Problem Set 8

Physics 304
Due: Wednesday, Nov. 2
Problem Set 8
(1) A conducting sphere, radius R, floats exactly half-submerged in a liquid dielectric with
relative permittivity !1 . Another less-dense fluid has relative permittivity ! 2 and floats on top so
that it surrounds the other half of the sphere. The fluids extend essentially to infinity.
(a) With total charge Q on the conductor, the electric field turns out to have the form, A / r 2 rˆ ,
with the same multiplier A in both dielectrics. Solve for A and show that the result satisfies the
expected boundary conditions along the flat surface between the two dielectrics.
(b) Find the bound charge density at radius R in each dielectric, and on the flat surface
separating the fluids.
(c) Find the energy required to charge this conducting sphere to the given charge Q.
(
)
(2) Griffiths problem 5.2 (page 207)
(3) (a) A cylindrical wire, length 2l and radius Ro, is positioned on the z axis between z = l and z
= –l. Initially there is a volume charge density ! = Az , where A is constant. At t = 0, the charge
"T !t%
z.
density begins to decrease linearly in time to zero at time t = T, in other words, ! = A $
# T '&
Notice that the total charge is zero at all times; the change occurs only through currents in the
wire. Find the current density in the wire (with direction) vs. z and time during this process, by
using the continuity equation and then integrating. Note that no current can be flowing into the
ends of the wire; there is a constant of integration.
(b) Similarly, solve for the case that ! = +Ae !" t for z > l and ! = "Ae"# t for z < l.
(c) There is an equation in problem 5.7 in the text, for the time-dependence of the dipole
moment. For the specific situation described part (a) show that this equation holds, using the
current density you obtained. (This can be important for calculating dipole radiation, a topic for
later chapters.)
(4) Suppose that a resistive sphere, radius R, is initially charged so that it has a uniform charge
density, !o , inside. Since the sphere can conduct current, this is a non-equilibrium situation and
transient currents will flow responding
!
! to internal electric fields. In this case the current density
is proportional to the field: J = ! ohm E [this is Ohm’s law; ! ohm is a constant which denotes the
conductivity].
(a) Find the electric field inside the sphere by standard means.
(b) Using !the continuity equation, find a differential equation for the charge density vs. time.
Solve for J and ! vs. time and show that ! remains spatially uniform but decreases with time.
(c) Show that at the same time a surface charge density ( ! ) must appear on the outer surface of
the sphere; you can use the integral form of the continuity equation to do so. Solve for ! vs.
time.
(d) Demonstrate that as t ! " the total charge appearing on the outside of the sphere equals the
charge that disappeared from the inside of the conductor.
(5) Consider a very long wire, radius R, which carries a uniform charge density ! on its outer
surface.
(a) If the wire is spinning about its long axis with constant angular velocity ! , find the
corresponding surface current density.
(b) Find the magnetic field at points along the axis of the wire by using the Biot-Savart law.
(6) Griffiths problem 5.39, page 247.