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Classical Electrodynamics

Chapter 6

Maxwell Equations ,

Macroscopic Electromagnetism,

Conservation Law

2011 Classical Electrodynamics Prof. Y. F. Chen

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Contents

§6.1Maxwell Equations

§6.2Conservation law


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§6.1Maxwell Equations

The basic laws of electricity and magnetism

↓ Displacement current

Maxwell equations

1. The basic laws of electricity and magnetism：


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2. Displacement current

：

3. Maxwell equations：


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4. In vacuum ( 、

)：

5. The definition of and in terms of the potentials and satisfies identically

the two homogeneous Maxwell equations. It is convenient to restrict the

consideration to the vacuum form to the Maxwell equations. Then the

inhomogeneous equations can be written in terms of the potentials as (6.10)(6.11)


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，Let

inserted into

(6.10)

(6.11)


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6. We have now reduced the set four Maxwell equations to two equations. But they are

still coupled equations. The uncoupling can be accomplished by exploiting the

arbitrariness involved in the definition of the potentials.

gauge transformation :

The freedom means that we can choose a set of potentials (A, Φ)to satisfy the Lorenz

condition

. This will uncouple the pair of equations (6.10) and (6.11)

and leave two inhomogeneous wave equations, one for Φ and one for A:


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7. Suppose that the potentials that satisfy (6.10) and (6.11) do not satisfy Lorenz

condition

. Then let us make a gauge transformation

→ and demand that satisfy the Lorenz condition:

Thus, provided a gauge function Λ can be found to satisfy


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Coulomb gauge(transverse gauge)：This is the gauge in which

From (6.10) we see that the scalar potential satisfies the Poisson equation, Φ

,

with solution, Φ

(6.23)

From (6.11)：

(6.24)

Let ，，


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With the help of the continuity equation

and Φ

(6.23) it is seen that

Φ

(6.24)


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※In passing we note a peculiarity of the Coulomb gauge. It is well known that

electromagnetic disturbances propagate with finite speed. Yet (6.23) indicates that the

scalar potential “propagates” instantaneously everywhere in space. The vector potential,

on the other hand, satisfies the wave equation (6.30), with its implied finite speed of

propagation c. At first glance it is puzzling to see how obviously unphysical behavior is

avoided. A preliminary remark is that it is the fields, not the potentials, that concern us.

A further observation is that the transverse current (6.28) extends over all space, even

if J is localized.


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8. Green functions for the wave equation

A. Green functions of time independent for the wave equation

The basic structure of the wave equations:

: source distribution

For the Poisson equation

, the solution


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Fourier transform :

When the representations wave function and source distribution are

inserted into the wave equation

it is found that

the Fourier transform satisfies the inhomogeneous Helmholtz wave

equation

、

inserted into


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The Green function appropriate to (6.35) satisfies the inhomogeneous

equation


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B. Green functions of time dependent for the wave equation

If


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The Green function is called the retarded Green function because it exhibits the

causal behavior associated with a wave disturbance. The argument of the delta function

shows that an effect observed at the point x at time t is caused by the action of a source a

distance R away at an earlier or retarded time, t’ = t − R/c. The time difference R/c is

just the time required for propagation of the disturbance from one point to the other.

Similarly, is called the advanced Green function.

(time independent)

(time dependent)


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To specify a definite physical problem, solutions of the homogeneous equation may be

added to either of and .

The presence of guarantees that at remotely early times, t, before the source has

been activated, there is no contribution from the integral. Only the specified wave

exists. The second situation is that at remotely late times (t → +∞) the wave is given as

, a known solution of the homogeneous wave equation. Then the complete

solution for all times is


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The commonest physical situation is described by the time retarded with = 0. It is

sometimes written with the Green function inserted explicitly:

9. Retarded solution for the fields: Jefimenko’s generalizations of the Coulomb and

Biot - Savart laws：

A. Electric field (time independent)

→ (time dependent)

(5.55)


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From Maxwell equations：

(6.49)

(6.51)


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Let

inserted into (6.51)


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(6.55)


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B. Magnetic field (time independent)

(5.14) →(time

dependent)

(5.56)

From Maxwell equations：

(6.50)

(6.52)


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Let

inserted into (6.52)


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(6.56)


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§6.2Conservation law：

1.

：

：

：

：


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2. Tensor：


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3.

and

, ,


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Let ( ting vector)

：


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4.

和

( ： )


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In vacuum ,

：


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If


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Let

,

,


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From divergence theorem：


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5.


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