Physics 304
Due: Weds., Nov. 9
Problem Set 9
!
(1) (a) Consider a magnetic field in a region of space given by, B = ! zx̂ + " x 2 ŷ . Find the force
on a straight segment of wire extending from (a, 0, 0) to (a, 0, a) carrying a current I in the
positive z direction.
(b) Show that the B field defined above satisfies both Gauss’ law for magnetism and
Ampere’s law, and determine the current density required! for these to be satisfied.
!
(c) Which of the following fields cannot occur? (i) B = ! (z " y) x̂ ; (ii) B = ! (z ! y) ŷ .
(2) Griffiths problem 5.9 (page 219).
(3) Griffiths problem 5.22 (page 239).
(4) Griffiths problem 5.23 (page 239).
(5) The figure shows two slabs of conductive material, above and below the x-y plane. The x-axis
extends out of the page toward you. Both slabs extend essentially
! to infinity in the x and y
directions. The upper region caries a uniform current density, J = +J o x̂ , with Jo a positive
!
constant, while the lower region carries the current density, J = !J o x̂ .
(a) Find the magnetic field inside and outside of the conductors, with direction, by using
Ampere’s circuital law (the integral version).
(b) Show by solving the corresponding equation that your solution also satisfies the differential
version of Ampere’s law.
(c) Find a vector potential corresponding to this field: equation (5.63) in the text is helpful to
establish the direction, but to find the functional form for A you can solve the differential
equations (5.59) and (5.62) directly. Also note that by symmetry of equation (5.63), A will vanish
at positions on the x-y plane.
(6) Consider a long hollow conductor with inner radius So and outer radius 2So. A current is
carried along the length of the conductor, which you can consider to be infinitely long. The
conductivity decreases as the radius increases in such a way that J increases linearly with s, from
Jo at s = So to 2 Jo at s = 2So.
(a) Find the total current.
(b) Find the magnetic field in the hollow region ( s < So ), in the region So < s < 2So , and for
s > 2So .