Mathematical Methods for Physicists: A concise introduction - Site Map

PROBLEMS
His work led to the understanding of electromagnetic radiation, of which
light is a special case.
Given the four Maxwell equations
rE ˆ =" 0 ;
…Gauss' law†;
rB ˆ 0 … j ‡ " 0 @E=@t† …Ampere's law†;
rB ˆ 0
rE ˆ@B=@t
…Gauss' law†;
…Faraday's law†;
where B is the magnetic induction, j ˆ v is the current density, and 0 is the
permeability of the medium, show that:
(a) the electric ®eld and the magnetic induction can be expressed as
E ˆr @A=@t;
B ˆrA;
where A is called the vector potential, and the scalar potential. It
should be noted that E and B are invariant under the following transformations:
A 0 ˆ A ‡r;
0 ˆ @=@t
in which is an arbitrary real function. That is, both (A 0 ;†, and
(A 0 ; 0 ) yield the same E and B. Any condition which, for computational
convenience, restricts the form of A and is said to de®ne a gauge. Thus
the above transformation is called a gauge transformation and is
called a gauge parameter.
(b) If we impose the so-called Lorentz gauge condition on A and :
rA ‡ 0 " 0 …@=@t† ˆ0;
then both A and satisfy the following wave equations:
r 2 A 0 " 0
@ 2 A
@t 2 ˆ 0j;
r 2 0 " 0
@ 2
@t 2 ˆ=" 0:
10.4 Given Gauss' law RR S
E ds ˆ q=", ®nd the electric ®eld produced by a
charged plane of in®nite extension is given by E ˆ =", where is the charge
per unit area of the plane.
10.5 Consider an in®nitely long uncharged conducting cylinder of radius l placed
in an originally uniform electric ®eld E 0 directed at right angles to the axis of
the cylinder. Find the potential at a point …> l† from the axis of the cylinder.
The boundary conditions are:
411