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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

    2011 Classical Electrodynamics Prof. Y. F. Chen

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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

    2011 Classical Electrodynamics Prof. Y. F. Chen

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    2. Displacement current

    3. Maxwell equations

    2011 Classical Electrodynamics Prof. Y. F. Chen

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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)

    2011 Classical Electrodynamics Prof. Y. F. Chen

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    Let

    inserted into

    (6.10)

    (6.11)

    2011 Classical Electrodynamics Prof. Y. F. Chen

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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, )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 and one for A:

    2011 Classical Electrodynamics Prof. Y. F. Chen

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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 can be found to satisfy

    2011 Classical Electrodynamics Prof. Y. F. Chen

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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,

    ,

    with solution,

    (6.23)

    From (6.11)

    (6.24)

    Let

    2011 Classical Electrodynamics Prof. Y. F. Chen

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    2011 Classical Electrodynamics Prof. Y. F. Chen

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    With the help of the continuity equation

    and

    (6.23) it is seen that

    (6.24)

    2011 Classical Electrodynamics Prof. Y. F. Chen

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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.

    2011 Classical Electrodynamics Prof. Y. F. Chen

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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

    2011 Classical Electrodynamics Prof. Y. F. Chen

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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

    2011 Classical Electrodynamics Prof. Y. F. Chen

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    The Green function appropriate to (6.35) satisfies the inhomogeneous

    equation

    2011 Classical Electrodynamics Prof. Y. F. Chen

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    B. Green functions of time dependent for the wave equation

    If

    2011 Classical Electrodynamics Prof. Y. F. Chen

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    2011 Classical Electrodynamics Prof. Y. F. Chen

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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)

    2011 Classical Electrodynamics Prof. Y. F. Chen

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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

    2011 Classical Electrodynamics Prof. Y. F. Chen

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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: Jefimenkos generalizations of the Coulomb and

    Biot - Savart laws

    A. Electric field (time independent)

    (time dependent)

    (5.55)

    2011 Classical Electrodynamics Prof. Y. F. Chen

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    From Maxwell equations

    (6.49)

    (6.51)

    2011 Classical Electrodynamics Prof. Y. F. Chen

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    Let

    inserted into (6.51)

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    (6.55)

    2011 Classical Electrodynamics Prof. Y. F. Chen

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    B. Magnetic field (time independent)

    (5.14) (time

    dependent)

    (5.56)

    From Maxwell equations

    (6.50)

    (6.52)

    2011 Classical Electrodynamics Prof. Y. F. Chen

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    Let

    inserted into (6.52)

    2011 Classical Electrodynamics Prof. Y. F. Chen

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    (6.56)

    2011 Classical Electrodynamics Prof. Y. F. Chen

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    6.2Conservation law

    1.

    2011 Classical Electrodynamics Prof. Y. F. Chen

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    2011 Classical Electrodynamics Prof. Y. F. Chen

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    2. Tensor

    2011 Classical Electrodynamics Prof. Y. F. Chen

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    2011 Classical Electrodynamics Prof. Y. F. Chen

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    3.

    and

    , ,

    2011 Classical Electrodynamics Prof. Y. F. Chen

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    Let (