Problem
7.
Four identical charges
q
, initially widely separated, are brought to the vertices of a tetrahedron of side
a
(Fig. 2626). Find the electrostatic energy of this configuration.
FIGURE 2626 Problem 7.
Solution
There are six different pairs of equal charges and the separation of any pair is
a
. Thus,
W
=
∑
pairs
kq
i
q
j
=
a
=
6
2
=
a
.
(See Problem 1.)
Problem
8.
A charge
Q
0
is at the origin. A second charge,
Q
x
=
2
Q
0
,
is brought to the point
x
=
a
,
y
=
0.
Then
a third charge
Q
y
is brought to the point
x
=
0,
y
=
a
.
If it takes twice as much work to bring in
Q
y
as it did
Q
x
,
what is
Q
y
in terms of
Q
0
?
Solution
The work necessary to bring up
Q
x
is
W
x
=
kQ
0
Q
x
=
a
=
2
kQ
0
2
=
a
,
while the work necessary to subsequently
bring up
Q
y
is
W
y
=
0
Q
y
=
a
+
x
Q
y
=
a
=
0
Q
y
(1
+
)
=
a
.
If
W
y
=
2
W
x
,
then
Q
y
+
)
=
4
Q
0
, or
Q
y
=
4
Q
0
=
+
1)
=
1.66
Q
0
.
(Note:
1
=
+
=
−
1.)
Problem
10. Two square conducting plates measure 5.0 cm on a side. The plates are parallel, spaced 1.2 mm apart,
and initially uncharged. (a) How much work is required to transfer
7.2
μ
C
from one plate to the
other? (b) How much work is required to transfer a second
7.2
C?
Solution
The separation is much smaller than the linear dimensions of the plates, so the discussion in Section 262
applies. (a) From Equation 262,
W
=
Q
2
d
=
2
ε
0
A
=
(7.2
C)
2
(1.2 mm)
=
2(8.85
×
10
−
12
F/m)(5 cm)
2
=
1.41 J.
(b) The additional work
required to double the charge on each plate is
Δ
W
=
(
2
Q
)
2
d
=
2
0
A
−
W
=
3
W
=
4
.
22 J.
Problem
13. A conducting sphere of radius
a
is surrounded by a concentric spherical shell of radius
b
. Both are
initially uncharged. How much work does it take to transfer charge from one to the other until they
carry charges
±
Q
?
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When a charge
q
(assumed positive) is on the inner sphere, the potential difference between the spheres is
V
=
kq
(
a
−
1
−
b
−
1
).
(See the solution to Problem 2563(a).) To transfer an additional charge
dq
from the
outer sphere requires work
dW
=
V
dq
,
so the total work required to transfer charge
Q
(leaving the
spheres oppositely charged) is
W
=
∫
0
Q
V
=
∫
0
Q
(
a
−
1
−
b
−
1
)
=
1
2
kQ
2
(
a
−
1
−
b
−
1
).
(Incidentally,
this shows that the capacitance of this spherical capacitor is
1
=
k
(
a
−
1
−
b
−
1
)
=
ab
=
k
(
b
−
a
)
; see Equation 26
8a.)
Problem
15. Two conducting spheres of radius
a
are separated by a distance
l À
a
;
since the distance is large,
neither sphere affects the other’s electric field significantly, and the fields remain spherically
symmetric. (a) If the spheres carry equal but opposite charges
±
q
,
show that the potential difference
between them is
2
=
a
.
(b) Write an expression for the work
dW
involved in moving an infinitesimal
charge
dq
from the negative to the positive sphere. (c) Integrate your expression to find the work
involved in transferring a charge
Q
from one sphere to the other, assuming both are initially uncharged.
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 Winter '10
 Hirsch
 Energy, Electric charge, Energy density

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