500
11 Electrostatics
By problem V 2 = V 1
∴
a +
√
a 2 + x 2
x
2
= 3 → x =
√
3 a
V 1 = V 2 = λ × 9 × 10
9
× ln 3
= 9.89 × 10
9 V
Fig. 11.36
11.37 U = U Qq + U Qq + U =
1
4πε 0
Qq
a
+
Qq
a
+
q 2
√
2a
= 0 by problem.
Therefore Q = −
q
2
√
2
·
11.38 Work done W = U 12 + U 23 + U 34 + U 41 + U 13 + U 24 where charges l and
3 are positive and 2 and 4 negative assuming that the potential energy is zero
for infinite separation of charges.
W =
q 2
4πε 0
−
1
a
−
1
a
−
1
a
−
1
a
+
1
√
2a
+
1
√
2a
=
−q 2
4πε 0 a
4 −
√
2
11.39 (a) The charge density is given by ρ =
3q
4π R 3 . Consider a shell of radius
r and thickness dr concentric with the sphere. The volume of the shell
is 4πr 2 dr and the charge in it will be dq = 4πr 2 drρ, Fig. 11.37. The
charge of the sphere of radius r is dq =
4
3
πr 3 ρ and may be considered
11 Electrostatics
By problem V 2 = V 1
∴
a +
√
a 2 + x 2
x
2
= 3 → x =
√
3 a
V 1 = V 2 = λ × 9 × 10
9
× ln 3
= 9.89 × 10
9 V
Fig. 11.36
11.37 U = U Qq + U Qq + U =
1
4πε 0
a
+
a
+
q 2
√
2a
= 0 by problem.
Therefore Q = −
q
2
√
2
·
11.38 Work done W = U 12 + U 23 + U 34 + U 41 + U 13 + U 24 where charges l and
3 are positive and 2 and 4 negative assuming that the potential energy is zero
for infinite separation of charges.
W =
q 2
4πε 0
−
1
a
−
1
a
−
1
a
−
1
a
+
1
√
2a
+
1
√
2a
=
−q 2
4πε 0 a
4 −
√
2
11.39 (a) The charge density is given by ρ =
3q
4π R 3 . Consider a shell of radius
r and thickness dr concentric with the sphere. The volume of the shell
is 4πr 2 dr and the charge in it will be dq = 4πr 2 drρ, Fig. 11.37. The
charge of the sphere of radius r is dq =
4
3
πr 3 ρ and may be considered