Physics tutorial #15
1. Show that the equivalent capacitance Ceq of a combination of capacitors C1 and C2 satisfies relation
-1 -1 -1
(a) Ceq = C1 + C2 when they are connected in series,
(b) Ceq = C1 + C2 when they are connected in parallel.
2. The capacitor C1 in figure below initially has a charge Q0, and the voltage between its plates is V0.
After the switch is closed, the capacitor C2 becomes charged. Find the ratio V/V0 where V is the
final voltage across the capacitors, and evaluate the ratio U/U0 of the final to the initial stored energy.
Q0
V0
C1 C2
3. Two concentric spherical conducting shells are separated by vacuum. The inner shell has total charge
+Q and outer radius ra = 9.5 cm, and the outer shell has charge -Q and inner radius ra = 10.5 cm.
(a) Use the Gauss s law to find the capacitance of this spherical capacitor.
(b) Find the electric potential energy using the capacitance from (a).
(c) Find the electric potential energy by integrating the electric-field energy density.
4. Is it possible to find two capacitors which connected in parallel give the same equivalent capacity as
when connected in series? Prove your answer.
5. An air capacitor is made by using two flat plates, each with area A, separated by a distance d. Then
a metal slab having thickness a (less than d) and the same shape and size as the plates is inserted
between them, parallel to the plates and not touching either plate. (a) What is the capacitance of
this arrangement? (b) Express the capacitance as a multiple of the capacitance C0 when the metal
slab is not present. (c) Discuss what happens to the capacitance in the limits a 0 and a d.
6. A parallel-plate capacitor has the space between the plates filled with two slabs of dielectric, one with
constant K1 and one with constant K2. Each slab has thickness d/2, where d is the plate separation.
Show that the capacitance is
2 A K1K2
0
C =
d K1 + K2
7. A parallel-plate capacitor has the space between the plates filled with two slabs of dielectric, one
with constant K1 and one with constant K2. The thickness of each slab is the same as the plate
separation d, and each slab fills half of the volume between the plates. Show that the capacitance is
A
0
C = (K1 + K2)
2d
Nivas Babu Selvaraj, Maciej Woloszyn
http://fatcat.ftj.agh.edu.pl/~woloszyn/phys/