copyright r. janow – fall 2014 1 physics 121: electricity and magnetism introduction syllabus,...

1Copyright R. Janow – Fall 2014

Physics 121: Electricity and Magnetism Introduction

Syllabus, rules, assignments, exams, etc. iClickers• Text: Young & Friedman, University Physics• Homework & Tutorial System: Mastering Physics

Content overview

Introductory First Lecture• Review of vector operations• Dot product, cross product• Scalar and vector fields in math and physics• Gravitation as an example of a vector field• Gravitational flux, shell theorems, flow fields• Methods for calculating fields

Fall 2014: • A “week of material” (referred to in the syllabus) includes a lecture that introduces new material, a recitation section covering material introduced during a previous week, and and on-line homework assignment • Recitations go over examples and homework assignments in comparatively small groups• Recitation class sessions are occasionally used for lectures.• Check page 4 of the Syllabus for details.

Copyright R. Janow – Fall 2014

Course Content

• 1 Week: Review of vectors & key field concepts– Prepares for electrostatic and magnetic fields, flux...

• 5 Weeks: Stationary charges – – Forces, fields, electric flux, Gauss’ Law, potential, potential energy, capacitance

• 2 Weeks: Moving charges – – Currents, resistance, circuits containing resistance and capacitance, Kirchoff’s

Laws, multi-loop circuits• 2 Weeks: Magnetic fields (static fields due to moving charges) –

– Magnetic force on moving charges, – Magnetic fields caused by currents (Biot-Savart’s and Ampere’e Laws)

• 2 Weeks: Induction – – Changing magnetic flux (field) produces currents (Faraday’s Law)

• 2 Weeks: AC (LCR) circuits, electromagnetic oscillations, resonance• Not covered:

– Maxwell’s Equations - unity of electromagnetism– Electromagnetic Waves – light, radio, gamma rays,etc – Optics

3Copyright R. Janow – Fall 2014

Physics 121 - Electricity and MagnetismLecture 01 - Vectors and Fields

Review of Vectors :• Components in 2D & 3D. Addition & subtraction• Scalar multiplication, Dot product, vector productField concepts:• Scalar and vector fields in math & physics• How to visualize fields: contours & field lines • “Action at a distance” fields – gravitation and electro-magnetics.• Force, acceleration fields, potential energy, gravitational potential• Flux and Gauss’s Law for gravitational field: a surface integral of

gravitational field More math:• Calculating fields using superposition and simple integrals• Path integral/line integral• Spherical coordinates – definition• Example: Finding the Surface Area of a Sphere• Example: field due to an infinite sheet of mass


Vector Definitions

Vectors in 2 Dimensions:

• Cartesian (x,y) coordinates

jA iA A yx

• Magnitude & direction

22yx A A A

x

y1-

A

Aant

- Experiments tell us which physical quantities are scalars and vectors - E&M uses vectors for fields, vector products for magnetic field and force

• Addition and subtraction of vectors:

yyy andxxx means BAC BAC B A C

yy andxx means AC AC A- C

y

x

A

A

Ax = A cos()

Ay = A sin()

k

j

i

z

• Notation for vectors:

amF

amF

aF

m


Definition: Right-Handed Coordinate Systems

• We always use right-handed coordinate systems.

• In three-dimensions the right-hand rule determines which way the positive axes point.

• Curl the fingers of your RIGHT HAND so they go from x to y. Your thumb will point in the positive z direction.

y

x

z

This course uses several right hand rules related to this one!


Vectors in 3 dimensions

• Unit vector (Cartesian) notation:

• Spherical polar coordinate representation:

1 magnitude and 2 directions

• Conversion into x, y, z components

• Conversion from x, y, z components

) , ,a( a

cosaasinsinaacossinaa

z

yx

xy

z

zyx

a/atana/acos

aaaa

1

1

222

kajaiaa zyx

Rene Descartes 1596 - 1650

y

x

a

z

az

ayax

a sin()


Right Handed Coordinate Systems

1-1: Which of these coordinate systems are right-handed?

A. I and II.B. II and III.C. I, II, and III.D. I and IV.E. IV only.

z

x

y II.

y

z

xIII.

z

y

x IV.

x

z

y I.

Ans: D


There are 3 Kinds of Vector Multiplication

jsAisA As yx

Multiplication of a vector by a scalar: A

As

vector times scalar vector whose length is multiplied by the scalar

Dot product (or Scalar product or Inner product):

B A B A BA AB ) ABcos( BA zzyyxxoo

- vector times vector scalar - projection of A on B or B on A - commutative

A

B

1kk 1,jj 1,ii

0ki 0, kj 0, ji

unit vectors measureperpendicularity:


Vector multiplication, continued

Cross product (or Vector product or Outer product):

A

B

C

- Vector times vector another vector perpendicular to the plane of A and B- Draw A & B tail to tail: right hand rule shows direction of C

e)commutativ(not A B - B A C

B to A from angle smaller the is where :magnitude

) ABsin( C

- If A and B are parallel or the same, A x B = 0- If A and B are perpendicular, A x B = AB (max)

Algebra:)B(sA B)A(s BAs :rules eassociativ

CA BA )CB( A :rule eistributivd

)CB(A C)BA(

k)BA-B(A j)BA-B(A i)BA-B(A

)kB jBi(B )k A jAiA( B A

xyyxzxxzyzzy

zyxzyx

0kk 0,jj 0,ii

j- ki ,i kj ,k ji

Unit vector representation:

i

kj

Fr

prL

Bvq EqF

Applications:


Example of Cross Product:

A force F = -8i + 6j Newtons acts on a particle with position vector r = 3i + 4j meters relative to the coordinate origin. What are a) the torque on the particle about the origin and b) the angle between the directions of r and F.

along z axis ˆ N.m k ˆ 5050

Fr

k k jj)( ij)( ji)( ii)(

)j6 i()j4 i(Fr

xxxx

321864846383

83

Use: to find magnitude

r

F

)sin( F r ||

10 ]6 8[ F 5 ]4 3[ r / 22/ 22 2121

1 )sin( )sin( 50 )sin( F r

r F 90 isthat o

)cos( 50 )cos( F r Fr

Better to Use:

0 jj)( ij)( ji)( ii)( Fr xxxx

242464846383

r F 90 0 )cos( 50 isthat o

so

Works for this case, but ambiguous: Why?

For angle try using:


What’s a “Field” - Mathematical View• A FIELD assigns a value to every point in space (2D, 3D, 4D,….)• It may have nice mathematical properties, like other functions:

• E.g. superposition, continuity, smooth variation, multiplication,..

• A scalar field f maps a vector into a scalar: f: R3->R1

• A scalar quantity is assigned to every point in 3D space• Temperature, barometric pressure, potential energy

ISOBARSEQUIPOTENTIALS

Example: map of the velocity of westerly winds flowing past mountains

Pick single altitudes and make slices to create maps

“FIELD LINES” (streamlines) show wind directionLine spacing shows speed: dense fastSet scale by choosing how many lines to drawLines begin & end only on sources or sinks

FIELD LINES

• A vector field g maps a vector into a vector: g: R3->R3

• Wind velocity, water velocity (flow), acceleration

• A 3D vector is assigned to every point in 3D space

• Taxing to the imagination, involved to calculate


Scalar field examples

• A scalar field assigns a simple number as the field value at every point in “space”.

• Temperature map portrays ground-level temperature as function of x-y position

• Maps R2 -> R1

• Scalar field: altitude at points on a mountain as function of x-y position.

• Contours follow constant altitude

Side View

steeper flatter

Contours

Contours closelyspaced

Contours far apart

• Grade (or slope) is related to the horizontal spacing of contours (vector field)


Slope, Grade, Gradients (another field) and Gravity

Height contours h can also portray potential energy U = mgh. The height and potential energy do not change along a contour. Motion perpendicular to a contour at a point is along the gradient.

• The steepness (or force) are related to the GRADIENTS of height (or gravitational potential energy) respectively, and are also fields.

• Are the GRADIENTS of scalar fields also scalar fields or are they vector fields?

15.0dx/dhx/hlim0x

• Slope and grade mean the same thing. A 15% grade is a slope of

• Gradient is measured along the path. For the case above it would be:

)sin(mgF

)sin(dl/dh

dl/dh mgdl/)mgh( ddl/dUF

• Gravitational force along path l is the gradient of potential energy

148.0 /101.115 dl/dhl/hlim0x

100

15

15%

h

l

x


Vector Fields

Side View

DIRECTION

• Gradient vectors point along the direction of steepest descent, which is also perpendicular to the contours.

• Imagine rain on the mountain. The vectors are also “streamlines.” Water running down the mountain will follow these streamlines.

• The value of a vector field at every

point in space is a vector – it has both magnitude and direction

• A vector field (like the gravitational force) can be generated by taking the gradient of a scalar field (such as potential energy).

• Gradient field lines are perpendicular to the contours (e.g., lines of constant potential energy)

• The steeper the gradient (e.g., rate of change of gravitational potential energy) the larger the field magnitude is.


Another scalar field – atmospheric pressure

How do the isobars affect air motion? What are the black arrows showing?

Isobars: linesof constant pressure


A related vector field: wind velocity

Wind speed and direction depend on the pressure gradient


Visualizing Physical Fields

Scalar field: lines of constant field magnitude• Altitude / topography – contour map• Pressure – isobars, temperature – isotherms• Potential energy (gravity, electric)

Vector field: field lines show a gradient • Direction shown by TANGENT to field line• Magnitude proportional to line density -

inversely to distance between lines• Lines start and end on sources and sinks of field (highs and lows)• Forces are fields, with direction related to

gravitational, electric, or magnetic field

Examples of scalar and vector fields in mechanics and E&M:

TYPE MECHANICS (GRAVITY) ELECTROSTATICS (CHARGE) MAGNETOSTATICS

(CURRENT)

FORCE (vector)

Gravitational Force = GMm / r2 Coulomb Force = kqQ / r2 Magnetic Force = q v X B

SCALAR

FIELDS

Gravitational Potential Energy

Gravitational Potential

(PE / UNIT MASS)

Electric Potential Energy

Electric Potential (volts)

(PE / UNIT CHARGE)

Magnetic P. E. (due to a current)

VECTOR

FIELDS

ag = Force / Unit Mass

= “Gravitational Field”

= Acceleration of Gravity “g”

E = Force / Unit Charge

= “Electric Field”

B = Force / Unit Current x Length

= “Magnetic Field”

Could be:• 2 hills, • 2 charges• 2 masses

Mass or negative charge

Magnetic field around a wire

carrying current


Some fields are used to explain “Action at a Distance”

Field Type Source Acts on Definition(dimensions)

Strength

gravitational massanother

mass

Force per unit mass at

test pointag = Fg / m

electrostatic chargeanother charge

Force per unit charge at test point

E = F / q

magneticelectric

current .length

another current .le

ngth

Force per unit current.length

B ~ F/qv or F/iL

• Place a test mass, test charge, or test current at some test point in a field • It feels a force due to the presence of remote sources of the field.• The sources “alter space” at every possible test point.• The forces (vectors) at a test point due to multiple sources add up via superposition (the individual field vectors add & form the net field).


Idea of a test mass• The field everywhere is proportional to

mass M at the origin

• The amount of force at some point due to M is proportional to the mass m at that point

• Use m as a test mass (it could be 1 kg for example) and measure the force on it as it moves around:

• g(r) is the “gravitational field”, also called the gravitational acceleration.

• The direction (only) is given by

• g(r) is a vector field, like the force.

rr

GMmF

2

r)r(g rr

GM

m

F2

r

M

Same idea for test charges & currents

m


Meaning of g(r):

1-2: What are the units of: ?

A. Newtons/meter (N/m)B. Meters per second squared (m/s2)C. Newtons/kilogram (N/kg)D. Both B and CE. Furlongs/fortnight

r)r(grr

GM

m

F

2

1-3: Can you suggest another name for ?

A. Gravitational constantB. Gravitational energyC. Acceleration of gravityD. Gravitational potentialE. Force of gravity

r)r(grr

GM

m

F

2


Superposition of fields (gravitational)• “Action-at-a-distance”: gravitational field permeates all of space with force/unit mass.• “Field lines” show the direction and strength of the field – move a “test mass” around to map it.• Field cannot be seen or touched and only affects the masses other than the one that created it.

• What if we have several masses? Superposition—just vector sum the individual fields.

The same ideas apply to electric & magnetic fields

M MM M

• The NET force vectors show the direction and strength of the NET field.


The gravitational field at a point is the acceleration of gravity g

(including direction) felt by a test mass at that point

Summarizing: Gravitational field of a point mass M

M

rb

rA

gAgB

gA

gB

surfaces ofconstant field & PE

inward forceon test mass m• Move test mass m around to map direction

& strength of force• Field g = force/unit test mass• Lines show direction of g is radially inward (means gravity is always attractive)• g is large where lines are close together

• Newton:

)m/s or g(Newtons/k2 r

r

GMg

2

Where do gravitational field lines BEGIN?• Gravitation is always attractive, lines BEGIN at r = infinity Why inverse-square laws? Why not inverse cube, say?

• Field lines END on masses (sources)


An important idea called Flux (symbol Basically a vector field magnitude x area

Definition: differential amount of flux dg of field ag crossing vector area dA

scalar) (adA n a Ad through a of flux d

g

gg

ag n

“unit normal”

outward andperpendicular to

surface dA

- fluid volume or mass flow - gravitational - electric - magnetic

Flux through a closed or open surface S: calculate “surface integral” of field over S

dA n a d S S

gS

Evaluate integrand at all points on surface S

EXAMPLE : FLUX THROUGH A CLOSED, EMPTY, RECTANGULAR BOX IN A UNIFORM g FIELD• zero mass inside• from each side = 0 since a.n = 0, from ends cancels• TOTAL = 0• Example could also apply to fluid flow ag

n

n

n

n

What if a mass (flux source) is in the box? Can field be uniform? Can net flux be zero.

“Phi”


FLUID FLUX EXAMPLE: WATER FLOWING ALONG A STREAMAssume: • constant mass density• incompressible fluid – constant • constant flow velocity parallel to banks• no turbulence (laminar flow)Flux measures the flow (current): • flow means amount/unit time across area• rate of volume flow past a point• rate of mass flow past point

2 related fields (currents/unit area):• velocity v represents volume flow/unit area/unit time• J = mass flow/unit area/unit time v J

Flux = amount of field crossing an area per unit time (field x area)

Av t

Al

t

V flux olumev

and A J Av A t

l

t

m luxf mass

A l V m chunk solid of mass

The chunk of mass moves l in time t:

v

tvl

A

Continuity: Net flux (fluid flow) through a closed surface = 0 ………unless a source or drain is inside

A n

AnA

area vector to

vectorunit outward the is

1A

2A

'n

n

Self Study


M

rb

rAgAgB

gA

gB

surfaces ofconstant field & PE

inward forceon test mass m

Find total flux through closed surface A

A2A 2AA Ad.r

r

GM Ad.r

r

GM Ad.g

)m/s or g(Newtons/k2 r

r

GMg

2

Flux depends only on the enclosed mass(same, say for surface B)

FLUX measures the strength of a field source that is inside a closed surface - “GAUSS’ LAW”

Gauss’ Law for gravitational field: The flux through a closed surface S depends only on the enclosed mass (source of field), not on the details of S or anything else

Example: spherically symmetric mass distribution, radial gravitational field

Ad.rr

GMAd . gAd . ieldf)flux(d 2

Field:

GM4 r4x r

GM Ad.g 2

A2A

AA

Integral for surface area of sphere


Shell Theorem follows from Gauss’s Law

1. The force (field) on a test particle OUTSIDE a uniform SPHERICAL shell of mass is the same as that due to a point mass concentrated at the shell’s mass center (use Gauss’ Law & symmetry or see section 13.6)

x

mr

xm r

Same for a solid sphere (e.g., Earth, Sun) via nested shells

2. For a test mass INSIDE a uniform SPHERICAL shell of mass m, the shell’s gravitational force (field) is zero

• Obvious by symmetry for center point• Elsewhere, integrate over sphere (painful) or apply Gauss’ Law & Symmetry

x

mx

3. Inside a solid sphere combine the above. The force on a test mass INSIDE depends only on mass closer to the CM than the test mass.

x• Example: On surface, measure acceleration g a

distance r from center

• Halfway to center, ag = g/2

33

4 rVsphere

xr

x

mr x

r

+ +

Self Study


Superposition Example: Calculate the field (gravitational) at a special point due to two point masses

Find the field at point P on x-axis due to two identical mass chunks m at +/- y0

• Superposition says add fields created at P by each mass chunk (as vectors!!)• Same distances r0 to P for both masses

• Same angles with x-axis

• Same magnitude ag for each field vector

• y components of fields at P cancel, x-components reinforce each other

• Result simplified because problem had a lot of symmetry

y

xP

m

m

+y0

-y0

+x0

ag

ag

r0

r0

20

20

20 y x r

00 r / x )cos(

n)gravitatio oflaw Newtons (fromg yx

mG a

20

20

2320

20

303

0

020

/ wherextot ]yx[ r

r

x m2G

r

)cos( m2G a a

Direction: negative x


Example: Calculate gravitational field due to mass distributed uniformly along an infinitely long line. Find the field at point P on x-axis

xP

dm = dy

y

x

dagr

y to

y to

= mass/unit length

-y

2 )dcos(/2

/2-

x

G d )cos(

x

G a

/

/x

2

2

2

Field of an infinite line falls off as 1/x not 1/x2

• Integrate over from –/2 to +/2 )tan( x y

)](tan[ xy x r 22222 1

)](tan[1 xd

)dtan(x

d

dy 2

d )cos(x

G d )cos(

)](tan[1x

)](tanx[1G da

2

2

x 2

)cos( r

mdG )cos(da da gx

2

d )](tanx[1 dy dm 2

ada x-

x

where

• Integrate over the source of field holding P fixed• Add differential amounts of field created at P by differential mass chunks at y (as vectors!!) • Include mass from y = – infinity to y =+ infinity

• For symmetrically located chunks:• y-components of fields cancel, • x-components of fields reinforce each other

• Mass per unit length is uniform, find dm in terms of :


Line integral (path integral) examples for a gravitational field

How much work is done on a test mass as it traverses a particular path

through a field? sdmasdFdUdW g

path along evaluate

B

A

sdF U

test mass

Gravitational field is conservative so U is independent of path chosen

B & A between path any for

A

B

B

A

sdF- sdF

S

chosen isthat path closed any for0 sdF U

circulation,or path integral

EXAMPLEuniform field

U= - mgh U= + mgh

B

Self Study


Spherical Polar Coordinates in 3 Dimensions

+x

+y

+z

P

r

)sin(r r |r| xy

)cos(rrz

90o90o

y

x

90o

zkz j y ix r

z)y,(x, r

),(r, r

21 /222radians][0,2in ,azimuth""

radians ][0,in ,olatitude"c"

)z y (x r

Cartesian

Polar, 3D

)(sinr yx r

)sin()sin(r )sin(r y222

xy

22

)cos()sin(r )cos(r x xy )cos(r z

Polar to Cartesian

)x/y(tan -1

)r/z(cos 1

21 /222 )z y (x r

Cartesian to Polar


Example: Show that the surface area of a sphere A= 4R2

by integrating over the surface of the sphere

x

z

y

r

Find dA – an area segment on the surface of the sphere, then integrate on angles (azimuth) and (co-latitude).

dhdl dA

Where:• dl is a curved length segment of the circle around the z-axis (along a constant latitude line)• dh is a segment along the direction (along a constant longitude line) d r dh d )sin( r dl

Self Study

Angle range for a full sphere: ]2 [0, ] [0,

]11)[(r2 )]cos([r2 d )sin( r2

} d )sin( { }d{ r dd )sin( r Ad.r A

2

0

2

0

2

2

0 0

22

0 0

2

surface

r4 A 2

Factors into 2 simple 1 dimensional integrations

Ad r Ad


Gravitational field due to an infinite sheet of mass

Constant - does not dependon distance from plane!Simple 2

dimensional integration

Self Study

copyright r. janow – fall 2014 1 physics 121: electricity and magnetism introduction syllabus,...

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