Physics 305
Problem set 8
Due: Weds, Apr 4
[1] Consider a sheet of current and charge, placed on the y-z plane (at x = 0). The charge and
current! density are zero for all times t < 0. At t = 0, a z-dependent current sheet is switched on,
with K = ! ( K o z ) ẑ . This current sheet remains on for times t > 0.
(a) Show from charge continuity that the given current sheet causes a position-independent
charge density K ot to accumulate in the sheet for times t > 0.
(b) Using the retarded-potential relationship, solve for the scalar potential, V, at a distance x
away from the y-z plane.
(c) Similarly, solve for the vector potential at a general position (x, y, z). Hint: the retarded time
relationship will cut out a circular area in the sheet within which the current contributes to the
vector potential. Look for a cancellation in the z-dependence that simplifies the result; the
integral should be relatively easy to evaluate.
(d) Solve for the electric and magnetic fields in the vicinity of the sheet, and verify that
!
! !
Ampere’s law, ! " B = µo# o $ E $t , holds as expected.
[2] Griffiths problem 10.10 (page 427).
[3] Griffiths problem 10.13 (page 434).
[4] Griffiths problem 10.25 (page 442).
[5] WiFi runs at 2.4 GHz, and some WiFi cards have small external antennas a few cm long.
(a) Consider an antenna working at this frequency that is 5.0 cm long, and approximate this
antenna as a perfect dipole (i.e. like that of figure 11.2) of the same length, emitting dipole
radiation in all directions. Find the value of the peak current flowing in the antenna during
transmission, if the average radiated power is 100 mW, a typical value for WiFi.
(b) Find the radiation resistance of this antenna, also using the perfect dipole approximation (and
see problem 11.3 in the text—the antenna appears as a resistor viewed from the electronics since
the power radiated is equivalent to ohmic power loss in a resistor R, for the same current).
[6] Griffiths problem 11.1 (page 449).