MANMOHAN DASH, PHYSICIST, TEACHER !

Physics for ‘Engineers and Physicists’

“A concise course of important results”

Lecture - 2

Electrodynamics

Lectures given around 9.Nov.2009 + further content developments this week; 21-23.Aug.2015 !

Recapitulation

“We discussed the vectors and fields in the last lecture”

Lecture - 1

Vector Calculus and Operations

I suggest ‘Lecture-1’ is followed first before Lecture-2’ for a better understanding ! Take a quick look >> Vector Calculus Lecture-1 <<

Electric Field

In the last lecture presentation we discussed vector fields. We saw how for every vector field there is a scalar field comprehensively related to each other. Eg for every potential energy field, which is a scalar field, there always exists a vector field called a Force vector or force field. Electric fields are the corresponding vector field of a scalar field called electric potential or electric potential field.

Electric fields are generated from various sources; static electric charge, electric current, magnetic induction etc.

Electric fields from static electric charges

The electric fields that are produced from static electric charges are determined by Coulomb’s Law of electrostatics force. These are called electrostatics field.

Electric fields are generated from static electric charge, out of the 3 types only +ve and –ve charges produce a Coulomb’s electrostatics force.

We have discussed recently that there are 3 kinds of electric charges and not just two. Here (slide-2) >> 3 types of electric charges <<

Accordingly only two types of charges give rise to electrostatics force, the positive and the negative.

Electrostatic Fields in vacuum

The Coulomb’s Force F is proportional to the two charges (q, Q) and inversely proportional to square of separation r. F is along line joining 2 charges

The central force F is produced by charge Q on charge q and electric field E is F/q. There are two scalar fields energy E, potential V. Accordingly there are two vector fields; F and E.

Coulomb’s Force Law, free space

We saw in last slide the Coulomb’s Force F is between 2 non-zero charges which we denote as (1, 2), (q, Q) or (q1, q2). F is a vector field, experienced by q, produced by Q. F is related to scalar field E, the energy of configuration of q and Q. Also F is deprecated to another vector field E by dividing the charge (q) that experiences the force. E is related to its own scalar field called the electric potential V. So we have; Its Q which produces electric force F and electric field E, on q.

rr

QEEqFr

r

qQF ˆ

4

1 ; ;ˆ

4

12

0

2

0

Coulomb’s Force; medium vs freespace !

We saw in last slide the Coulomb’s Force F is between 2 non-zero charges in free space. Its slightly modified if the charges in consideration are present in any media rather than freespace. We see that; we have two definitions of epsilon, the epsilon with zero subscript, read as epsilon_nut, is the ‘permittivity of free-space’ and the regular epsilon is the ‘permittivity of medium’. The factor by which ‘permittivity of medium’ is larger than ‘permittivity of free-space’ is called as ‘relative permittivity’ and is denoted by epsilon_r. epsilon_r is also called as ‘dielectric constant’ denoted by K.

rr

QEEqFr

r

qQF ˆ

4

1 ; ;ˆ

4

122

100 Kr

Electric Displacement and Electric Flux!

Now we should define two more quantities in relation to the electric field vector E. Before that we need to state the values of the constants that we discussed in last slide. We note that the epsilon has particular values for particular media. But epsilon_nut which specifies the permittivity of free space has a very well specified value;

2

212

02

29

0

1085.8 ;100.94

1

Nm

C

C

Nm

Electric Displacement and Electric Flux!

We already know E is the electric field vector. Here the charge in the denominator is arbitrarily vanishing so as not to intefere with the field F or E produced by source charge Q. Similarly D is deprecated from E and called electric displacement vector. It depends on the permittivity of the space in which the E is being decided. The electric flux Phi is defined as a surface integral (Lecture-1)

S

E

S

generalq

SdESdAEDEDq

FE

)( , , ,lim 0

0

Electric Flux, Gauss Law of electrostatics!

Total electric flux Phi over a closed surface is equal to 1/eps times the net charge enclosed by the surface. The surface is called Gaussian surface.

We evaluate the flux for a point charge Q from the spherical symmetry of the electric field E and find it to be;

22

0

2

4

1 ;4

r

q

rEor

qEr net

E

0 net

S

E

qSdE

Gauss Law in differential form !

Gauss law in the last slide was in the integral form. We like to cast it into a differential form. We also like to discuss its form in the dielectric medium. Gauss law can also be stated in terms of electric Displacement Vector D (from above); >> Total charge is from volume distribution of charge. Now apply Gauss Divergence Theorem on the surface integral in Gauss law for E. See GDT in (Lecture-1)

vacuummedium

net

S

E EDEDq

SdE

0

0

/ ,

net

S

qSdD

>> Gauss Law in Differential Form

V

net dVq

Gauss Law in differential form !

By applying GDT we would have >> So Gauss Law can be written as

Gauss Law in Differential Form <<

VS

dVESdE )(

0)( 1

)(00

VVV

dVEsodVdVE

DEEE

mediumvacuum

0

00

Magnetic field.

A moving charge produces a magnetic force. And in turn a moving charge experiences magnetic force, produced by any other moving charge. The stationary charges do not interact magnetically. Similar to electric field vector one can define a magnetic field vector. We defined electric field vector from electric force vector. Hence the clue is to find the magnetic force vector to define the magnetic field vector.

Magnetic field is also known as magnetic induction or magnetic flux intensity. Its SI unit is Tesla. Its lines of force/flux are shown below.

Magnetic field and magnetic force

A magnetic field B is defined in terms of the magnetic force F which a moving test charge q experiences when it moves in the field with velocity v. As we mentioned the SI unit of B is Tesla. Biot-Savart Law; The Biot-Savart law gives the elemental magnetic field created, when an element of current induces such a field. Like before Magnetic Permeability of free space is denoted as; mu_nut. Mu is magnetic permeability in any medium.

gaussm

Wb

mA

NT 4

2101

.11

BvqFB

A

Tm

r

rlIdBd )104( ,

4

7

03

0

Magnetic intensity and magnetic flux

A magnetic field B is defined in terms of the magnetic force. Like we defined an electric-displacement D vector from an electric field vector E which we had defined from its electric force vector F we can do the same for the magnetic field vector B. We define the magnetic intensity H from magnetic field (We had defined B from magnetic force F) So F >> B >> H. Like electric flux we define magnetic flux; its the surface integral of the magnetic field B over the surface S. It has a SI unit Weber (Wb). B is also called magnetic flux density.

met

AmpunitSI

BH ; ,

0

S

B WbWeberunitSISdB . ,

Magnetic flux and Gauss law of magnetism

There are no magnetic mono-poles or magnetic charges from which a magnetic force emanates. The magnetic fields, forces, flux emerge from electric charges in relative motion. Charges in motion produce, electric forces, fields and flux as well. Charges in rest produce electric fields, but in motion produce electric & magnetic fields.

S

SdB 0

0 VS

dVBSdB

The absence of magnetic mono-poles is represented by Gauss Law of magnetism as stated above. Like we did for electric flux, we convert the surface integral to a volume integral by Gauss Divergence Theorem. This leads to the differential form of Gauss law of magnetism. 0 B

We are in a pickle

Since there are no mag. mono-poles or magnetic charges from which a magnetic force emanates we are in a pickle. As we see, the Gauss law in magnetism does not correspond to any kind of charge now. We can not then apply the same treatment to magnetic field that we applied on electric fields?

Fortunately we have what it takes for us to be getting out of the pickle. We have what we call Ampere’s Circuital Law which would do the same for us that Gauss law in electrostatics did. We could evaluate magnetic fields if we have a symmetric situation; we have this amazing law.

Ampere’s Circuital Law

Line integral of a magnetic field B along a closed loop is equal to mu_nut times the net electric current I enclosed by the loop. We also note the same law for the H field we defined some slides ago.

Consider an electric current flowing in a straight line as-if emerging from the floor of the CCD where you are drinking your coffee. Now Put your thumb (RH) along it upwards and curl your palm. (so nails comes closer to thumb) In the next slide we will see how the Ampere’s circuital law we just discussed helps us determine the H, B.

net

C

net

C

IldHorIldB , 0

Application

Magnetic field by Ampere’s Circuital Law

We have applied here the Ampere’s circuital law, by recognizing the circular symmetry of the situation from the straight current. We are out of the pickle now.

So the curl of the palm gives the direction of the magnetic field as shown, ‘tangents’ to the circle. B is uniform across the loop, the integral simply is; circumference of the loop.

Differential form of Ampere’s Circuital Law

To convert the integral form of Ampere’s law that we have stated shortly, we have to use a similar trick as we used in the case of electric field Gauss law. We convert the current I into a surface integral, by recognizing the current density j is the current per unit area. [j is a vector field, but I is not]

S

SdjI

ISdBldBC

0

S

)(

SC

SdjSdBldB

0

S

)(

We apply the Stoke’s Theorem that we discussed in (Lecture-1) to convert the line integral in Ampere’s circuital law to the curl of the magnetic field B. This leads to the differential form of the Ampere’s circuital law.

jB

0

Equation of Continuity !

Now that we see the flow of electric charge is what causes magnetic forces (as well as electric forces) and these flow are defined by current I and current density vector field j we want to see the relation that establishes the familiar concept fo conservation of electric charge.

S

SdjI

V

)( dVjSdjS

VV

dVt

dVt

)(

t

q-I

In the second step we apply the Gauss Divergence Theorem to convert the surface integral to a volume integral. In the 3rd step we recognize the rate of flow of electric charge from the volume V, through surface S, as the electric current.

0

tj

Faraday’s Law of electromagnetic induction

Michael Faraday was an experimental Physicist who stumbled upon the discovery that electric phenomena induces magnetic effects as well as magnetic phenomena induces the electric effects. Electricity and Magnetism were considered separately until then. His work led to the unification and this was together called as electromagnetism. The basic observation that an emf (electromotive force, we called electric potential field V, a scalar field, in Lect-1) is developed when there is a variation in the magnetic flux is known as Faraday’s law.

t

emf B

)(

Faraday’s Law of electromagnetic induction

The emf is so induced that it opposes its own cause, much like love developed in us opposes the very person who loves us. The emf induced in a conducting loop is equal to the negative of the rate of change of magnetic flux through the surface enclosed by the loop. Its also called as Lenz’s law.

t

emf B

)(

Faraday’s Law of electromagnetic induction

In slide 17 of (Lecture-1) we saw that electric field vector E is the first derivative (in space) of the electric potential V. In other words V is the path integral of the field vector E. [path integral on slide 32-33 of (Lecture-1) ] We also know the magnetic flux. This leads to an integral form of the Faraday’s Law (of electromagnetic induction) and called as “integral form of Maxwell-Faraday equation”. We use Stoke’s theorem (like we did in case of Ampere’s Law) to reduce the integral form to the differential form of Maxwell-Faraday Eq!

SC

S

B

C

SdBt

ldEEqnsFaraday

wellMaSdBldE

'

x form; integral , 0 ;

)( Th; sStoke'

t

BEeqnMFofformalDifferenti

SdBt

SdEldESSC

Maxwell’s Displacement Current

Maxwell’s displacement current is a current associated with a time varying electric field even if there is no actual flow of charges. Consider a parallel plate capacitor as shown. The displacement current I_d b/w plates is found.

A current is produced between the plates due to a variation in the electric field E due to current I.

Modified Ampere’s Circuital Law

Now we have a reason why we must modify the Ampere’s circuital law as we discussed it. This is because apart from an actual current I (from motion of charge) there is a current from the variation of E field as given above. This pseudo-current also produces an actual magnetic field. So in effect the variation in E produces the B field, as we mentioned at beginning. Modified form is known as Ampere-Maxwell Law.

We saw that the displacement current flowing between the plates is This can also be written as a surface integral, for the general case, by converting the area A for the case of capacitor into dS.

dt

dEAId 0

t

IsoSdEt

I Ed

S

d

00 ; ,

Modified Ampere’s Circuital Law

Instead of repeating the steps that we have followed before we simply add the displacement current term at the right place. Its easy; We also realize that the term to the right of j is the displacement current density. (the vector field) Its also easy to see that by dividing mu on both sides of differential form of M.A.C. law or Ampere-Maxwell law we get the following differential law in terms of H.

)( ; .

)( ;

00

0

t

EjBformDiff

IIldBformIntegralC

d

t

Ejd

0

t

DjH

Maxwell electromagnetic equations

When we summarize all the equations we discussed what we get are called as Maxwell’s electromagnetic equations. These are 4 in number but due to the fact that these can be stated in terms of E or B field, D or H field and differential or integral forms, they look like a lot. Lets summarize.

Differential Form [E, H, D, B appear. No Mu, eps appear]

jt

DH

t

BE

Varyingtime

,0

0B

D

statesteady

Maxwell electromagnetic equations

Here are the differential and integral forms in terms of E and B field only.

Differential and Integral Form [E, B only]

jt

EB

t

BEBE

0,0,

SCSC

SVS

Sdt

EjldBSdB

tldE

SdBdVSdE

)(,

0,1

0

0

Thank you

We discussed extensively the electromagnetism course for an engineering 1st year class. This is also useful for ‘hons’ and ‘pass’ Physics students.

This was a course I delivered to engineering first years, around

9th November 2009. I added all the diagrams and many explanations only now; 21-23 Aug 2015.

Next; Lectures on ‘electromagnetic waves’ and ‘Oscillations

and Waves’. You can write me at g6pontiac@gmail.com or visit my website at http://mdashf.org