THE UNIVERSITY OF BRITISH COLUMBIA

Physics 401 Assignment # 10:

RETARDED
POTENTIALS

SOLUTIONS:

Wed. 15 Mar. 2006 - finish by Wed. 22 Mar.
1. (p. 426, Problem 10.8) - Retarded Gauge: Confirm that the RETARDED POTENTIALS satisfy the LORENTZ GAUGE condition,
 (1)

 (2)

ANSWER Following the hint, we first show
 (3)

where , denotes derivatives with respect to , and denotes derivatives with respect to :   The identity
 (4)

and the (hopefully by now familiar) results
 (5)

 (6)

 (7)

Adding together Eqs. (6) and (7) gives Eq. (3).
Next, noting that depends on both explicitly and through , whereas it depends on only through , we confirm that

 (8)

 (9)

Derivatives of with respect to (on which it does not depend explicitly) mix in the time derivative through the implicit dependence of tr on . That is,
 (10)

because, for a given , , and .
However, depends explicitly and implicitly upon , and must locally satisfy the EQUATION OF CONTINUITY (i.e. charge conservation) at any instant of time in terms of the source coordinates , so we have

 (11)

because .
Finally we use this to calculate the divergence of in Eq. (10.19):

 (12)

The DIVERGENCE THEOREM tells us that

Now, if the closed surface encloses all the charges and currents in the source volume, over the whole surface and the surface integral is zero, leaving

or .

2. (p. 427, Problem 10.10) - Weird Loop:

A piece of wire bent into a weirdly shaped loop, as shown in the diagram, carries a current that increases linearly with time:

1. Calculate the retarded vector potential at the center.   ANSWER Choose the origin at the same place as the field point: the centre. Thus and . The source region is uncharged, so V = 0.

where . Now, by symmetry there is as much current going "up" as "down" at the same and tr, so the components cancel. This leaves

where

or .

2. Find the electric field at the center.

ANSWER Since V=0 we have just
.

3. Why does this (neutral) wire produce an electric field?   ANSWER Because the vector potential is changing with time, "Doh!" I think this is meant as a retroactive hint in case you got hung up on the preceding question.

4. Why can't you determine the magnetic field from this expression for ?

ANSWER Finding requires knowledge of the dependence of on ; but we have calculated only at one point in space! If you want a differentiable you will have a far more difficult calculation to perform.

3. (p. 434, Problem 10.13) - Circulating Charge: A particle of charge q moves in a circle of radius a at constant angular velocity . [Assume that the circle lies in the plane, centered at the origin, and that at time t=0 the charge is at (a,0), on the positive x axis.] Find the LIÉNARD-WIECHERT POTENTIALS for points on the z axis.   ANSWER In general,

where means that the quantities in the square brackets are to be evaluated at the retarded time . Relative to the origin, .
For a point on the z axis, and so , independent of time. We also have and . Thus and where . Then , leaving
and
.

4. (p. 441, Problem 10.19) - Sliding String of Charges: An infinite, straight, uniformly charged string, with charge per unit length, slides along parallel to its length at a constant speed v.
1. Calculate the electric field a distance d from the string, using Eq. (10.68):

where .

ANSWER Suppose the field point is a perpendicular distance s from the string; measure z from the nearest point on the string, as shown in the diagram. Equation (10.68), in which we do not need to evaluate anything at a retarded time, gives the contribution to from a single charge q. We need to superimpose such contributions from all charge elements at positions down the string: for each of these we use :

For each element dz at z there is an equal element dz at -z; thus the "horizontal" components cancel, leaving only the x component of , namely . Meanwhile, since , ; and since , . So and

Let so that and :

just as for a line charge at rest!

2. Find the magnetic field of this string, using Eq. (10.69):

where .   ANSWER Well, and , so this is trivial:1

where . (Again, the same result as for a steady current in magnetostatics.)

Jess H. Brewer
2006-03-21