2049 Spring 2005
Quiz 3 Solution
Name:
Period:
Question 1
The figure below shows three charged particles located on a horizontal axis.
d
d
P
−q
+q
−q
r
(a) What is the potential at point P ? (1 point)
V (r) =
X kqi
i
ri
=
kq
kq
kq
+
−
.
r
r−d r+d
(1)
(b) Approximate your answer in part (a) for the situation r À d. Use the expansion (1 + x)p ≈ 1 + px for
x ¿ 1 and thus write the potential at point P as the sum of two terms (1 point).
µ
¶
µ
¶
kq
1
1
kq
kq
2d
V (r) =
1+
−
≈
(1 + 1 + d/r − 1 + d/r) =
1+
(2)
r
1 − d/r 1 + d/r
r
r
r
(c) In one sentence state the physical meaning of each term from part (b) (1 point).
The first term corresponds to the potential of a monopole at the origin and the second term corresponds to
the potential of a dipole of charge separation 2d.
Question 2
Two different capacitors are to be considered: a parallel plate capacitor of width a, length b, and separation d; and a spherical capacitor of inner radius a, outer radius b, and the concentric shell separation is
b − a = d.
(a) If each capacitor has the identical potential between its plates, which, if any, has the greater charge (1
point)?
The parallel and spherical plate capacitances are,
A²0
ab²0
=
,
d
d
4πab²0 4πab²0
=
,
=
b−a
d
Cpp =
(3)
Csp
(4)
respectively. Comparing the two capacitances shows that the spherical plate capacitance is greater by a factor
of 4π, and since C = Q/V , the charge on the spherical plate capacitor must be greater if the potentials are
the same.
(b) What is the ratio of the spherical plate capacitance to the parallel plate capacitance (1 point)?
Csp
= 4π .
Cpp
(5)
(c) From the spherical plate capacitance, determine the capacitance of an isolated sphere of radius R (Hint:
take b → ∞) (1 point).
4πab²0
lim
= 4π²0 R .
(6)
b→∞ b − a
a→R