Physics 304
Problem Set 3
Due: Monday, Sept. 19
(1) The “self capacitance” is the capacitance of a single conductor positioned very far from other
conductors or charges. The second plate can be considered to be the “sphere” at infinity, and ∆V
for the capacitor is the voltage difference between the conductor and infinity. The self
capacitance is a useful concept in estimating the stray capacitance on wires and conductors.
(a) Find the self capacitance of a sphere of radius R. Also evaluate the value specifically for the
case that R = 10 cm.
(b) The charge per unit length for an infinite wire of radius a is well-defined, however for this
case the self capacitance cannot be defined. Explain why.
(2) Griffiths problem 2.36 (page 101)
(3) A sphere, radius R, is constructed from thin foil, and given total charge Q. It is located far
from any other charges. Find the surface tension (force per length) in the foil. Hint: imagine that
the sphere is cut around its equator, and find the force on the top and bottom parts.
(4) Consider a wide, flat conducting plate (essentially infinite in extent). Due to a suitable
arrangement of external charges, just above the surface of the conductor the electric field has the
qd
form, E! =
, with d a constant. (The parameter s is the cylindrical radius.)
2"# o (d 2 + s 2 )3/2
(a) Find the total charge on the surface of the conductor.
(b) Find the force on the conductor.
(c) Show that this field is equivalent to the field at the mid-plane of a dipole, with charges ±q
located a distance ±d above/below the surface.
(d) Compare the force calculated in (b) to the force on one of the charges of the equivalent
dipole.
(5) Find the capacitance of a long “triaxial” cable, with an inner wire of radius a, surrounded by
a thin foil conductor of radius b, which is in turn surrounded by an outer conducting shield with
inner radius c. Suppose that the middle foil occupies negligible radius, and that the space
between conductors is vacuum.
(a) Find the capacitance per length if the outer and inner conductors are grounded, and the
middle conductor is connected to a voltage source referenced to ground.
(b) Find the capacitance per length if the inner and outer conductors are connected to a voltage
source, and the middle conductor is left “floating” (unconnected).
(6) Two air-gap capacitors are initially unconnected, and have equal charges Q, arranged so that
the positive charges are on the right and the negative charges are on the left.
(a) After the wires are connected top to bottom as in the picture, find the final
voltage across each capacitor, and find the change in stored energy. Show that
the energy will always decrease, unless C1 = C2. [Note that once connected,
the charges will rapidly oscillate; in the end the energy is lost through Ohmic
losses in the wires and/or electromagnetic wave generation.]
(b) After the process described in (a) is completed, the plates of C2 are pulled
far apart so that the capacitance goes essentially to zero. Find the work
required to pull the plates apart in this way.
(7) Griffiths problem 2.52 (page 109).