Electromagnetism II (spring term 2020)

E Goudzovski eg@hep.ph.bham.ac.uk

http://epweb2.ph.bham.ac.uk/user/goudzovski/Y2-EM2

Lecture 3

Maxwell’s equations in free space

Previous lecture (1)

1

Divergence (Gauss) theorem:

Integral of its divergence over the enclosed volume

KelvinStokes (curl) theorem:

Flux of a vector field through a closed surface

Line integral of a vector field around a closed curve

Flux of its curl through

any surface enclosed by the curve

Previous lecture (2)

2

Gauss’s law

Absence of magnetic poles

Faraday’s law

Ampere’s law

The principal EM laws in free space in the differential form:

This lecture

3

Lecture 3: Maxwell’s equations in free space

The continuity equation

Modification of Ampere’s law for non-steady currents

The displacement current

Maxwell’s equations of electrodynamics in free space

The continuity equation

4

Conservation of electric charge: any variation in the charge within

a closed surface is due to charges that flow across the surface

Differential form

(using the divergence theorem):

Divergence of both sides of Ampere’s law:

T1 from lecture 2 Continuity equation

Ampere’s law applies to steady currents only ( ),

and requires generalization

Modification to Ampere’s law

5

To generalize Ampere’s law for non-steady currents,

need to modify its RH side to make its divergence zero.

Gauss’ law: . Differentiation over t:

Displacement current density We found a quantity which has

zero divergence, and is suitable to modify Ampere’s law:

Not a “proof”: infinite number of ways to remove the contradiction.

Any solenoidal vector field (i.e. such that ) can be added to the RH side of the modified Ampere’s law.

(the AmpereMaxwell law)

The displacement current

6

Alternating current in a copper wire [conductivity =6×107 (·m)1]

Electric field: E = E0sin(t)

Conduction current density: JC = E = E0sin(t)

Displacement current density: JD = 0∂E/∂t = 0E0cos(t)

Ratio of maximum max conduction to displacement current:

JCmax / JD

max = E0 / 0E0 ~ 1019 s1 1

is very small for all frequencies used in practice.

A changing E-field produces B-field: Maxwell’s decisive step

towards electrodynamics (1865).

It took 30 years from Faraday’s discovery of induction to postulate the displacement current as the source of magnetic field.

Example 1

7

Q

S0

A charged sphere discharging

into external conductive medium.

By symmetry, is directed radially.

What about the field?

By symmetry, tangential component

BT=0, and radial component BR is

the same in each point of sphere S.

Assuming BR≠0 leads to

Conclude that B=0.

Ampere’s law leads to JC=0.

E-field at the surface of the sphere (lecture 1):

Conduction current at surface S0 is balanced by displacement current:

S0

Example 2

8

Discharge of a capacitor:

S1

For the surface S1, only the

conduction current IC contributes.

For the surface S2, there is no conduction current; conclude that ID=IC.

S2

By definition, Lecture 1: E=/0

Check the equality ID=IC:

Example 3

9

A thin parallel plate capacitor with

circular plates of radius R is being

charged. Find the magnetic field B(r).

Using axial symmetry, in the absence of conduction current (JC=0),

Inside the capacitor (r<R),

Outside the capacitor (r≥R),

Maxwell’s equations in free space

10

(M1)

(M2)

(M3)

(M4)

(remember them)

Displacement current density:

The laws of electrodynamics:

The continuity equation follows from (M1) and (M4).

Discussion and summary

11

Maxwell’s equations in free space (2 scalar + 2 vector) are

equivalent to 8 scalar equations with 10 variables

(Ex, Ey, Ez, Bx, By, Bz, Jx, Jy, Jz, ).

They need to be complemented by the equations characterizing

the media. This will be discussed in lectures 59.

They are not symmetric wrt electric and magnetic fields:

no magnetic poles as sources of B field (M2);

no magnetic currents as sources of E field (M3).

For constant fields (∂E/∂t = ∂B/∂t = 0), two independent groups,

electrostatics: div E = /0; curl E = 0;

magnetostatics: div B = 0; curl B = 0J.