Physics 305
Problem set 2
Due: Friday, Feb. 3
1) In a region of space between x = +d and x = –d , forming a flat slab of thickness 2d parallel to
!
" 3x 3 ! xd 2 %
the y-z plane, the electric field varies according to E = Eo $
'& ŷ . Eo is a constant.
d3
#
(a) Find the magnetic
field in this region, using the differential form of Faraday’s law, and
!
assuming that B = 0 everywhere at time t = 0.
(b) Find the current density in this region, using the differential form of Ampere’s law.
(c)
! Assume that the current density calculated in (b) is confined to the region x = ±d, so that
J = 0 outside. (Also assume there is no sheet of current at the boundary.) Show that the magnetic
field will be zero outside the slab (x > +d and x < –d) using Ampere’s circuital law.
(d) Although the magnetic field is zero outside, the electric field cannot be zero. Show that the
electric field must be uniform and constant vs. time, both above and below the slab, by using the
circuital Faraday’s law. To do this, use a rectangular loop in the x-y plane extending above and
below the slab, calculating the flux starting with the magnetic field obtained in part (a).
2) Griffiths problem 7.29 (page 320).
3) Consider a long winding composed of two single-layer solenoids, wound concentrically along
the same axis. The spool that supports them is nonmagnetic, with relative permeability 1, so it
does not affect the resulting field. The outer solenoid has radius b, and the inner has radius a.
Both have the same length , and the same number of turns per length equal to n. Treat each in
the long solenoid limit, neglecting end effects.
(a) Find the self-inductance of each solenoid, and find the mutual inductance between the two
coils.
(b) Now suppose that the two coils are connected in series and fed current I in such a way that,
looking one direction along the length of the winding, current in the outer coil runs clockwise in
each turn, while it runs counterclockwise in the inner coil. Find the magnetic field in the region
inside the inner coil, and in the region between the coils. (Use superposition.)
(c) Find the self-inductance of the entire winding, by adding up the flux in each turn and dividing
by I. Show that the inductance obeys the (generally applicable) relation, L = L1 + L2 ± 2M.
4) A spherical coil can be made to create a uniform B field inside if the windings circulate in the
!ˆ direction with turn spacing proportional to sin ! . Consider such a winding, radius R, with turn
spacing given by n = no sin ! . Assume that the wires are very thin and no is large, such that a
current I will approximate a sheet of surface current of magnitude K = no I sin ! . This current will
lead to a uniform field inside along z, and a pure dipole field outside with the same r and !
dependence as in 5.86.
(a) Given that these are the forms of the solutions, assume that the uniform field inside has
magnitude Bo, and write this field as a vector in terms of r and ! . Also write down a dipole field
outside, giving it an unknown magnitude B1 just outside the equator of the sphere. The vector
boundary condition given in (5.74) can be used to solve for the two unknown magnitudes, if the
current is given. Do this and find the internal field in terms of no I .
(b) Consider a circular turn at angle ! , and find the magnetic flux through this turn.
(c) Find the self-inductance of the winding by combining to find the total flux through all turns.
5) Griffiths problem 7.50 (page 337).
6) Griffiths problem 7.54 (page 338).