25
4 · · Four point charges of magnitude 2
µ
C are at the corners of a square of side 4 m. Find the electrostatic potential energy if (a) all the charges are negative, (b) three of the charges are positive and one is negative, and (c) two are positive and two are negative.
1. Determine the distances between charges
(a), (b), (c) Use Equ. 25-1 r
1,2
= r
1,3
= r
2,4
= r
3,4
= 4 m; r
1,4
= r
2,3
= 5.657 m
(a) U = 48.7 mJ; (b) U = 0; (c) U = –12.7 mJ or
–23.2 mJ, depending on the configuration
5* · · Four charges are at the corners of a square centered at the origin as follows: q at (–a, +a);
2q at (a, a); –3q at (a, –a); and 6q at (–a, –a). A fifth charge +q is placed at the origin and released from rest. Find its speed when it is a great distance from the origin.
V(0) = (kq/a 2 )(1 + 2 – 3 + 6) = 6kq/a 2
2. mv
2
= qV(0)
v = q 6 2 k/m a
36 · · Five identical capacitors of capacitance C
0
are connected in a bridge network as shown in
Figure 25-26. (a) What is the equivalent capacitance between points a and b? (b) Find the equivalent capacitance if the capacitance between a and b is changed to 10C
0
.
(a) Here we have three parallel paths between a and b. The equivalent capacitances of the top and bottom paths are each C
0
/2. Thus C equ
= 2C
0
.
(b) If the central capacitance is 10C
41* · An electric field of 2
×
10
4
0
, then C equ
= 11C
0
.
V/m exists between the plates of a circular parallel–plate capacitor that has a plate separation of 2 mm. (a) What is the voltage across the capacitor? (b)
What plate radius is required if the stored charge is 10
µ
C?
(a) V = Ed V = 2
×
10
4 ×
2
×
10
–3
V = 40 V
(b) C = Q/V =
ε
0
(
π
R
2
)/d; R = (Qd/V
πε
0
) 1/2 R = 4.24 m
46 · · A cylindrical capacitor consists of a long wire of radius R
1
and length L with a charge +Q and a concentric outer cylindrical shell of radius R
2
, length L, and charge -Q. (a) Find the electric field and energy density at any point in space. (b) How much energy resides in a cylindrical shell between the conductors of radius r, thickness dr, and volume 2
π
rL dr ? (c)
Integrate your expression from part (b) to find the total energy stored in the capacitor, and compare your result with that obtained using U = 1
2
C V
2
.
(a) Between the wire and shell, E = 2k
λ
/r, where
λ
= Q/L. u = 1/2
ε
0
E
2
= 2k
2 ε
0
Q
2
/r
2
L
2
.
(b) dU = 2
π
rL u(r) dr = (kQ
2
/rL)dr.
(c) U
= kQ
L
2
R
2
R
1
dr r
= kQ
L
2
1 n


R
R
2
1

 . Using Equ. 25-11 and U = 1/2 Q
2
/C, one obtains the same result.

49* · · A spherical capacitor has an inner sphere of radius R
1
with a charge of +Q and an outer concentric spherical shell of radius R
2
with a charge of –Q. (a) Find the electric field and the energy density at any point in space. (b) Calculate the energy in the electrostatic field in a spherical shell of radius r, thickness dr, and volume 4
π r
2
dr between the conductors? (c)
Integrate your expression from part (b) to find the total energy stored in the capacitor, and compare your result with that obtained using U = 1
2
QV.
(a) From Gauss’s law, E = kQ/r
2
and u = 1/2
ε
0
E
2
= 1/2 k
2 ε
0
Q
2
/r
4
.
(b) dU = 4
π r
2
u(r)dr = (kQ
2
/2r
2
)dr.
(c) U
= kQ
2
2
R
2
R
1 dr r
2
= kQ
2
2
(
R
R
1
2
R
−
2
R
1
)
=
1
2
Q
2



R
2
4
πε
0
−
R
1
R
1
R
2



. Note that the quantity within the parentheses is 1/C
(see Problem 48), so U = 1/2 Q
2
/C.
70 · · Find the capacitance of the parallel-plate capacitor shown in Figure 25-29.
This can be considered a combination of three capacitors as follows. C
1
and C
2
in series, with that series combination in parallel with C
3
. Let the series combination be C s
. Following the procedure of Problem 65, one obtains C s
= (
ε
0
A/d)[
κ
1
κ
2
/(
κ
1
+
κ
2
)]. C
3
=
κ
3
ε
0
A/2d. Thus, C
= C s
+ C
3
= (
ε
0
A/2d)[
κ
3
+ 2
κ
1
κ
2
/(
κ
1
+
κ
2
)].
78 · · Determine the capacitance of each of the networks shown in Figure 25-30.
(a) Proceed as in Problem 76 C equ
= 2C
0
2
/3C
0
= 2C
0
/3
(b) The three C
0
are in parallel
(c) Two (2C
0
) in series combination
C equ
= 3C
0
C equ
= C
0