . . . . . . . . . . . . 233.1.8 A given electric potential #1 . . . . . . . . . . . . . . . 243.1.9 A given electric potential #2 . . . . . . . . . . . . . . . 25

3.2 Example problems – vacuum and charged metal bodies . . . 253.2.1 Metal sphere . . . . . . . . . . . . . . . . . . . . . . . . 253.2.2 Metal tube . . . . . . . . . . . . . . . . . . . . . . . . . 263.2.3 Two concentric metal spheres . . . . . . . . . . . . . . 273.2.4 Three concentric metal spheres . . . . . . . . . . . . . 283.2.5 Parallel thin metal wires . . . . . . . . . . . . . . . . . 29

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3.2.6 Three parallel metal wires with a battery . . . . . . . 303.2.7 Metal disc . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.3 Example problems – dielectric materials . . . . . . . . . . . . 313.3.1 Electric dipole . . . . . . . . . . . . . . . . . . . . . . . 313.3.2 Electric dipole and a point charge . . . . . . . . . . . 323.3.3 Polarized sphere . . . . . . . . . . . . . . . . . . . . . 333.3.4 Metal sphere with dielectric shell . . . . . . . . . . . . 343.3.5 Dielectric slab in external field . . . . . . . . . . . . . 34

3.4 Example problems – capacitors . . . . . . . . . . . . . . . . . 363.4.1 Parallel plate capacitor – part #1 . . . . . . . . . . . . 363.4.2 Parallel plate capacitor – part #2 . . . . . . . . . . . . 373.4.3 Two parallel wires . . . . . . . . . . . . . . . . . . . . 38

3.5 Example problems – energy . . . . . . . . . . . . . . . . . . . 383.5.1 Metal sphere . . . . . . . . . . . . . . . . . . . . . . . . 383.5.2 Electric circuit . . . . . . . . . . . . . . . . . . . . . . . 393.5.3 A given electric potential . . . . . . . . . . . . . . . . 40

3.6 Example problems – image theory . . . . . . . . . . . . . . . 413.6.1 An electric dipole . . . . . . . . . . . . . . . . . . . . . 413.6.2 A thick and a thin cylinder . . . . . . . . . . . . . . . 413.6.3 Two wires in a metal tube . . . . . . . . . . . . . . . . 423.6.4 A small sphere close to a corner . . . . . . . . . . . . 433.6.5 A thin wire close to a corner . . . . . . . . . . . . . . . 44

4 Time-invariant electric currents 46

4.1 Example problems – power dissipation . . . . . . . . . . . . 464.1.1 Power dissipation in a metal cylinder . . . . . . . . . 46

4.2 Example problems – resistance . . . . . . . . . . . . . . . . . 474.2.1 Brick shaped tank with conducting fluid . . . . . . . 474.2.2 Plate with a hole . . . . . . . . . . . . . . . . . . . . . 484.2.3 Cube with inhomogeneous conductivity . . . . . . . 484.2.4 Conducting plate #1 . . . . . . . . . . . . . . . . . . . 504.2.5 Conducting plate #2 . . . . . . . . . . . . . . . . . . . 514.2.6 Conducting plate #3 . . . . . . . . . . . . . . . . . . . 524.2.7 Conducting plate #4 . . . . . . . . . . . . . . . . . . . 54

5 Magnetostatics 56

5.1 Example problems – vacuum and current distributions . . . 565.1.1 Tubular current density . . . . . . . . . . . . . . . . . 565.1.2 Plate current density . . . . . . . . . . . . . . . . . . . 575.1.3 Current in a wire . . . . . . . . . . . . . . . . . . . . . 585.1.4 Current in three connected wires . . . . . . . . . . . . 595.1.5 Circular wire loop in a coil . . . . . . . . . . . . . . . 615.1.6 A charged metal sphere that rotates . . . . . . . . . . 615.1.7 Magnetic force on a liquid . . . . . . . . . . . . . . . . 62

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5.1.8 Magnetic flux through loop . . . . . . . . . . . . . . . 635.1.9 Current carrying wire close to superconductor . . . . 64

5.2 Example problems – magnetic materials . . . . . . . . . . . . 655.2.1 A magnetized plate . . . . . . . . . . . . . . . . . . . . 655.2.2 Current inside magnetic tube . . . . . . . . . . . . . . 655.2.3 Magnetic circuit . . . . . . . . . . . . . . . . . . . . . . 665.2.4 Self and mutual inductance . . . . . . . . . . . . . . . 675.2.5 Two circular loops . . . . . . . . . . . . . . . . . . . . 685.2.6 Toroid core . . . . . . . . . . . . . . . . . . . . . . . . . 695.2.7 Inductance . . . . . . . . . . . . . . . . . . . . . . . . . 695.2.8 Two square loops . . . . . . . . . . . . . . . . . . . . . 70

6 Quasistatics 72

6.1 Example problems – induction . . . . . . . . . . . . . . . . . 726.1.1 Metal ring in magnetic field . . . . . . . . . . . . . . . 726.1.2 Three parallel wires . . . . . . . . . . . . . . . . . . . 736.1.3 A ring transformer . . . . . . . . . . . . . . . . . . . . 746.1.4 A unipolar machine . . . . . . . . . . . . . . . . . . . 756.1.5 Conducting plate in external field #1 . . . . . . . . . . 766.1.6 Heating of conducting cylinder . . . . . . . . . . . . . 776.1.7 Power line and a barn . . . . . . . . . . . . . . . . . . 78

7 Electrodynamics 80

7.1 Example problems – real fields . . . . . . . . . . . . . . . . . 807.1.1 A charged conducting body . . . . . . . . . . . . . . . 807.1.2 Wave propagation in vacuum . . . . . . . . . . . . . . 817.1.3 Determination of electric field from magnetic field . . 817.1.4 Plane wave in a dielectric medium . . . . . . . . . . . 82

7.2 Example problems – complex fields . . . . . . . . . . . . . . 837.2.1 Sinusoidal wave in vacuum . . . . . . . . . . . . . . . 837.2.2 An evanescent wave . . . . . . . . . . . . . . . . . . . 837.2.3 A plane wave in lossy medium . . . . . . . . . . . . . 857.2.4 A plane wave in lossless medium . . . . . . . . . . . 857.2.5 Wave propagation in a good conductor . . . . . . . . 867.2.6 Energy densities of a plane wave . . . . . . . . . . . . 877.2.7 Wire with alternating current . . . . . . . . . . . . . . 877.2.8 Alternating current in coaxial cable . . . . . . . . . . 887.2.9 Conducting plate in external field #2 . . . . . . . . . . 89

7.3 Example problems – reflection and diffraction . . . . . . . . 907.3.1 Wave incident on perfect electric conductor . . . . . . 907.3.2 Plane wave incident on a dielectric half-space . . . . 917.3.3 Plane wave incident on metal plate . . . . . . . . . . . 927.3.4 A plane wave incident on a copper plate . . . . . . . 927.3.5 A plane wave incident on a dielectric . . . . . . . . . 93

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7.3.6 Oblique incidence on dielectric plate . . . . . . . . . . 947.3.7 Prism with total reflection . . . . . . . . . . . . . . . . 957.3.8 Prism without reflection losses . . . . . . . . . . . . . 96

A Answers to some problems 98A.1 Electrostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98A.2 Time-invariant electric currents . . . . . . . . . . . . . . . . . 102A.3 Magnetostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . 103A.4 Quasistatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105A.5 Electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 107

B Study plan 111

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Chapter 1

Introduction

It is often very useful to challenge yourself when you solve problems. Inthis way, you can often find good ways of solving the problem, avoid dif-ficult calculations and detect errors in your own solution without askinganyone else. Altogether, this saves substantial amounts of time in thelonger perspective. Here’s a list of questions that you can ask yourselfwhen you solve a problem – often these questions reveal potential prob-lems in a solution and can be used to avoid many mistakes. The list is byno means complete and probably you can find many other ways that helpyou do correct and reliable calculations.

• Type of problem

– What’s the underlying theory required to solve the problem?(This question is much more important than you may think...)

– What physical quantities are involved? Charges? Currents?Metals? Dielectric materials? Magnetic materials?

– Is the problem static or dynamic? Can it be treated in the fre-quency domain?

– How do the phyical quantities behave based on simple physicalreasoning?

– Can you visualize the situation and its qualitative features?

• Possible simplifications

– Can you find a simplified version of the problem that can beeasily solved for a quick comparison?

– Could you simplify the problem without loosing too much ac-curacy?

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– What sort of approximations can you use?

– Can you exploit spatial or other types of symmetries?

• Expected results

– What are you supposed to calculate? (It’s hard to find the rightanswer if you’re working on the wrong question...)

– What do you expect as a result? Is the result a vector or a scalar?Is the result a field?

– What’s the unit of the final result?

• Related problems and other solution alternatives

– Do you know about similar problems? How do they differ?What do they have in common? What can you use for the prob-lem that you are currently working on?

– What are the possible ways of solving the problem do you have?Can you make a list of different options? Which option is thebest?

– Can you solve the problem in two different ways without toomuch work? Do the answers compare well?

• Available aids

– Where can you find the relevant help in the tables of formulas?(Here you can save time if you know beforehand where to seekthe information.)

– Can you combine a few pieces of information from the availabletables of formulas to solve the problem? (Here it is very usefulto know how different pieces of information, such as formulas,relate to each other – a link that is often established by the un-derlying theory.)

– What assumptions must be fulfilled for the usage of a particularformula or expression?

• Intermediate results

– How can you check the intermediate steps and intermediate re-sults?

– Can you identify special cases where you already know the an-swer?

– What are the units of intermediate results?

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– Should the intermediate results be scalars or vectors? Did youmix scalars and vectors such that you have intermediate resultsthat state that “a scalar is equal to a vector”? (If so something isvery wrong...)

• Sanity checks

– What’s the expected order of magnitude of the result?

– Does the result have the expected sign and/or direction?

– What special cases could you use for checking the result?

– Is the unit of the result correct?

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Chapter 2

Vector analysis

2.1 Review questions

2.1.1 Scalars and vectors

In everyday life just as well as in physics, we describe different quanti-ties in different ways. For example, we speak about the temperature andgive a value such as 10C. A rather different example is the wind which ischaracterized by its strength (say 10 m/s) and its direction (say the wind isblowing to the north).

S-2.1 What characterizes a scalar?

S-2.2 What characterizes a vector?

S-2.3 Give at least five examples of scalar quantities.

S-2.4 Give at least five examples of vector quantities.

S-2.5 What characterizes the dot product of two vectors? How do youinterpret the dot product? How can you use the dot product?Can you visualize such a situation?

S-2.6 List all possible ways that you can compute the dot product oftwo vectors. How do they differ? When is it useful to one type offormula instead of another? Do they all give the same answer?

S-2.7 What is the dot product of two parallel vectors?

S-2.8 What is the dot product of two perpendicular vectors?

S-2.9 What is the dot product of two anti-parallel vectors?

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S-2.10 What characterizes the cross product of two vectors? How doyou interpret the cross product? How can you use the cross prod-uct? Can you visualize such a situation?

S-2.11 List all possible ways that you can compute the cross product oftwo vectors. How do they differ? When is it useful to one type offormula instead of another? Do they all give the same answer?

S-2.12 What is the cross product of two parallel vectors?

S-2.13 What is the cross product of two perpendicular vectors?

S-2.14 What is the cross product of two anti-parallel vectors?

2.1.2 Fields

Often a quantity such as the temperature varies, e.g. the temperature in onecity may be different from the temperature in another city. Another exam-ple is the wind that blows in different directions with different strengths indifferent parts of a country.

S-2.15 What characterizes a field?

S-2.16 Give at least five examples of scalar fields. Visualize the differentfields for some given situation. How do they vary with respect tospace? How many different independent variables do you have?

S-2.17 How can you describe a given scalar field by means of its typicalbehavior? Use the examples that you have in your list. Thinkabout different circumstances that may arise.

S-2.18 Give at least five examples of vector fields. Visualize the differentfields for some given situation. How do they vary with respect tospace? How many different independent variables do you have?

S-2.19 How can you describe a given vector field by means of its typicalbehavior? Use the examples that you have in your list. Thinkabout different circumstances that may arise.

2.1.3 Geometry

In order to describe the position of objects in relation to each other, we needtools of different type. Think about when you give directions to a friend onhow to go from the bus stop to the restaurant. For example, you may use

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language such as 50 m north of the bus stop there is a coffee shop and 5 mabove the coffee shop there is a bridge.

S-2.20 Give at least five examples of objects that you see around you.How can you describe their position in the room?

S-2.21 Introduce a Cartesian coordinate system and describe the posi-tion of different objects that you see with respect to the coordi-nate system of your choice.

S-2.22 Choose another Cartesian coordinate system where the origin isplaced somewhere else. Again, describe the position of differentobjects that you see with respect to the coordinate system of yourchoice.

S-2.23 Choose yet another Cartesian coordinate system where the coor-dinate axes are rotated. Again, describe the position of differentobjects that you see with respect to the coordinate system of yourchoice.

S-2.24 How can you compute the distance between some of the objectsthat you see around you? Does the distance depend on the coor-dinate system that you use?

S-2.25 How can you describe a line with respect to a Cartesian coordi-nate system? For example, such a line can be the straight linewhere two walls in a room meet, i.e. a corner.

S-2.26 How can you describe a surface with respect to a Cartesian coor-dinate system? For example, such a surface can be the ceiling inthe room.

S-2.27 How can you describe a volume with respect to a Cartesian coor-dinate system? For example, such a volume can be volume thata room occupies.

2.1.4 Coordinate systems

A useful tool for describing geometry is a coordinate system. The map ofManhattan resembles quite a bit a Cartesian coordinate systems, with itsstreets in the west-east direction and its avenues in the south-north direc-tion. The three most important coordinate systems for three space dimen-sions are the Cartesian coordinate system, the cylindrical coordinate systemand the spherical coordinate system.

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S-2.28 How do you describe the coordinates in these three coordinatesystems? What is the meaning of the coordinates in these threecoordinate systems? Can you visualize the surfaces where eachcoordinate is constant?

S-2.29 How do you describe the directions in these three coordinate sys-tems? What are the characteristic features of such direction de-scriptions?

S-2.30 Can you describe the position of a point in space by means of thethree coordinate systems? How do the descriptions differ? Whydo the descriptions differ now that they all describe the positionof the same point? Can you visualize the situation with the posi-tion vector that starts at the origin and ends at the position of thepoint?

S-2.31 Try to describe the surface of a brick in the three coordinate sys-tems. One coordinate system will be much easier to use as com-pared to the other two. Why?

S-2.32 Try to describe the surface of a cylinder can in the three coordi-nate systems. One coordinate system will be much easier to useas compared to the other two. Why?

S-2.33 Try to describe the surface of a sphere in the three coordinatesystems. One coordinate system will be much easier to use ascompared to the other two. Why?

S-2.34 What’s the purpose of a coordinate system?

S-2.35 Do you always need a coordinate system? When do you need itand when can you do without?

2.1.5 Integration of fields

It is often desirable to sum or integrate a quantity that varies. For example,if you’re in a car and you know its speed and direction you can use integra-tion to compute the final destination, should you know the position whereyou started. Do you know how?

S-2.36 Given a copper wire of varying diameter, i.e. some parts of thewire are thicker than other parts of the wire. How can you com-pute the total mass of the wire? The wire also has a given totallength.

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S-2.37 Given a copper sheet of varying thickness, i.e. some parts of thesheet is thicker than other parts of the sheet. How can you com-pute the total mass of the sheet? Say the sheet has the shape of arectangle and that the lengths of its sides are given.

S-2.38 Given the mass density field of say a pile of snow, i.e. the snowmay be more densely packed in some regions as compared toother regions. How can you compute the total mass of the snow?

S-2.39 Take a vector field such as the water flowing in a river. How canyou compute the total amount of water (per second) that passesthrough a fisherman’s landing-net?

S-2.40 Take a vector field such as the water flowing in a river. Imaginethat you’re able to suddenly make the water in the river deep-frozen, except for a thin tube that forms a closed contour e.g. acircle or a rectangle. Will the water in the tube stand still or willit move? If it moves, in which direction does it move? How canyou compute the velocity of the water in the tube, should it notbe standing still?

2.1.6 Differentiation of fields

It is often useful to know how a vector field varies with respect to space.The rate of change for a function is described by the derivative. For fields,we typically use the gradient, the divergence and the curl.

S-2.41 What type of field (scalar field or vector field) can you apply thegradient to? What type of field do you get as a result?

S-2.42 What type of field (scalar field or vector field) can you apply thedivergence to? What type of field do you get as a result?

S-2.43 What type of field (scalar field or vector field) can you apply thecurl to? What type of field do you get as a result?

S-2.44 Pick a field of your choice and apply the gradient to this field.What are the characteristic features of the result? Can you visu-alize the result? What’s your interpretation of the result? Whatsort of special cases can you find?

S-2.45 Pick a field of your choice and apply the divergence to this field.What are the characteristic features of the result? Can you visu-alize the result? What’s your interpretation of the result? Whatsort of special cases can you find?

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S-2.46 Pick a field of your choice and apply the curl to this field. Whatare the characteristic features of the result? Can you visualizethe result? What’s your interpretation of the result? What sort ofspecial cases can you find?

S-2.47 What are the expressions for these three differential operators inthe three typical coordinate systems that are most common forthree space dimensions?

2.1.7 The usage of BETA

For practical calculations, it is very useful to have a table of formulas suchas BETA, which you also are allowed to use at the exam!

S-2.48 Where do you find trigonometric formulas?

S-2.49 Where do you find closed form expressions for integrals? Howdo you use them?

S-2.50 Where do you find information about the Cartesian coordinatesystem? How do you use this information? Can you with theaid of the formulas calculate the gradient, the divergence and thecurl of a given field?

S-2.51 Where do you find information about the cylindrical coordinatesystem? How do you use this information? Can you with theaid of the formulas calculate the gradient, the divergence and thecurl of a given field?

S-2.52 Where do you find information about the spherical coordinatesystem? How do you use this information? Can you with theaid of the formulas calculate the gradient, the divergence and thecurl of a given field?

S-2.53 What other types of useful information can you find in the tableof formulas?

2.2 Example problems

It is useful for understanding to visualize fields. Here are some examplesthat may help you to relate mathematical expressions of some canonicalfields to their appearance.

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2.2.1 A scalar field

Problem description

Consider the scalar field φ = x2 + y2.

Qualitative questions

S-2.54 Evaluate the field at some points in the xy-plane.

S-2.55 What characteristic features does the field have?

S-2.56 Also, plot equipotential surfaces. What are their shape?

S-2.57 Evaluate the gradient of this field and visualize the result. Whatare the characteristic features of the gradient field? How does thegradient field relate to the equipotential surfaces?

S-2.58 Can you reformulate the scalar field in another coordinate systemthat may be more suitable?

Quantitative questions

T-2.1 Given the vector field ~F = −x2x − y2y, calculate the scalar fieldφ such that ~F = −∇φ.

2.2.2 A vector field #1

Problem description

Consider the vector field ~F = xx+ yy.




S-2.61 Calculate the divergence of this field and visualize the result.What are the characteristic features of the divergence field?

S-2.62 Calculate the curl of this field and visualize the result. What arethe characteristic features of the curl field?

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S-2.63 Can you suggest a coordinate system that may be more suitablefor this particular field?


T-2.2 Calculate the divergence and curl of the field in Cartesian coordi-nates. Reformulate the result in terms of cylindrical coordinates.

T-2.3 Reformulate the original vector field in cylindrical coordinates.Given the expression in cylindrical coordinates, calculate againthe divergence and curl of the field.

T-2.4 Compare the results above. What is the conclusion?

2.2.3 A vector field #2

Problem description

Consider the vector field ~F = −xy + yx.




S-2.66 Calculate the curl of this field and visualize the result. What arethe characteristic features of the curl field?

S-2.67 Calculate the divergence of this field and visualize the result.What are the characteristic features of the divergence field?

S-2.68 Can you suggest a coordinate system that may be more suitablefor this particular field?


T-2.5 Calculate the divergence and curl of the field in Cartesian coordi-nates. Reformulate the result in terms of cylindrical coordinates.

T-2.6 Reformulate the original vector field in cylindrical coordinates.Given the expression in cylindrical coordinates, calculate againthe divergence and curl of the field.

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T-2.7 Compare the results above. What is the conclusion?

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Chapter 3

Electrostatics

3.1 Example problems – vacuum and charge distribu-

tions

3.1.1 Force between two point charges

Problem description

Consider a point charge q1 located at ~r1 and another point charge q2 locatedat ~r2.


S-3.1 What force do you expect acts on the point charge q1? What’s thedirection and magnitude of this force?

S-3.2 What force do you expect acts on the point charge q2? What’s thedirection and magnitude of this force?

S-3.3 How do the two forces compare with each other?

S-3.4 How do the two forces depend on q1 and q2?

S-3.5 How do the two forces depend on ~r1 and ~r2?


T-3.1 Calculate the force that acts on q1 when ~r1 = x3 and ~r2 = y5.What’s the direction and magnitude of the force?

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T-3.2 Calculate the force that acts on q2 when ~r1 = x3 and ~r2 = y5.What’s the direction and magnitude of the force?

T-3.3 Visualize the two point charges in the xy-plane together with theforces that act on them.

3.1.2 Electric field given two point charges

Problem description

Consider a point charge q1 located at ~r1 and another point charge q2 locatedat ~r2.


S-3.6 What electric field do you expect from the point charge q1?What’s the direction and magnitude of this electric field?

S-3.7 What electric field do you expect from the point charge q2?What’s the direction and magnitude of this electric field?

S-3.8 What’s the superposition of the electric field from q1 and q2, i.e.the total electric field?

S-3.9 How does the electric field depend on q1 and q2?

S-3.10 How does the electric field depend on ~r1 and ~r2?

S-3.11 What type of symmetries do you expect from the electric field?

S-3.12 Visualize the electric field lines when q1 = q2 together with thetwo point charges. Visualize also the equipotential surfaces.

S-3.13 Visualize the electric field lines when q1 = −q2 together with thetwo point charges. Visualize also the equipotential surfaces.


T-3.4 Calculate the total electric field along the line that goes throughboth ~r1 and ~r2. What’s the direction and magnitude of the electricfield? Visualize the result. How does the result depend on thedistance between the two point charges? How does the resultdepend on the value of q1 and q2?

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T-3.5 Visualize the electric field lines two cases when q1q2 > 0 andq1q2 < 0 given that (i) |q1| > |q2|, (ii) |q1| = |q2| and (iii) |q1| <|q2|, which yields 12 different situations. Are some of the casessimilar? Make a table that displays the possible combinations.Also, include the two point charges in the visualization.

T-3.6 Under what circumstances can the electric field be zero along theline that goes through both ~r1 and ~r2? At which points may thisoccur?

3.1.3 Uniform line charge

Problem description

An infinitely long line charge is aligned with the z-axis and its charge den-sity ρl is constant.


S-3.14 How do you expect that the magnitude of the electric field varywith respect to space?

S-3.15 How do you expect the electric field to be directed? Divide theline charge into short segments and consider the contributionfrom each such segment by means of superposition – it is as-sumed that the segments are so short that they can be consideredpoint sources when viewed from the field point.

S-3.16 Visualize the electric field in a plane perpendicular to the linecharge, together with the line charge itself.

S-3.17 How does the electric field depend on the charge density ρl?

S-3.18 What type of symmetries can you identify?

S-3.19 What is a suitable coordinate system for this problem?

S-3.20 What happens if the line charge is of finite length? What type ofsimilarities and differences can you identify?


T-3.7 Calculate the electric field everywhere given a line charge bymeans of superposition.

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T-3.8 Calculate the electric field everywhere given a line charge bymeans of Gauss law.

3.1.4 Two parallel line charges of finite length

Problem description

Consider two thin and straight line charges of length a that are positionedalong the opposite sides of a square, where the sides of the square are oflength a. One line charge has the total charge Q uniformly distributedalong its length. The other line charge has the total charge −Q uniformlydistributed along its length.


S-3.21 Visualize the two uniformly charged lines in the plane that coin-cides with them.

S-3.22 Visualize the electric field for this situation, both between the linecharges and in the region around them.

S-3.23 Which direction do you expect the electric field point at the centerpoint of the square?

S-3.24 What type of symmetries does the field lines exhibit?

S-3.25 Can you estimate the magnitude of the electric field at the centerpoint of the square? For example, what happens if you approxi-mate the charge distribution of a line charge with its total chargelocated at the center of that line charge? How does such an ap-proximation change the resulting electric field? Does the direc-tion change? Does the magnitude change? Can you use this for arough estimate of the electric field?


T-3.9 Compute the exact value of the electric field at the center of thesquare.

21

3.1.5 Circular line charge

Problem description

A circular line charge of radius a and constant charge density ρl is placedwith its center at the origin of a Cartesian coordinate system such that itcoincides with the xy-plane.


S-3.26 What type of symmetry does the charge distribution display?

S-3.27 Visualize the electric field and the electric potential around theline charge.

S-3.28 How do you expect that the electric field and the electric potentialvary along the z-axis?

S-3.29 What type of symmetries can you identify for the electric fieldand the electric potential?

S-3.30 Could you calculate the electric field along the z-axis only basedon the corresponding electric potential along the z-axis? If so,why is it possible to do this type of calculation in this particularcase?


T-3.10 Calculate the electric field along the z-axis.

T-3.11 Calculate the electric potential along the z-axis.

T-3.12 Calculate the electric field along the z-axis based on the corre-sponding expression for the electric potential.

3.1.6 Line charge of finite length

Problem description

A line charge of length L is aligned with the z-axis and it extends fromz = −L/2 to z = +L/2. The charge density along the line charge is constantand it equals ρl.

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S-3.31 Visualize the electric potential around the line charge in a planethat coincides with the line charge.

S-3.32 What type of symmetries does the electric potential display?

S-3.33 Visualize the electric field around the line charge in a plane thatcoincides with the line charge.

S-3.34 What type of symmetries does the electric field display with re-spect to magnitude and direction of its field vector?

S-3.35 How does the electric field and the electric potential relate to eachother?

S-3.36 What’s the characteristic features of the electric potential and theelectric field for field points where the distance to the line chargeis small as compared to the distance to both its end points.

S-3.37 What’s the characteristic features of the electric potential and theelectric field for field points where the distance to the line chargeis large as compared to the length L of the line charge.


T-3.13 Compute the electric potential everywhere.

T-3.14 Compute the electric field everywhere.

3.1.7 Electric flux integral

Problem description

A point charge Q is placed at the origin. The electric flux ψ through theopen surface S is given by

ψ =

∫

Sǫ0 ~E · d~s. (3.1)

The unit normal vector of the surface S points away from the origin.

23


S-3.38 What does the quantity ψ represent?

S-3.39 How can you compute ψ? Do you have several alternatives?

S-3.40 What’s the unit of ψ?


T-3.15 Calculate ψ for the square surface S with its corners at the points(a, 0, 0), (a, a, 0), (a, a, a) and (a, 0, a). (Here, the constant a ispositive.)

T-3.16 Calculate ψ for the curved surface S that fulfills the requirementsx2 + y2 + z2 = a2, x ≥ 0, y ≥ 0 and z ≥ 0. (Here, the constant a ispositive.)

T-3.17 Calculate ψ for the triangular surface S with its corners at thepoints (a, 0, 0), (0, a, 0) and (0, 0, a). (Here, the constant a is posi-tive.)

3.1.8 A given electric potential #1

Problem description

A spherical symmetric charge distribution in vacuum yields a sphericalsymmetric potential that is given by

V (R) =Q

4πǫ0(R + a)

where a > 0 is a constant.


S-3.41 Visualize the electric potential as a function of R.

S-3.42 Visualize the electric field as a function of R.

S-3.43 What type of charge distribution do you expect?

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T-3.18 Calculate the charge distribution.

T-3.19 Calculate the total charge.

3.1.9 A given electric potential #2

Problem description


V (R) =

V0[1− (R/a)2] for R ≤ a0 for R > a

in spherical coordinates.






T-3.20 Calculate the charge distribution.

T-3.21 Calculate the total charge.

3.2 Example problems – vacuum and charged metal

bodies

3.2.1 Metal sphere

Problem description

A metal sphere of radius a = 10 cm is located in air at a large distance fromother objects. The sphere has the total charge Q. The so-called dielectric

25

strength of air is Emax = 3 · 106 V/m. When this value of the electric fieldis exceeded, the air ionizes and the sphere is discharged by a spark, whichis referred to as dielectric breakdown.


S-3.47 What’s the charge distribution on the sphere? What type of sym-metries can you identify?

S-3.48 How do you expect that the electric field is directed around themetal sphere? How do you expect that the magnitude of the elec-tric field depends on the spatial coordinates? What type of sym-metries can you identify?

S-3.49 Visualize the metal sphere together with the electric field aroundthe sphere.

S-3.50 Where do you have the highest values for the electric field?


T-3.22 What’s the maximum value of the total chargeQ before dielectricbreakdown?

T-3.23 What’s the potential of the sphere at the limit of dielectric break-down?

3.2.2 Metal tube

Problem description

The charge Q is uniformly distributed on the surface of a very long metalcylinder of radius a and length L. The metal tube will be referred to as ametal cylinder below.


S-3.51 What is the charge density in this case? What’s the unit of thecharge density?

S-3.52 How do you expect that the magnitude and the direction of theelectric field vary with respect to space?

26

S-3.53 Visualize the electric field in the plane perpendicular to the metalcylinder’s axis of revolution, where the plane intersects the cylin-der at its midpoint. Also, include the cross-section of the metalcylinder itself.

S-3.54 What type of symmetries can you identify? How does the sym-metries manifest themselves regarding the direction and magni-tude of the electric field? What’s a suitable coordinate system fordetailed calculations?

S-3.55 What do you think happens close to the end points of the cylin-der? What’s the impact of your findings on the results when youare far away from the end points?

S-3.56 Given the electric field, how do you expect the electric potentialto vary with respect to space?

S-3.57 What type of symmetries does the potential exhibit?

S-3.58 What type of electric field and electric potential do you expect faraway from the tube?


T-3.24 Compute the electric field in the region close to the midpoint ofthe metal cylinder.

T-3.25 Compute the electric potential in the region close to the midpointof the metal cylinder, should the potential be zero at r = b > a.

3.2.3 Two concentric metal spheres

Problem description

A metal sphere of radius awith the total charge +Q is placed concentricallyinside another metal sphere of radius b > a and the total charge −Q.


S-3.59 Visualize the two metal spheres. Also, introduce the dimensionsand indicate the total charge associated with each sphere.

S-3.60 What is the charge density in this case? What’s the unit of thecharge density?

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S-3.61 How do you expect that the magnitude and the direction of theelectric field vary with respect to space?

S-3.62 Visualize the electric field in the plane that coincides with thecenter of the two spheres, together with the metal spheres them-selves.

S-3.63 What type of symmetries can you identify? How does the sym-metries manifest themselves regarding the direction and magni-tude of the electric field? What’s a suitable coordinate system fordetailed calculations?

S-3.64 Given the electric field, how do you expect the electric potentialto vary with respect to space?

S-3.65 What type of symmetries does the potential exhibit?

S-3.66 How does this problem compare with the example prob-lem 3.2.2? What are the similarities and differences?


T-3.26 Compute the electric field everywhere.

T-3.27 Compute the electric potential everywhere, should the potentialbe zero at r = b.

3.2.4 Three concentric metal spheres

Problem description

Three concentric metal spheres of radii a, b and c are placed in free space,where a < b < c. They have the total charge qa, qb and qc, respectively.


S-3.67 How does the charge distribute on the three metal spheres? Whattype of symmetries can you identify?

S-3.68 How does the charge distribution influence the electric field andthe electric potential? What type of symmetries can you expectfor the electric field and the electric potential?

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S-3.69 What happens if the the spheres of radii b and c are connected toeach other by a metal wire? How does that influence the electricfield and the electric potential?


T-3.28 Calculate the electric potential at the center of the three spheres,should the potential at infinity be set to zero.

T-3.29 Calculate the electric potential at the center of the three spheresfor the case when the two spheres of radii b and c are connectedby a metal wire. Again, the potential is set to zero at infinity.

3.2.5 Parallel thin metal wires

Problem description

Four thin, long, straight and parallel conductors are charged according tothe figure. The length of the conducting wires is L and their radii is b. Theseparation distance between two adjacent wires is a as shown in the figure.


S-3.70 Visualize the electric field between the line charges.

S-3.71 Visualize the electric potential between the line charges.

S-3.72 How will the charge distribute on the surface of the wires? Howdoes the fact that the wires are thin influence the charge distribu-tion?


T-3.30 Compute the force on one of the outer conductors.

T-3.31 Compute the potential difference between the outer conductors.

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3.2.6 Three parallel metal wires with a battery

Problem description

Three thin, long, straight, parallel conducting wires of radius a and lengthL are placed at the distance b and d from each other according to the figurebelow. Here, a is much smaller than the other dimensions L, b and d. Abattery of voltage U0 is connected with its positive electrode to wire A andB, while the negative electrode is connected to wire C.


S-3.73 Assume that the wires are uncharged from the beginning. Whathappens when the battery is connected to the wires?

S-3.74 How are the wires charged? What do the wires correspond to inan electrical circuit context?

S-3.75 What’s the expected charge distribution on each wire? What’s thetotal charge on each wire? Is the total charge positive or negative?

S-3.76 What’s the significance of the small radius of the wires?


T-3.32 Calculate the total charge on the three individual wires.

30

3.2.7 Metal disc

Problem description

A thin circular metal disc of radius a is located in free-space, far away fromother objects. The disc is charged with the total charge Q, which distributeson both sides of the disc. The variation of the surface charge density as afunction of the radius r (from the center of the disc) can be expressed as

ρs(r) =Q

4πa√a2 − r2

(3.2)


S-3.77 Visualize the charge distribution on the surface of the disc. Howdoes the charge distribution change close to the edge of the disc?What type of symmetries can you identify?

S-3.78 How do you expect the electric potential to depend on the spacecoordinates? Can you identify any particular symmetries?

S-3.79 How does the electric potential vary in the metal? How can youexploit this for the calculation of the electric potential of the metaldisc?


T-3.33 What’s the potential of the metal disc, should the potential faraway from the disc be set to zero?

3.3 Example problems – dielectric materials

3.3.1 Electric dipole

Problem description

An electric dipole with the dipole moment ~p is placed at the origin accord-ing to the figure. The dipole is perpendicular to the z-axis and α denotesthe angle between the dipole and the y-axis.

31


S-3.80 Visualize the electric field around the electric dipole. What char-acteristic features does the electric field display?



T-3.34 Calculate the electric field at the point P, which is located on thepositive x-axis at a distance a from the origin.

3.3.2 Electric dipole and a point charge

Problem description

A point charge is placed at the distance d from an electric dipole with thedipole moment ~p.


S-3.82 Visualize the electric field around the electric dipole. What char-acteristic features does the electric field display?


S-3.84 What happens to the point charge in this situation?

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S-3.85 What happens to the electric dipole in this situation?


T-3.35 Calculate the force on the point charge, given the electric fieldfrom the dipole.

T-3.36 Calculate the net force on the electric dipole, given the electricfield from the point charge.

3.3.3 Polarized sphere

Problem description

A dielectric sphere of radius a has the constant polarization ~P = RP0, i.e.the polarization vector points in the radial direction of a spherical coor-dinate system where its origin coincides with the center of the dielectricsphere.


S-3.86 Visualize this situation, where you include both the sphere andthe polarization vector.

S-3.87 How do you expect that the electric field varies with respect tospace in this situation? What about the electric flux density?What about the fields outside and inside the sphere? Do theydiffer?

S-3.88 How could you calculate the electric field and the electric fluxdensity?

S-3.89 How could you calculate the electric potential?


T-3.37 Calculate the potential at the center of the sphere based on theconstitutive relation that relates the electric flux density to theelectric field.

T-3.38 Calculate the potential at the center of the sphere based on theequivalent polarization charge distribution expressed in terms ofthe polarization vector.

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3.3.4 Metal sphere with dielectric shell

Problem description

A metal sphere of radius a has a dielectric shell of inner radius a and outerradius b > a. The relative permittivity of the dielectric shell is ǫr and themetal sphere has the total charge Q.


S-3.90 How will the charge on the metal sphere be distributed?

S-3.91 Visualize the problem: the metal sphere; the dielectric shell; andthe charge distribution on the metal sphere.

S-3.92 How do you expect that the electric flux density varies with re-spect to space? What about the electric field and the electric po-tential?

S-3.93 How does the dielectric shell polarize? What’s the effect of theinduced equivalent polarization charge distributions in the di-electric shell?


T-3.39 Calculate the electric flux density everywhere.

T-3.40 Calculate the electric field everywhere.

T-3.41 Calculate the electric potential everywhere.

T-3.42 Calculate the polarization vector everywhere. What are theequivalent polarization charge densities for this setting?

3.3.5 Dielectric slab in external field

Problem description

A large dielectric slab of relative permittivity ǫr =√3 is subject to an exter-

nal homogeneous electric field according to the figure below. The electricfield lines outside the dielectric slab make the angle α1 = 30 with respectto the normal of the slab’s surface.

34


S-3.94 What happens in the dielectric slab when the external field is ap-plied?

S-3.95 What type of equivalent polarization charge densities do you ex-pect? How will the equivalent polarization charge densities de-pend on the space coordinates?

S-3.96 How could you generate the external field?


T-3.43 Calculate the electric field vector inside the slab.

T-3.44 Calculate the angle α2 between the electric field lines inside theslab and the normal to the surface of the slab.

T-3.45 Calculate the equivalent surface polarization charge density.

35

3.4 Example problems – capacitors

3.4.1 Parallel plate capacitor – part #1

Problem description

Two square metal plates of side a are placed at a distance d from each other.Let’s introduce a coordinate system such that the first plate coincides withthe plane z = 0 and occupies the region defined by 0 ≤ x ≤ a and 0 ≤y ≤ a. Then the other plate coincides with the plane z = d and occupiesthe region defined by 0 ≤ x ≤ a and 0 ≤ y ≤ a. The volume defined by0 ≤ x ≤ a, 0 ≤ y ≤ a, 0 ≤ z ≤ h is occupied by a dielectric slab with therelative permittivity ǫr, where h < d. A battery of voltage U0 is connectedwith its positive electrode to the upper plate and its negative electrode tothe lower plate, which is also grounded.


S-3.97 Visualize the geometry with the battery and the grounding.

S-3.98 What happens in this situation?

S-3.99 Where will there be charge? On the metal plates?

S-3.100 How do you expect that the electric flux density varies with re-spect to space? Can visualize the electric flux density?

S-3.101 What do you expect from the corresponding electric field andelectric potential?

S-3.102 What boundary conditions must be fulfilled by the electric fluxdensity and the electric field?

S-3.103 How is the capacitance influenced by the dielectric slab?


T-3.46 Calculate the electric flux density everywhere between the metalplates.

T-3.47 Calculate the electric field everywhere between the metal plates.

T-3.48 Calculate the electric potential everywhere between the metalplates.

T-3.49 Calculate the capacitance for this situation.

36

3.4.2 Parallel plate capacitor – part #2

Problem description

Two square metal plates of side a are placed at a distance d from each other.Let’s introduce a coordinate system such that the first plate coincides withthe plane z = 0 and occupies the region defined by 0 ≤ x ≤ a and 0 ≤y ≤ a. Then the other plate coincides with the plane z = d and occupiesthe region defined by 0 ≤ x ≤ a and 0 ≤ y ≤ a. The volume defined by0 ≤ x ≤ a, 0 ≤ y ≤ b, 0 ≤ z ≤ d is occupied by a dielectric slab with therelative permittivity ǫr, where b < a. A battery of voltage U0 is connectedwith its positive electrode to the upper plate and its negative electrode tothe lower plate, which is also grounded.


S-3.104 Visualize the geometry with the battery and the grounding.


S-3.106 Where will there be charge? On the metal plates?

S-3.107 How do you expect that the electric flux density varies with re-spect to space? Can visualize the electric flux density?

S-3.108 What do you expect from the corresponding electric field andelectric potential?

S-3.109 What boundary conditions must be fulfilled by the electric fluxdensity and the electric field?

S-3.110 How is the capacitance influenced by the dielectric slab?


T-3.50 Calculate the electric flux density everywhere between the metalplates.

T-3.51 Calculate the electric field everywhere between the metal plates.

T-3.52 Calculate the electric potential everywhere between the metalplates.

T-3.53 Calculate the capacitance for this situation.

37

3.4.3 Two parallel wires

Problem description

Two thin metal wires of radii a and length L are placed parallel to eachother at a distance d≪ L, where a≪ d.


S-3.111 Assume that the two wires are initially uncharged. What hap-pens if a battery is connected to the two wires, with its positiveelectrode to one wire and its negative electrode to the other wire?

S-3.112 Where is the charge located? How is the charge distributed?What type of charge distribution will we have?

S-3.113 Visualize the electric field lines around and in-between the wiresfor a cross-section that is perpendicular to the wires and inter-sects the wires somewhere in the middle of the wires. Also, in-clude the wires themselves together with the battery.

S-3.114 How does the fact that the wires are thin influence the chargedistribution and the electric field?

S-3.115 Visualize also the electric potential.

S-3.116 How does the electric field and the electric potential vary withrespect to the space coordinates for field points that are far awayfrom the wires?


T-3.54 Calculate the capacitance between the metal wires.

3.5 Example problems – energy

3.5.1 Metal sphere

Problem description

A metal sphere of radius a and the total charge Q is located in a mediumwith the constant permittivity ǫ.

38


S-3.117 How will the charge distribute on the metal sphere?

S-3.118 What types of symmetries do you expect that the electric field,the electric flux density and the electric potential exhibit?

S-3.119 Suggest two different ways to calculate the electrostatic energy ofthis system. How do the differ?


T-3.55 Calculate the electrostatic energy of the system by means of thecharge and electric potential.

T-3.56 Calculate the electrostatic energy of the system by means of theelectric flux density and the electric field.

3.5.2 Electric circuit

Problem description

Three capacitors are from the beginning equally charged with the charge±Q, according the figure below. First, the capacitors C1 = C and C2 = 2Care connected by a switch until a new stationary state is reached. Then,the switch is used to connect the capacitors C1 = C and C3 = 3C instead,which yields yet another stationary state.


S-3.120 What happens during the switching?

39

S-3.121 Do you expect that the electrostatic energy changes as a result ofthe switching?

S-3.122 Do you expect that the charge on the capacitors changes as a re-sult of the switching?

S-3.123 What’s the condition for the new stationary states that occur dueto the switching?


T-3.57 Calculate the energy dissipated as heat in the resistance R as aresult of the switching.

3.5.3 A given electric potential

Problem description


V (R) =

V0[1− (R/a)2] for R ≤ a0 for R > a

in spherical coordinates.





S-3.127 Suggest two different ways to calculate the electrostatic energyof the system. How do they differ? Which one do you expectyields the easiest calculations? Are the two results expected to beequal?


T-3.58 Calculate the electrostatic energy of the system.

40

3.6 Example problems – image theory

3.6.1 An electric dipole

Problem description

An electric dipole ~p is located at the height z = h over a large metal groundplane, that coincides with z = 0. The angle between the electric dipolemoment ~p and z is π/4.


S-3.128 How can you model the electric dipole by means of two pointcharges? How are these charges imaged in the ground plane?

S-3.129 Which direction of the electric field do you expect at the pointwith the coordinates x = h and y = 0 that coincides with theplane spanned by ~p and z?

S-3.130 What boundary conditions must be fulfilled on the metal plane?

S-3.131 How can you calculate the electric field and/or the electric po-tential for this situation?


T-3.59 Calculate the electric field ~E at the point with the coordinatesx = h and y = 0 that coincides with the plane spanned by ~p andz.

3.6.2 A thick and a thin cylinder

Problem description

Two cylinders of radii a and b≪ a are parallel, where the distance betweenthe cylinder axes is d. The cylinders are placed in air.


S-3.132 What type of symmetries do you expect that the electric field andthe electric potential exhibit?

41

S-3.133 How do you think the charge distributes on the two cylinders?How does this influence your modeling of the two cylinders?How does this influence your calculation of the electric fieldand/or the electric potential?

S-3.134 What boundary conditions must be fulfilled on the metal cylin-ders?



T-3.60 Calculate the capacitance between the two cylinders.

3.6.3 Two wires in a metal tube

Problem description

Two equally long, thin and parallel wires are placed inside a long metaltube of radius a. The distance between the wires is b and they are locatedon opposite sides of the tube’s cylinder axis at a distance b/2. The two wireshave the charge +Q and −Q. The metal tube is uncharged.


S-3.136 What types of symmetries can you expect from the electric fieldand the electric potential?

S-3.137 How do you expect that the induced charge on the metal tubedistributes? How can this charge distribution influence the forceon the two wires?

S-3.138 What boundary conditions must be fulfilled on the metal wiresand the metal tube?



T-3.61 Calculate b such that the electrostatic force on the two wires iszero.

42

3.6.4 A small sphere close to a corner

Problem description

A small metal sphere of radius a carries the total charge Q. The metalsphere is located at the distance b ≫ a from two large metal planes thatform a corner according to the figure below.


S-3.140 How do you expect that the charge distributes on the small metalsphere? Why?

S-3.141 What happens with the charge on the metal planes? How is thischarge distributed?

S-3.142 What boundary conditions must be fulfilled on the metal sphereand on the metal planes?



T-3.62 Calculate the force on the metal sphere.

43

T-3.63 Calculate the electric potential on the metal sphere, should theelectric potential be zero at infinity.

3.6.5 A thin wire close to a corner

Problem description

Problem description

A long and thin metal wire of radius a and length L carries the total chargeQ. The metal wire is located at the distance b ≫ a from two large metalplanes that form a corner according to the figure below.


S-3.144 How do you expect that the charge distributes on the thin metalwire? Why?

S-3.145 What happens with the charge on the metal planes? How is thischarge distributed?

S-3.146 What boundary conditions must be fulfilled on the metal wiresand on the metal planes?

44



T-3.64 Calculate the force per unit length on the metal wire.

T-3.65 Calculate the electric potential on the metal wire, should the elec-tric potential be zero at infinity.

45

Chapter 4

Time-invariant electric currents

4.1 Example problems – power dissipation

4.1.1 Power dissipation in a metal cylinder

Problem description

A metal cylinder of radius a and length L has the conductivity σ. Inside

the metal cylinder, the electric field is given by ~E = ϕE0r/a, where E0 is aconstant that corresponds to the electric field at r = a.


S-4.1 Visualize the electric field together with the metal cylinder.

S-4.2 How does the electric current density relate to the electric field?

S-4.3 How does the power dissipation depend on the spatial coordi-nates?


T-4.1 Calculate the total power dissipation in the metal cylinder.

46

4.2 Example problems – resistance

4.2.1 Brick shaped tank with conducting fluid

Problem description

A brick shaped tank has a metal bottom that connects two of the oppositewalls, where these two walls of width w are made of metal. The remain-ing two walls, which are opposite to each other, are made of an electricallyinsulating material. The brick shaped tank is filled with an electrically con-ducting fluid with the conductivity σ, which is significantly smaller thanthe conductivity of the metal. At the depth b and the distance a from oneof the metal walls, a metal wire of radius c is placed perpendicular to theinsulating walls. The wire is connected to the positive electrode of a batterywith the voltage U0, where the negative electrode is connected to the metalpart of the tank. The depth of the tank is much larger than the distance bbetween the wire and the surface of the conducting fluid.


S-4.4 What happens when the battery is connected to the wire and themetal part of the tank?

S-4.5 What type of current density do you expect in this problem?

S-4.6 What boundary condition should be fulfilled on the metal wireand the metal tank?

S-4.7 What boundary condition should be fulfilled on the surface ofthe conducting fluid?

S-4.8 How does the fact that the wire is thin influence the field solutionand the calculation of the electric current density? How close canthe wire be to the wall or the surface of the conducting fluid?


T-4.2 Calculate the resistance between the wire and the metal part ofthe tank.

47

4.2.2 Plate with a hole

Problem description

Two parallel and thin metal wires of radius b are parallel and the distancebetween the wires is 4a. The wires pass through a very large conductingplate such that the wires are perpendicular to the plate. The conductingplate has the thickness d and a conductivity σ, which is significantly smallerthan the conductivity of the metal wires. A circular hole of radius a isdrilled through the plate and the center of the hole is located midway be-tween the two wires. A battery of voltage U0 is connected to two wires.


S-4.9 What happens when the battery is connected to the two wires?

S-4.10 What type of current density do you expect in this problem?

S-4.11 What boundary condition should be fulfilled on the metal wires?

S-4.12 What boundary condition should be fulfilled on the surface ofthe hole?

S-4.13 How does the fact that the wires are thin influence the field so-lution and the calculation of the electric current density? Howclose can the wire be to the surface of the hole?


T-4.3 Calculated the resistance between the two wires.

4.2.3 Cube with inhomogeneous conductivity

Problem description

A cube of side a has an inhomogeneous conductivity σ(x) = σ0(1 + x/a),where the axes of a Cartesian coordinate system are aligned with threeedges of the cube according to the figure below. The positive electrode of abattery is connected to a metal plate of side a and the negative electrode toanother square metal plate of the same size.

48


S-4.14 Visualize the space variation of the conductivity together withthe cube.

S-4.15 What happens when the metal plates of the battery is connectedto two opposite surfaces of the cube? How do the different op-tions compare?

S-4.16 What type of current density distribution do you expect? Canyou relate this situation to a similar situation that may occur inan electric circuit?

S-4.17 How does the inhomogeneous conductivity influence the electriccurrent density once the battery is connected to the cube?


T-4.4 Calculate the resistance of the cube when the battery is con-nected to the surfaces of the conducting cube that have constantx-coordinates.

T-4.5 Calculate the resistance of the cube when the battery is con-nected to the surfaces of the conducting cube that have constanty-coordinates.

49

4.2.4 Conducting plate #1

Problem description

A conductive plate of thickness d with the conductivity σ is shaped asshown in the figure below. The electrodes of a battery is connected to theleft side (denoted A) and the right side (denoted B).


S-4.18 What happens when the battery is connected to the plate?

S-4.19 Visualize the current density in the plate together with the plateitself.

S-4.20 What happens if you force the current to flow along paths thatare easy to use if you should calculate the resistance?

S-4.21 What happens if you introduce regions of infinite conductivitysuch that the resistance is easy to calculate?


T-4.6 Calculate an lower bound for the resistance between the two endsA and B of the plate.

T-4.7 Calculate an upper bound for the resistance between the twoends A and B of the plate.

50


Problem description

A conductive plate of thickness t with the conductivity σ is shaped as aquarter of an annular disc of inner radius a and outer radius b, as shown inthe figure below. A battery is connected to (i) the sides labeled A and B inthe figure or (ii) the sides labeled C and D in the figure.


S-4.22 What happens when the battery is connected to the plate alongthe sides A and B?

S-4.23 What happens when the battery is connected to the plate alongthe sides C and D?

S-4.24 Visualize the current density distribution for the two cases: (i)battery connected to the sides A and B; and (ii) battery connectedto the sides C and D. How do the current densities compare?What type of coordinate system is natural to use for the descrip-tion of these current densities?

S-4.25 What boundary conditions should be fulfilled on the sides of theplate when the battery is connected to the sides A and B?

51

S-4.26 What boundary conditions should be fulfilled on the sides of theplate when the battery is connected to the sides C and D?

S-4.27 Given a guess for the current density, how can you validate thatyour guess solves this problem exactly?

S-4.28 How can you calculate the resistance for the two different waysof connecting the battery?


T-4.8 Calculate the resistanceRAB when the battery is connected to thesides A and B.

T-4.9 Calculate the resistanceRCD when the battery is connected to thesides C and D.

T-4.10 Calculate the product RABRCD.


Problem description

A conductive plate of thickness t with the conductivity σ is shaped as twoquarters of an annular disc of inner radius a and outer radius b = 2a, asshown in the figure below. A battery is connected to the two sides labeledA and B in the figure.

52




S-4.31 Can you compute the exact current density distribution for thisproblem?

S-4.32 What happens if you force the current to flow along paths thatare easy to use if you should calculate the resistance?

S-4.33 What happens if you introduce regions of infinite conductivitysuch that the resistance is easy to calculate?


T-4.11 Calculate an lower bound for the resistance between the two endsA and B of the plate.

T-4.12 Calculate an upper bound for the resistance between the twoends A and B of the plate.

53


Problem description

A conductive plate of thickness t with the conductivity σ is shaped as twoquarters of an annular disc of inner radius a and outer radius b = 2a, asshown in the figure below. A battery is connected to the two sides labeledA and B in the figure. The figure also shows some equipotential surfaces(−1000,−800,−600, . . . , 600, 800, 1000 V) when a battery of voltage U0 =2000 V is applied to the sides A and B.




S-4.36 What boundary conditions do you have on the sides of the plate?Does the equipotential surfaces and the current density satisfythe boundary conditions?

S-4.37 How can you calculate the resistance of the plate? List all theoptions that you can find. Which option is the easiest? Whichoption would yield the most accurate answer?

54


T-4.13 Compute the resistance of the plate based on the equipotentialsurfaces shown in the figure.

55

Chapter 5

Magnetostatics

5.1 Example problems – vacuum and current distribu-

tions

5.1.1 Tubular current density

Problem description

The total current I flows in the z-direction in a tubular shell of inner radiusa and outer radius b as shown in the figure below.

56


S-5.1 What type of symmetries does the current density exhibit?

S-5.2 Which direction do you expect for the magnetic flux density?How does the magnitude of the magnetic flux density vary withrespect to space?

S-5.3 Visualize the magnetic flux density together with the currentdensity.

S-5.4 What type of symmetries do you expect that the magnetic fluxdensity exhibit?

S-5.5 What coordinate system is appropriate for describing the currentdensity and the magnetic flux density?


T-5.1 Calculate the magnetic flux density everywhere.

5.1.2 Plate current density

Problem description

A very long, thin and flat metal plate of width 2a carries the total currentI in the z-direction, as shown in the figure below. The current density isuniformly distributed over the metal plate. A point P of particular interestis also shown in the figure, where the point has the position vector ~r =xa+ ya.

57


S-5.6 What type of symmetries does the current density exhibit?

S-5.7 Visualize the magnetic flux density together with the currentdensity.

S-5.8 Which direction do you expect for the magnetic flux density atthe point P? What approximate magnitude of the magnetic fluxdensity do you expect at the point P?

S-5.9 How can you calculate the magnetic flux density at the point P?Do you have several options? In such a case, which one do youfind most useful?


T-5.2 Calculate the magnetic flux density at the point P.

5.1.3 Current in a wire

Problem description

A conducting wire is shaped such that it has two half-infinite sectionsjoined by a half-circle of radius a, as shown in the figure below. The wire

58

carries the current I . A point P of particular interest is also shown in thefigure. The point P is located at the center of the half-circle.


S-5.10 Visualize the magnetic flux density together with the current inthe wire.





5.1.4 Current in three connected wires

Problem description

Three metal wires are connected at the origin of a Cartesian coordinate sys-tem, as shown in the figure below. Two of the wires coincide with thex-axis. The wire that occupies the part of the x-axis with x < 0 carriesthe current 2I and the other wire (that occupies the part of the x-axis withx > 0) carries the current I , where both these currents flow in the positive

59

x-direction. The third wire coincides with the positive y-axis and it carriesthe current I in the positive y-direction. A point P of particular interest isalso shown in the figure, where the point has the position vector ~r = za.


S-5.13 What condition must be satisfied by the currents at the origin? Isthis condition fulfilled? Can you express this condition by meansof a surface integral?

S-5.14 Visualize the magnetic flux density together with the currents inthe wires.





60

5.1.5 Circular wire loop in a coil

Problem description

As shown in the figure below, a small circular wire loop of radius b is placedat the center of a coil of length L and radius a ≫ b with N turns that carrythe current I . The axes of the circular wire loop and coil coincide.


S-5.17 Visualize the magnetic flux density together with the currents inthe coil.

S-5.18 Which direction do you expect for the magnetic flux density atthe circular wire loop? What are the characteristic features of themagnetic flux density on the surface of the circular wire loop?

S-5.19 How can you calculate the magnetic flux density at the center ofthe circular wire loop? Do you have several options? In such acase, which one do you find most useful?


T-5.5 Calculate the magnetic flux through the circular wire loop.

5.1.6 A charged metal sphere that rotates

Problem description

A metal sphere of radius a has the total charge Q. The metal sphere rotatesat an angular velocity ω around the z-axis. It is assumed that the charge is

61

uniformly distributed on the surface of the sphere, despite the fact that thesphere is rotating.


S-5.20 Visualize the magnetic flux density together with the charge dis-tribution on the surface of the metal sphere that rotates.

S-5.21 What does the moving charge correspond to?

S-5.22 Which direction do you expect for the magnetic flux density atthe center of the sphere? What approximate magnitude of themagnetic flux density do you expect at this point?

S-5.23 How can you calculate the magnetic flux density at the center ofthe sphere? Do you have several options? In such a case, whichone do you find most useful?

S-5.24 Why is it an approximation to assume that the charge is uni-formly distributed on the surface of the rotating sphere?


T-5.6 Calculate the magnetic flux density at the center of the sphere.

5.1.7 Magnetic force on a liquid

Problem description

A conducting liquid of mass density ρ is placed in a U-shaped tube. As thefigure below shows, the liquid is displaced as a result of the electric currentdensity and the magnetic flux density present in the brick shaped regionof dimensions a, b and c. Electrodes of sides b and c are placed inside thetube and a total current I is conducted through the fluid. The magnetic flux

density ~B is perpendicular to the rectangular surface of sides a and c. Theheight difference of the fluid in the two vertical tubes is denoted h.

62



S-5.26 Visualize the electric current density and the magnetic flux den-sity. Visualize also the force that acts on the fluid.

S-5.27 How can you calculate the height difference h in the two verti-cal tubes? Do you have several options? In such a case, whichoptions is the most convenient?


T-5.7 Calculate the height difference h in the two vertical tubes?

5.1.8 Magnetic flux through loop

Problem description

A magnetic dipole ~m = zm is located at the origin of a Cartesian coordinatesystem. A thin metal wire shaped as a circular loop of radius a is placed inthe plane z = b with its center on the z-axis.


S-5.28 Visualize the magnetic flux density from the magnetic dipole to-gether with the circular loop.

63

S-5.29 How does the magnetic flux density from the magnetic dipolevary over the surface of the circular loop?

S-5.30 How could you calculate the magnetic flux through the circularloop? Do you have several options? If so, which option is easiest?


T-5.8 Calculate the magnetic flux through the circular loop.

5.1.9 Current carrying wire close to superconductor

Problem description

A straight current carrying metal wire with a circular cross section is ele-vated (without mechanical support) parallel over a large horizontal super-conducting plane. The mass density of the wire is ρ and its radius is a. Thecurrent in the wire is I and all the fields are zero in the superconductor.


S-5.31 Visualize the magnetic flux density from the current carryingwire, together with the wire itself and the superconducting plane.

S-5.32 What boundary condition is satisfied at the surface of the super-conducting plane?

S-5.33 How can you calculate the force that lifts the metal wire?


T-5.9 Calculate the distance h between the current carrying metal wireand the superconducting plane.

T-5.10 Calculate the smallest current necessary for the wire to elevatefrom the superconducting plane.

64

5.2 Example problems – magnetic materials

5.2.1 A magnetized plate

Problem description

A circular plate of radius a and thickness d. The plate consists of a magneticmaterial and it is homogeneously magnetized in the direction of the plate’s

axis of revolution, i.e. ~M = zM0.


S-5.34 Visualize the magnetization vector together with the plate.

S-5.35 How is the magnetic flux density directed inside and outside theplate?

S-5.36 How is the magnetic field directed inside and outside the plate?

S-5.37 Visualize the magnetic flux density and the magnetic field bymeans of field lines for a plane that coincides with the plate’saxis of rotation.



T-5.11 Calculate the magnitude and direction of the magnetic flux den-sity and the magnetic field at the center of the plate.

T-5.12 Calculate the demagnetizing factor |H/M | for a thin and verylarge plate.

5.2.2 Current inside magnetic tube

Problem description

A long straight copper wire of radius a carries the current I . The wire isenclosed by a long cylindrical shell of iron with the permeability µ, wherethe inner radius of the shell is b and its outer radius is c.

65



S-5.40 Visualize this situation together with the magnetic flux densityand the magnetic field. How is the magnetic flux density and themagnetic field directed?

S-5.41 What type of symmetries can you identify for the field solutionsand the geometry?

S-5.42 What boundary conditions must be fulfilled at the different inter-faces in this problem?

S-5.43 What options do you have to calculate the magnetic flux densityand the magnetic field? If you find several possibilities, whichoptions is the easiest to use?


T-5.13 Calculate the magnetic field everywhere.

T-5.14 Calculate the magnetic flux density everywhere.

T-5.15 Calculate the equivalent magnetization current densities in theiron.

5.2.3 Magnetic circuit

Problem description

An iron ring of average radius a and cross section area A1 has a diametricalbridge of cross section area A3. Both the ring and its bridge are made ofa magnetic material with the (constant) permeability µ. The two halves ofthe ring are equipped with coils with N1 and N2 turns, respectively. Thetwo coils are connected in series and a current I flows through both coils,such that fluxes in the ring are co-directed. Leakage can be neglected.


S-5.44 Compare this situation with an electric circuit. What do the coilsrepresent? What do the magnetic materials correspond to?

66

S-5.45 Electrical circuits are often analyzed by means of Kirchhoff’s cur-rent and voltage laws. What are the corresponding laws for thissituation?

S-5.46 How do you expect that the magnetic field is directed in the dif-ferent parts of the magnetic circuit. What about the magnetic fluxdensity and the magnetic flux?

S-5.47 What’s the significance of the neglected leakage? How does thatinfluence the analysis of the problem?


T-5.16 Calculate the flux Φ3 through the bridge.

5.2.4 Self and mutual inductance

Problem description

Three iron cylinders with the reluctance R are connected by large ironblocks of negligible reluctance. To coils with N1 and N2 turns enclose theiron cylinders as shown by the figure below. Leakage can be neglected.

67





T-5.17 Calculate the self-inductance of the two coils.

T-5.18 Calculate the mutual inductance between the two coils.

T-5.19 Calculate the coupling factor between the two coils.

5.2.5 Two circular loops

Problem description

Two circular current loops of radii a and b ≪ a have a common axis. Theyare located at a distance c from each other. Each loop consists of one singleturn of a thin wire. The current Ib flows in the small loop.


S-5.50 Visualize the two loops and the current in the small loop. Also,visualize the magnetic flux density that results from the currentcarrying loop.

S-5.51 What type of symmetries can you identify? How can you use thisto solve the problem?

S-5.52 Are there several ways to solve this problem? How does the dif-ferent alternatives compare?


T-5.20 Calculate the flux through the large loop.

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5.2.6 Toroid core

Problem description

A winding with N turns encloses a toroidal core with a rectangular crosssection of sides a and b. The average radius of the toroid is c and the relativepermeability µr. The winding carries the current I .


S-5.53 What happens when the current flows in the winding?

S-5.54 Visualize the magnetic flux density and magnetic field in this sit-uation.

S-5.55 How does the magnetic flux density and the magnetic field varywith respect to space? What about the direction of the field solu-tion?

S-5.56 What happens if c≫ b?


T-5.21 Calculate the magnetic energy stored in the toroid.

5.2.7 Inductance

Problem description

A magnetic circuit consists of a ring with three spokes, as shown in thefigure below. The spokes have the cross section area A and the averagelength l1 for the flux. Two of the spokes have windings with N and 2Nturns, respectively. The ring has the cross section area A and the averagelength between the spokes is l2 = 2πl1/3. The permeability of the magneticcircuit is µ. The magnetic leakage can be neglected.

69





T-5.22 Calculate the inductance L for the two port.

5.2.8 Two square loops

Problem description

Two square shaped wire loops are placed in the plane z = 0, as shown inthe figure below. The largest loop is 20a along the side and the smaller loop4a. The sides are parallel with the x- and y-axes and the centers of the two

70

loops coincide. The reference direction is in positive ϕ-direction for bothloops.


S-5.59 What happens when a current flows in the small loop? What isthe flux through the large loop?

S-5.60 What happens when a current flows in the large loop? What isthe flux through the small loop?

S-5.61 How can you calculate the mutual inductance between the loops?Do you have several options? In such a case, which one is theeasiest option?


T-5.23 Calculate an approximate value for the mutual inductance be-tween the two loops.

71

Chapter 6

Quasistatics

6.1 Example problems – induction

6.1.1 Metal ring in magnetic field

Problem description

A metal ring of radius a is located in a region with the homogeneous mag-

netic flux density ~B = zB0 cos(ωt). The metal ring coincides with the planez = 0. The frequency ω is very low.


S-6.1 What happens in with the charges in the metal ring for setting?

S-6.2 How does the external magnetic flux density change due to thepresence of the ring?

S-6.3 Can you separate the total magnetic flux density into one partthat consists of the original magnetic flux density and the otherpart that corresponds to the perturbations due to the metal ring?

S-6.4 What type of symmetry do you expect from the total magneticflux density?

S-6.5 Visualize the magnetic flux density in this situation, togetherwith the geometry.

S-6.6 What’s the significance of the very low frequency?

72


T-6.1 Calculate the induced voltage in the metal ring.

T-6.2 Use Faraday’s law to determine the electric field where the metalring is located.

T-6.3 Use the relation Vind =∮

L~E · d~l to calculate the induced voltage,

based on the electric field calculated from Faraday’s law.

6.1.2 Three parallel wires

Problem description

Three very long parallel wires are shown in the figure below. The wire Acarries a time-harmonic current i(t) = i0 cos(ωt), where the frequency ω isvery low. The distance between the pair of wires B-C is b and their positionwith respect to the current carrying wire is given by a and h according thefigure.



S-6.8 What’s the magnetic flux density given the current that flows inwire A?


73

S-6.10 How can you calculate the induced voltage in the pair of wiresB-C? Do you have several options? Which option is the best?

S-6.11 How do you expect that the induced voltage in the pair of wiresB-C depends on the dimensions a, b and h?


T-6.4 Calculate the induced voltage per unit length in the pair of wiresB-C.

6.1.3 A ring transformer

Problem description

A ring shaped transformer core of rectangular cross section is equippedwith two windings according to the figure below. The primary windinghas a negligible and it is fed by a sinusoidal voltage v1(t), which in turnyields the current i1(t) = i0 cos(ωt) in the primary winding. The primarywinding has N1 turns. The transformer core is also equipped with a sec-ondary winding of N2 turns, which is open circuited. The permeability ofthe transformer core is µ and its inner and outer radii are a and b, respec-tively. Consequently, the width of the cross section is b− a and its height isd. The frequency is very low and leakage is negligible.


S-6.12 What happens when the current i1(t) flows in the primary wind-ing?

74

S-6.13 What type of field distribution can you expect? How does thefield vary with respect to the space?

S-6.14 What type of symmetries do you expect that the field solutionexhibit? Which coordinate system do you find appropriate forthe analysis of this problem?

S-6.15 Visualize the magnetic flux density and the magnetic field in thissituation, together with the geometry.

S-6.16 What’s the significance of the fact that leakage is negligible?Which parameter influence the leakage in this situation?


T-6.5 Calculate the magnetic flux Φ(t) in the transformer core.

T-6.6 Calculate the voltages v1(t) and v2(t).

6.1.4 A unipolar machine

Problem description

A unipolar machine is constructed by a permanent magnet that consists oftwo coaxial iron cylinders connected at one of the ends according to thefigure below. The inner iron cylinder is the north pole and the outer ironcylinder is the south pole, which yields a magnetic flux density that is ra-dially directed between the two iron cylinders. The magnetic flux betweenthe inner and outer cylinder is Φ. Coaxially with the iron cylinders, a rotat-ing copper tube is placed and it makes n revolutions per minute.

75


S-6.17 What happens when the copper rotates?


S-6.19 How can you calculate the induced voltage between the ends ofthe copper tube? Do you have several options? Which option isthe best?


T-6.7 Calculate the magnitude of the induced voltage between the endsof the copper tube.

6.1.5 Conducting plate in external field #1

Problem description

A thin circular plate of radius a and the thickness h has the conductivity σ.The conducting plate is located in a region with an external magnetic flux

density ~B(t) = ~B0 cos(ωt), where the magnetic flux density is parallel tothe plate’s axis of revolution. Perturbations of the external magnetic fluxdensity can be neglected and the frequency is very low.



S-6.21 How is the magnetic flux density be influenced by the presenceof the plate? Do you expect induced currents? How could suchcurrents influence the magnetic flux density?

S-6.22 How do you expect the induced currents to distribute in theplate? How does such a current flow? How does it depend onthe spatial coordinates?

S-6.23 What type of symmetries do you expect from the magnetic field?Which coordinate system is appropriate for such a situation?

S-6.24 Visualize this situation with the external magnetic flux densityand the induced current. Also, indicate the magnetic flux densitydue to the induced current.

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S-6.25 What’s the significance of that the perturbations of the externalmagnetic flux density can be neglected?


T-6.8 Calculate the time-averaged power dissipation in the conductingplate.

6.1.6 Heating of conducting cylinder

Problem description

A cylinder of radius a and the length l has the conductivity σ, as shown inthe figure below. The cylinder is supposed to be heated by induction and,therefore, it is placed coaxially inside a coil of length L with N windingsthat carry the current i(t) = i0 cos(ωt). The coil is very long compared tothe size of the conducting cylinder, i.e. L ≫ l and L ≫ a. Perturbations ofthe magnetic flux density due to the coil can be neglected and the frequencyis very low.



77

S-6.27 How is the magnetic flux density be influenced by the presenceof the cylinder? Do you expect induced currents? How couldsuch currents influence the magnetic flux density?

S-6.28 How do you expect the induced currents to distribute in thecylinder? How does such a current flow? How does it dependon the spatial coordinates?


S-6.30 Visualize this situation with the external magnetic flux densityand the induced current.



T-6.9 Calculate the power dissipation in the cylinder.

6.1.7 Power line and a barn

Problem description

A barn is located parallel to a three-phase power line according to the figurebelow. Suppose a large rectangular coil is placed under the roof of the barn.The length of the barn is l and the other dimensions are shown in the figurebelow. The currents in the three-phase power line are i1 = i0e

−j2π/3, i2 = i0and i3 = i0e

+j2π/3. The currents in the power line oscillates at a frequencyω, which is very low given this situation.

78



S-6.33 How is the induced voltage influenced by the size of the barn?

S-6.34 How do you incorporate the effect of all the three phases of thepower line?

S-6.35 What happens if the distance of power-line wires decreases?


T-6.10 Calculate the induced voltage per turn in the coil placed in theroof of the barn.

79

Chapter 7

Electrodynamics

7.1 Example problems – real fields

7.1.1 A charged conducting body

Problem description

A conducting body has the conductivity σ and the permittivity ǫ. Initially,the charge distribution in the body is ρ(~r, t = 0).


S-7.1 What happens for t > 0?

S-7.2 How does the charge move?

S-7.3 What happens after long time, i.e. in the limit t→ ∞?

S-7.4 Where does the charge end up?


T-7.1 Derive the differential equation for ρ(~r, t).

T-7.2 Solve the differential equation for ρ(~r, t), given its initial condi-tion.

80

7.1.2 Wave propagation in vacuum

Problem description

An electromagnetic propagates in vacuum. The electric field of the wave isgiven by

~E(z, t) = xE0,x cos(ω(t− z/c0)) + yE0,y sin(ω(t− z/c0))

where c0 is the speed of light in vacuum.


S-7.5 Visualize the electric field with respect to the space coordinatesfor t = 0.


S-7.7 Visualize the electric field for ωt = π/4, π/2, 3π/2, . . .. How doesthe electric field change? What’s your interpretation of the be-havior of the electric field?


T-7.3 Derive the wave equation for the electric field from Maxwell’sequations.

T-7.4 Verify that the electric field satisfies the wave equation for theelectric field.

7.1.3 Determination of electric field from magnetic field

Problem description

An electromagnetic plane wave of sinusoidal shape propagates in vacuum.Its magnetic field is given by

~H(z, t) = xH0 cos

(

ω

[

t− 1

c0(y sinα+ z cosα)

])

where c0 is the speed of light in vacuum and α is an angle.

81


S-7.8 What’s the significance of the angle α?





T-7.5 Calculate the electric field associated with the given magneticfield.

7.1.4 Plane wave in a dielectric medium

Problem description

An electromagnetic plane wave of linear polarization has a sinusoidalshape propagates in a homogeneous medium with the permittivity ǫ andthe permeability µ0. The power density of the wave is S0.


S-7.12 How does the permittivity influence the electromagnetic wave?

S-7.13 How is the electric field and magnetic field of the plane waverelated to the power density?


T-7.6 Calculate the peak value of the electric field.

T-7.7 Calculate the peak value of the magnetic field.

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7.2 Example problems – complex fields

7.2.1 Sinusoidal wave in vacuum

Problem description

A sinusoidal electromagnetic wave in vacuum is described by the electricfield

~E(y, z) = (y4 + z3) exp(−j(6y − 8z))





S-7.17 What characterizes a plane wave? How can you calculate themagnetic field given the corresponding electric field for a planewave? Do you have several options? How do the options com-pare?


T-7.8 Calculate the direction of propagation for the wave.

T-7.9 Is the wave a plane wave?

T-7.10 Calculate the magnetic field associated with the given electricfield.

7.2.2 An evanescent wave

Problem description

A time-harmonic electromagnetic wave is generated by a source that oscil-lates with the frequency ω. The wave propagates in vacuum is described

83

by the electric field

~E(x, z) = yE0 exp(−αz) exp(−jβx)

where α and β are real constants.


S-7.18 What’s the significance of E0? How does the electric field changewhen E0 is increased?

S-7.19 What’s the significance of α? How does the electric field changewhen α is increased?

S-7.20 What’s the significance of β? How does the electric field changewhen β is increased?






T-7.11 Calculate the magnetic field associated with the given electricfield.

T-7.12 Express the electric and magnetic field in the time domain.

T-7.13 Derive the relation between α, β and ω such that the wave equa-tion in vacuum is satisfied.

84

7.2.3 A plane wave in lossy medium

Problem description

A time-harmonic electromagnetic plane wave is generated by a source thatoscillates with the frequency ω. It propagates in a medium described bythe permeability µ0, the relative permittivity ǫr = 5 and the conductivityσ = 10−5 S/m.







T-7.14 Calculate the phase velocity of the plane wave.

T-7.15 Calculate the damping constant of the plane wave.

7.2.4 A plane wave in lossless medium

Problem description

An electromagnetic plane wave is generated by a source with that oscillateswith the frequency ω. It propagates in a lossless medium with the relativepermittivity ǫr.

85


S-7.29 How is the electromagnetic wave influenced by the value of therelative permittivity?

S-7.30 How do you expect the wavelength change with respect to thefrequency and the relative permittivity?

S-7.31 What characterizes the wave impedance? How is it influencedby the relative permittivity?

S-7.32 What attenuation do you expect in this situation?


T-7.16 Calculate the damping constant.

T-7.17 Calculate the wave number.

T-7.18 Calculate the wave impedance.

T-7.19 Express the wave number in terms of the speed of light c0.

T-7.20 Express the wave impedance in terms of Z0, i.e. the waveimpedance of vacuum.

7.2.5 Wave propagation in a good conductor

Problem description

A metal is described by the relative permeability µr and the conductivityσ. An electromagnetic plane wave is generated by a source that oscillateswith the frequency ω. Here, the material parameters and the frequency arerelated to each other by σ ≫ ωǫ0µ0µr.


S-7.33 How is the electromagnetic wave influenced by the value of therelative permeability?

S-7.34 What’s the significance of the wavelength in this situation?

S-7.35 What characterizes the wave impedance? What’s the expectedcharacteristics of the wave impedance for this situation?

86

S-7.36 What attenuation do you expect in this situation?

S-7.37 What’s the significance of the relation σ ≫ ωǫ0µ0µr?


T-7.21 Calculate the damping constant.

T-7.22 Calculate the wave number.

T-7.23 Calculate the wave impedance.

T-7.24 Express the wave impedance in terms of Z0, i.e. the waveimpedance of vacuum.

7.2.6 Energy densities of a plane wave

Problem description

An electromagnetic plane wave that varies with respect to time with thefrequency ω propagates in a homogeneous medium with the relative per-mittivity ǫr, the relative permeability µr and the conductivity σ.


S-7.38 How can the electric energy density be calculated for this situa-tion? What do you expect from the result?

S-7.39 How can the magnetic energy density be calculated for this situ-ation? What do you expect from the result?


T-7.25 Calculate the ratio of the magnetic and electric average energydensities for an arbitrary point in space.

7.2.7 Wire with alternating current

Problem description

An iron wire of radius a = 0.1 mm and length l has the conductivity σ =5 · 106 S/m and the relative permeability µr = 100. The wire conducts a

87

time-harmonic current that is given by i(t) = i0 cos(ωt).


S-7.40 How is the current directed inside the wire?

S-7.41 How does the current vary with respect to the space coordinatesinside the wire?

S-7.42 The current causes a magnetic field that, according to Faraday’slaw, yields an induced electric field. How is this magnetic fieldand electric field directed? How do they depend on the spatialcoordinates? What’s the impact of the induced electric field?

S-7.43 What happens when the frequency is very low?

S-7.44 What happens when the frequency is very high?


T-7.26 Calculate the resistance per unit length of the wire at the fre-quency 50 Hz.

T-7.27 Calculate the resistance per unit length of the wire at the fre-quency 10 MHz.

7.2.8 Alternating current in coaxial cable

Problem description

A coaxial cable is constructed by an inner conductor of radius a and a tubu-lar outer conductor of inner radius b and outer radius c. The coaxial cablecarries a time-harmonic current that oscillates at a frequency such that thepenetration depth δ is much smaller than both a and c−b. The conductivityof both the inner conductor and the tubular outer conductor is σ.



S-7.46 What’s the significance of the relations δ ≪ a and δ ≪ c− b?

88

S-7.47 Where is the electric current flowing? How is it directed? Howdoes it depend on the space coordinates?

S-7.48 Should it be necessary, how can you calculate the electric current?If you have access to the electromagnetic field in the coaxial cable,could you use that result to compute the electric current?


T-7.28 Calculate the alternating-current resistance per unit length forthe coaxial cable.

7.2.9 Conducting plate in external field #2

Problem description

A thin circular plate of radius a and the thickness d has the conductivity σ.The conducting plate is located in a region with an external magnetic flux

density ~B(t) = ~B0 cos(ωt), where the magnetic flux density is parallel tothe plate’s axis of revolution.



S-7.50 How is the magnetic flux density be influenced by the presenceof the plate? Do you expect induced currents? How could suchcurrents influence the magnetic flux density?

S-7.51 How do you expect the induced currents to distribute in theplate? How does such a current flow? How does it depend onthe spatial coordinates?


S-7.53 Visualize this situation with the external magnetic flux densityand the induced current. Also, indicate the magnetic flux densitydue to the induced current.


89


T-7.29 Calculate the induced eddy current density ~J(~r, t) in the conduct-ing plate, under the assumption that the magnetic field due to theinduced eddy currents themselves can be neglected.

T-7.30 Calculate the magnetic flux density ~Bec at the center of the plate,

where ~Bec is due to the induced eddy currents.

T-7.31 Is it realistic to neglect the contribution ~Bec to the magnetic fluxdensity from the eddy current given a situation with the numericvalues ω = 2π · 103 rad/s, σ = 107 S/m, a = 3 cm and d =0.1 mm?

7.3 Example problems – reflection and diffraction

7.3.1 Wave incident on perfect electric conductor

Problem description

An electromagnetic plane wave in vacuum is characterized by an electricfield with the peak value E0. The plane wave is perpendicularly incidenton a perfect electric conductor with σ = ∞.



S-7.56 What conditions must be fulfilled at the metal surface? Howcan you use these condition to calculate the total electric field?What’s the characteristic features of the electric field that must beadded to the incident electric field?

S-7.57 How can you calculate the magnetic field for this situation? Doyou have several options? Which is the easiest option to use?

S-7.58 How do you expect the total electric field to vary with respect tospace? Can you visualize this situation for a couple of represen-tative time instants? What about the magnetic field?

S-7.59 What’s the significance of σ = ∞?

90


T-7.32 Calculate the frequency-domain surface current ~J s at the surfaceof the metal.

7.3.2 Plane wave incident on a dielectric half-space

Problem description

An electromagnetic plane wave is incident from air perpendicularly on alarge flat surface, which separates the air region from a half space of a ma-terial with the conductivity σ = 0, the relative permeability µr = 1 and therelative permittivity ǫr.



S-7.61 What happens when the incident field hits the flat interface be-tween the air region and the dielectric region? Is there an elec-tromagnetic wave in the dielectric? In which direction does thiswave propagate? What’s the polarization of such a transmittedelectromagnetic wave? What happens in the air region? Is it suf-ficient to use the incident plane wave or do you need to introduceanother wave?

S-7.62 What conditions must be fulfilled at the flat interface betweenthe air region and the dielectric region? How do these conditionsrelate the fields in the air region to the fields in the dielectric re-gion? How many equations and how many unknowns do youhave?


T-7.33 Calculate the relative permittivity ǫr, should the surface reflect10% of the energy associated with the incident wave.

91

7.3.3 Plane wave incident on metal plate

Problem description

An electromagnetic plane wave in vacuum is perpendicularly incident ona metal plate. The skin depth for the wave in the plate is δ = 1 mm. Atthe interface between vacuum and metal, the incident wave is described by

the time-domain electric field ~E = x100√2 sin(ωt) µV/m, where the unit

vector x is perpendicular to the normal n = −z of the metal surface. At thisinterface, 99% of the incident wave’s energy is reflected.


S-7.63 What happens at the surface of the metal plate?

S-7.64 Why is such a large amount of the incident wave’s energy re-flected? How could you use this? How is this related to thetransmitted field and its energy?

S-7.65 What’s the wave impedance of the metal? How does it compareto the free-space wave impedance? How do they compare?

S-7.66 Can you use approximations in this situation? How could youdo that? How could the reflection and transmission coefficientbe approximated?

S-7.67 How does the electric field depend inside the metal with respectto the distance from the surface?


T-7.34 Calculate the time-domain expression of the electric field at thedepth z.

7.3.4 A plane wave incident on a copper plate

Problem description

An electromagnetic plane wave is linearly polarized. It is generated by asource that oscillates at the frequency f = 1GHz. The wave propagates invacuum and is perpendicularly incident on a 0.5 mm thick copper platewith the conductivity σ = 58 · 106 S/m.

92



S-7.69 What’s the skin depth in this situation? How does the skin depthcompare with the thickness of the plate? What’s the significanceof this result?

S-7.70 How can you calculate the transmitted field and the reflectedfield? What type of result do you expect? Can you use approxi-mations?

S-7.71 What’s the wave impedance of the copper plate? How does itcompare to the wave impedance of vacuum?


T-7.35 Calculate the amount of the incident power that is absorbed bythe plate.

7.3.5 A plane wave incident on a dielectric

Problem description

An electromagnetic plane wave in vacuum with the electric field ~E =x10 cos(ωt − kz) is perpendicular incident on a plane interface at z = 0.In the region z > 0, the relative permittivity is ǫr = 9 and, consequently, thevacuum region is described by z < 0.



S-7.73 What conditions must be fulfilled at the interface between thevacuum and the dielectric? How can you make these conditionsfulfilled given the incident electric field? What’s the field insidethe dielectric?

S-7.74 How can you describe the reflection and transmission at the sur-face? What’s the impact of the reflected wave? How does it in-fluence the total electric field?

93


T-7.36 Calculate where the electric field has its maxima in the vacuumregion, should the frequency be 300 MHz.

T-7.37 Calculate the maximum value of the electric field in the vacuumregion, should the frequency be 300 MHz.

7.3.6 Oblique incidence on dielectric plate

Problem description

An electromagnetic plane wave is incident on a lossless dielectric plate. Theelectric field vector coincides with the incident plane, which is spanned bythe normal of the surface of the dielectric and the wave vector. The incidentangle θi is equal to the Brewster angle.



S-7.76 How can you calculate the reflection and transmission at the in-terface where the incident wave impinges the dielectric?

S-7.77 What happens at the other interface? What about the reflectionat this interface?

94


T-7.38 Show that the reflection from the dielectric plate is zero.

7.3.7 Prism with total reflection

Problem description

A light ray is diffracted and totally reflected in a prism with a cross sec-tion of an isosceles triangle, according to the figure. The diffraction takesplace at the Brewster angle and the reflection under the condition for totalreflection. The incoming and returning rays are parallel.


S-7.78 What’s the condition for the Brewster angle? What characterizesthe Brewster angle?

S-7.79 What’s the condition for total reflection? What characterizes totalreflection?

S-7.80 How does the diffraction for the incoming ray compare with thediffraction that results in the outgoing ray?

S-7.81 Can you list all conditions that must be satisfied simultaneouslyfor this problem? How do they compare? What’s the impli-cations of equality conditions as compared to inequality condi-tions?

95

S-7.82 How many unknowns do you have? How many equations doyou have? How do they compare?

S-7.83 Is it possible to eliminate all the unknowns to arrive at an equa-tion in terms of only the index of refraction?


T-7.39 Calculate bounds on the index of refraction n which makes thissituation possible.

7.3.8 Prism without reflection losses

Problem description

A prism has a cross section that is shaped as an isosceles triangle, withthe angle α between the equally long sides of the triangle. The index ofrefraction of the prism is denoted n. A linearly polarized ray of light isdiffracted a total angle of φ without any reflection losses.


S-7.84 What conditions must be fulfilled at the surfaces of the prism?

96

S-7.85 How many equations do you have? How many unknowns doyou have? How do the number of equations compare with thenumber of unknowns?

S-7.86 What’s the significance of the fact that the incident is linearly po-larized? What happens if the direction of the linear polarizationchanges?


T-7.40 Calculate the angle α(φ) and the index of refraction n(φ).

97

Appendix A

Answers to some problems

A.1 Electrostatics

T-3.9

E =

√2Q

πǫ0a2

T-3.13

V =ρl

4πǫ0ln

(

|z + L/2 +√

r2 + (z + L/2)2||z − L/2 +

√

r2 + (z − L/2)2|

)

T-3.14

Er =ρl

4πǫ0r

(

z + L/2√

r2 + (z + L/2)2− z − L/2√

r2 + (z − L/2)2

)

Ez =ρl

4πǫ0

(

1√

r2 + (z − L/2)2− 1√

r2 + (z + L/2)2

)

T-3.15

ψ =Q

24

T-3.16

ψ =Q

8

T-3.17

ψ =Q

8

98

T-3.18

ρv =Qa

2πR(R+ a)3

T-3.19Total charge = Q

T-3.20

ρv =6ǫ0V0a2

for R < a

ρs = −2ǫ0V0a

for R = a

ρv = 0 for R > a

T-3.21Total charge = 0

T-3.22Q < 4πǫ0a

2Emax = 3.3 · 10−6 C

T-3.23

V =Q

4πǫ0a= 3 · 105 V

T-3.26

~E = ~0 for R < a

~E = RQ

4πǫ0R2for a < R < b

~E = ~0 for R > b

T-3.27

V =Q

4πǫ0

(

1

a− 1

b

)

for R ≤ a

V =Q

4πǫ0

(

1

R− 1

b

)

for a ≤ R ≤ b

V = 0 for R ≥ b

T-3.28

V =1

4πǫ0

(qaa

+qbb+qcc

)

99

T-3.29

V =1

4πǫ0

[

qa

(

1

a− 1

b

)

+qa + qb + qc

c

]

T-3.30

V1 − V4 =q1πǫ0L

ln

(

3a

b

)

+q2πǫ0L

ln (2)

T-3.31

F =−q1(2q1 − 3q2)

12πǫ0La

T-3.32

QA = QB = 2πǫ0LU0

/

ln

(

d4

a3b

)

QC = −2QA

T-3.33

V =Q

8ǫ0a

T-3.34~E = x

2p sinα

4πǫ0a3− y

p cosα

4πǫ0a3

T-3.37

V = −Paǫ0

T-3.38

V = −Paǫ0

T-3.43

E2 =E1√2

T-3.44α2 =

π

4

T-3.45

|ρps| =√3− 1

2ǫ0E1

T-3.54

C =πǫ0L

ln(d/a)

100

T-3.55

W =Q2

8πǫa

T-3.56

W =Q2

8πǫa

T-3.57

W =Q2

8C

T-3.58

W =8

5πǫ0aV

20

T-3.59~E = −z p

8πǫ0h3

T-3.60C

L= 2πǫ0

/

ln

(

d2 − a2

ab

)

T-3.61

b = 2a(√

5− 2)1/2

T-3.62

~F = xQ2

16πǫ0b2

(

1

2√2− 1

)

+ yQ2

16πǫ0b2

(

1

2√2− 1

)

T-3.63

V =Q

4πǫ0

(

1

a− 1

b+

1

2√2b

)

T-3.64

~F = −x Q2

8πǫ0Lb− y

Q2

8πǫ0Lb

T-3.65

V =Q

2πǫ0Lln

(√2b

a

)

101

A.2 Time-invariant electric currents

T-4.1P =

π

2σLE2

0a2

T-4.2

R =1

2πσwln

(

2a√a2 + b2

bc

)

T-4.3

R =1

πσdln

(

20a

3b

)

T-4.4

R =ln 2

σa

T-4.5

R =2

3σa

T-4.6

Rlower =7

σd

T-4.7

Rupper =8

σd

T-4.8RAB =

π

2σd ln(b/a)

T-4.9

RCD =2 ln(b/a)

πσd

T-4.10

RABRCD =

(

1

σd

)2

T-4.11Rlower =

π

σd ln(b/a)

T-4.12

Rupper =π

2σd· b+ a

b− a

102

T-4.13R = 5900 Ω

A.3 Magnetostatics

T-5.1

~B = ~0 for r ≤ a

~B = ϕµ0I

2πr· r

2 − a2

b2 − a2for a ≤ r ≤ b

~B = ϕµ0I

2πrfor r ≥ b

T-5.2~B =

µ0I

4πa

[

−x arctan(2) + y ln(√

5)]

T-5.3

~B = zµ0I

4a

(

2

π+ 1

)

T-5.4~B =

µ0I

4πa(x− 3y)

T-5.5

Φ =µ0NIπb

2

2√

a2 + (L/2)2

T-5.6~B = z

µ0Qω

6πa

T-5.7

h =IB

bρg

T-5.8

Φ =µ0ma

2

2(a2 + b2)3/2

T-5.9

h =µ0I

2

4π2a2ρg

103

T-5.10

Imin = πa2√

4ρg

µ0a

T-5.11

~B = µ0 ~Md/2

√

a2 + (d/2)2

~H = ~M

(

d/2√

a2 + (d/2)2− 1

)

T-5.12Demagnetizing factor = 1

T-5.13

~H = ϕIr

2πa2for r ≤ a

~H = ϕI

2πrfor r ≥ a

T-5.14

~B = ϕµ0Ir

2πa2for r ≤ a

~B = ϕµ0I

2πrfor a ≤ r < b

~B = ϕµI

2πrfor b < r < c

~B = ϕµ0I

2πrfor r > c

T-5.15

~Jms = z(µr − 1)I

2πbfor r = b

~Jmv = ~0 for b < r < c

~Jms = −z (µr − 1)I

2πcfor r = c

104

T-5.16

Φ3 =(N1 −N2)I

R1 + 2R3

R1 =πa

µA1

R3 =2a

µA3

T-5.17

L1 =2N2

1

3R

L2 =2N2

2

3R

T-5.18

M =N1N2

3R

T-5.19

k =1

2

T-5.20

Φ =µ0πa

2b2Ib2(a2 + c2)3/2

T-5.21

W =I2µN2b

4πln

(

2c+ a

2c− a

)

T-5.22

L =µA18N2

l1(9 + 2π)

T-5.23

M =µ0a8

√2

5π

A.4 Quasistatics

T-6.1Vind = B0πa

2ω sin(ωt)

105

T-6.2~E = ϕ

ωB0r

2sin(ωt)

T-6.3Vind = B0πa

2ω sin(ωt)

T-6.4

Vind =i0ωµ0 sin(ωt)

2πln

(

√

h2 + (a+ b)2

h2 + a2

)

T-6.5

Φ =µN1i0d

2πln

(

b

a

)

cos(ωt)

T-6.6

v1 = −ωµN21 i0d

2πln

(

b

a

)

sin(ωt)

v2 =ωµN1N2i0d

2πln

(

b

a

)

sin(ωt)

T-6.7Vind =

n

60Φ

T-6.8

〈P 〉 = πha4σB20ω

2

16

T-6.9

〈P 〉 =(

µ0ωNi0L

)2 πσla4

16

T-6.10

V ind = −jωµ0i02π

[

ln

(√

(2a+ b+ w)2 + (H − h)2

(2a+ b)2 + (H − h)2

)

e−j2π/3

+ ln

(√

(a+ b+ w)2 + (H − h)2

(a+ b)2 + (H − h)2

)

+ ln

(√

(b+ w)2 + (H − h)2

b2 + (H − h)2

)

e+j2π/3

]

106

A.5 Electrodynamics

T-7.1∂ρv(~r, t)

∂t= −σ

ǫρv(~r, t)

T-7.2ρv(~r, t) = ρv(~r, 0)e

−σt/ǫ

T-7.3

−∇2 ~E +1

c20

∂2 ~E

∂t2= ~0

T-7.5

~E = Z0H0 (−y cosα+ z sinα) cos

(

ω

[

t− 1

c0(y sinα+ z cosα)

])

T-7.6

Emax =

√

2Z0S0√ǫr

T-7.7

Hmax =

√

2√ǫrS0Z0

T-7.8

k =y3− z4

5

T-7.9Plane wave.

T-7.10~H = x

5

Z0e−j(6y−8z)

T-7.11~H = x

αE0

ωµe−αze−jβx+jπ/2 + z

βE0

ωµe−αze−jβx

T-7.12

~E = yE0e−αz cos(ωt− βx)

~H = xαE0

ωµe−αz cos(ωt− βx+ π/2) + z

βE0

ωµe−αz cos(ωt− βx)

107

T-7.13

β2 − α2 =

(

ω

c0

)2

T-7.14vphase =

c0√ǫr

T-7.15

α =σZ0

2√ǫr

T-7.16α = 0

T-7.17β = ω

√µ0ǫ0ǫr

T-7.18

Z =

√

µ0ǫ0ǫr

T-7.19

β =ω√ǫr

c0

T-7.20

Z =Z0√ǫr

T-7.21

α =

√

ωσµ0µr2

T-7.22

β =

√

ωσµ0µr2

T-7.23

Z = (1 + j)

√

ωµrµ02σ

T-7.24

Z = (1 + j)

√

ωµrǫ02σ

Z0

(

where

√

ωµrǫ02σ

≪ 1

)

108

T-7.25〈wm〉〈we〉

=

√

1 +( σ

ωǫ

)2

T-7.26R

L=

1

σπa2= 6.4Ω/m

T-7.27R

L=

1

2πa

√

ωµ

2σ= 45Ω/m

T-7.28R

L=

1

2π

√

ωµ

2σ

(

1

a+

1

b

)

T-7.29~J(r, t) = ϕ

ωσ

2B0r cos(ωt− π/2)

T-7.30~B(0, t) = z

µ0ωσad

4B0 cos(ωt− π/2)

T-7.31

Yes, should a relative error of a couple of per cent be acceptable.

T-7.32

Js =2E0

Z0

T-7.33

ǫr =

(√10 + 1√10− 1

)2

= 3.71

T-7.34E = 10−6 exp

(

−103z)

cos(

ωt− 103z − π/4)

V/m

T-7.35Pabs

Pinc≈ 4

√

ωǫ02σ

= 8.8 · 10−5

T-7.36zmax = −0.25(1 + 2n) where n = 0, 1, 2, . . .

109

T-7.37Emax = 15

T-7.39n ≥ 1.839

T-7.40

α =π − φ

2

n = tan

(

π

4+φ

4

)

110

Appendix B

Study plan

Here, the course is subdivided into study blocks. Each study block consistsof a few concepts that can be studied from (i) reading few sections in thebook, (ii) studying a theory problem, and (iii) solving a number of problemsby pen and paper calculations. When the course is given, each study blockcorresponds to one lecture and one tutorial. Here, problems intended forpen and paper calculations are denoted with letters: T – a problem fromthis collection of problems; P – a problem from the course book by Cheng;and E – a solved exercise from the course book by Cheng.

• Study block #1 – Electrostatics #1

– Concepts: electrostatics, point charge, Coulomb’s law, superpo-sition, electric field

– Reading in Cheng: 1-1, 1-2, 1-3, 2-1, 2-2, 2-3, 2-4, 2-5, 2-6, 2-7, 2-8,2-9, 2-10, 2-11, 2-12, 3-1, 3-2, 3-3

– Theory problem: 1

– Tutorial problems: T-3.7, T-3.8, P2-19, P2-26

– Homework problems: E2-18, E2-16, E2-19, P2-21


– Concepts: charge distribution, Gauss’ law, Gauss’ theorem, di-vergence, field lines, conservative field, Stoke’s theorem and curl

– Reading in Cheng: 3-4

– Tutorial problems: T-3.24, T-3.26, T-3.10, P3-8

– Homework problems: E2-22, P3-5, P3-7, P3-12, T-3.16, T-3.17, T-3.18, T-3.19

111


– Concepts: electric potential, conductors, boundary conditions,Faraday’s cage

– Reading in Cheng: 3-5, 3-6


– Tutorial problem: T-3.11, T-3.12, T-3.25, T-3.27, T-3.32

– Homework problem: T-3.30, T-3.31, T-3.18, T-3.19, T-3.20, T-3.21,T-3.22, T-3.23


– Concepts: electric dipole, insulators, polarization, electric fluxdensity



– Tutorial problem: T-3.34 T-3.39, T-3.40, T-3.41, T-3.42, T-3.43, T-3.44, T-3.45,

– Homework problem: T-3.37, T-3.38, P3-22

– Laboratory work: Questions on part A


– Concepts: capacitance, boundary conditions, Poisson’s equa-tion, Laplace equation

– Reading in Cheng: 3-9, 3-10, 4-1, 4-2


– Tutorial problem: T-3.46, T-3.47, T-3.48, T-3.49, T-3.50, T-3.51, T-3.52, T-3.53, P3-30

– Homework problem: T-3.54, P3-35, P3-32


– Concepts: work, electrostatic energy, electrostatic force, forceand torque for dipoles

– Reading in Cheng: 3-11

– Theory problem: 5, 6

– Tutorial problem: T-3.55 T-3.56 T-3.57 T-3.58

– Homework problem: P3-44, P3-48


112

– Concepts: uniqueness, image theory

– Reading in Cheng: 4-3, 4-4.1, 4-4.2, 4-5, 4-6

– Tutorial problem: T-3.60, T-3.61, T-3.62, T-3.63

– Homework problem: T-3.59, T-3.64, T-3.65

• Study block #8 – Time-invariant electric currents #1

– Concepts: electric current density, Ohm’s law, equation of conti-nuity, boundary conditions, image theory

– Reading in Cheng: 5-1, 5-2, 5-4, 5-6


– Tutorial problem: T-4.1, T-4.2, T-4.3,

– Homework problem: E5-4, E5-6 T-4.8, T-4.9, T-4.10,

• Study block #9 – Time-invariant electric currents #2

– Concepts: electromotive force, Joule’s low, resistance, upper andlower bounds for resistance, Kirchhoff’s laws

– Reading in Cheng: 5-3, 5-5, 5-7

– Tutorial problem: T-4.6, T-4.7, T-4.8, T-4.9, T-4.10,

– Homework problem: T-4.4, T-4.5, T-4.11, T-4.12, T-4.13,

• Study block #10 – Magnetostatics #1

– Concepts: magnetostatics, magnetic flux, magnetic flux density,Lorentz’s force, force on conductor, Ampères law


– Tutorial problem: T-5.1, E6-2 T-5.5



– Concepts: vector potential, Biot-Savart’s law, magnetic dipole


– Tutorial problem: T-5.2, T-5.3, T-5.8, T-5.20,

– Homework problem: T-5.4, T-5.5, E6-4


– Concepts: magnetic materials, ferromagnetism, permanentmagnets, magnetization, equivalent magnetization currents,magnetic field

– Reading in Cheng: 6-6, 6-7, 6-8, 6-9

113


– Tutorial problem: T-5.11, T-5.12, T-5.17, T-5.18, T-5.19, P6-22, E6-14

– Homework problem: T-5.13, T-5.14, T-5.15, T-5.16, T-5.22,


– Concepts: boundary conditions, shielding, self inductance, mu-tual inductance, magnetic energy, magnetic forces

– Reading in Cheng: 6-10, 6-11, 6-12, 6-13


– Tutorial problem: T-5.7, T-5.21, T-5.23 T-5.9, T-5.10, P6-46


• Study block #14 – Quasistatics #1

– Concepts: Quasistatics, Faraday’s law, jω-method for the fre-quency domain, complex fields

– Reading in Cheng: 7-1, 7-2, 7-7.1, 7-7.2

– Tutorial problem: T-6.1, T-6.2, T-6.3, T-6.4, T-6.7, P7-3


• Study block #15 – Quasistatics #2

– Concepts: eddy currents, skin effect, transformers, generators,motors

– Tutorial problem: T-6.5, T-6.6, T-6.8

– Homework problem: T-6.9, T-6.10

• Study block #16 – Electrodynamics #1

– Concepts: wave phenomena, displacement current, Maxwell’sequations, boundary conditions, wave equation, electromag-netic waves in vacuum

– Reading in Cheng: 7-3, 7-4, 7-5, 7-6, 7-7.3, 7-7.4

– Theory problemer: 10, 11

– Tutorial problem: T-7.1, T-7.2, T-7.3, T-7.4, T-7.5,

– Homework problem: P7-23


– Concepts: plane waves, polarization, Doppler effect, skin effect


114


– Tutorial problem: T-7.11, T-7.12, T-7.13, T-7.21, T-7.22, T-7.23, T-7.24, T-7.26, T-7.27

– Homework problem: T-7.8, T-7.9, T-7.10, T-7.14, T-7.15, T-7.16,T-7.17, T-7.18, T-7.19, T-7.20, T-7.28


– Concepts: phase velocity, group velocity, Poynting’s vector, re-tarded potentials, reflection and transmission for normal inci-dence



– Tutorial problem: T-7.6, T-7.7, T-7.25, T-7.34, T-7.36, T-7.37

– Homework problem: T-7.32, T-7.33, T-7.35, P8-16, P8-19


– Concepts: oblique incidence for plane interface, Snell’s law, Fres-nel’s formulas, Brewster’s angle, total reflection


– Tutorial problem: T-7.39, T-7.40, P8-41

– Homework problem: T-7.38

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electromagnetism-a collection of problems 20101026

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