Question 764: Electrostatics & Capacitance

Question

Three concentric spherical shells have radii a, b and c (a < b < c) anf have surface charge densities $$\sigma $$, $$-$$$$\sigma $$ and $$\sigma $$ respectively. If V
A
, V
B
and V
C
denote the potentials of the three shells, then, for c = a + b, we have

Accepted Answer

$${V_A} = {1 \over {4\pi {\varepsilon _0}}}\left\{ {{{qA} \over a} + {{qB} \over b} + {{qC} \over c}} ight\}$$
$$ = {{4\pi } \over {4\pi {\varepsilon _0}}}\left\{ {{{{a^2}\sigma } \over a} - {{{b^2}\sigma } \over b} + {{{c^2}\sigma } \over c}} ight\}$$
$${V_A} = {1 \over {{\varepsilon _0}}}\left\{ {{{{a^2}\sigma } \over a} - {{{b^2}\sigma } \over b} + {{{c^2}\sigma } \over c}} ight\}$$
$${V_B} = {1 \over {{\varepsilon _0}}}\left\{ {{{{a^2}\sigma } \over a} - {{{b^2}\sigma } \over b} + {{{c^2}\sigma } \over c}} ight\}$$
$${V_C} = {1 \over {{\varepsilon _0}}}\left\{ {{{{a^2}\sigma } \over a} - {{{b^2}\sigma } \over b} + {{{c^2}\sigma } \over c}} ight\}$$
Given c = a + b.
If a = a, b = 2a and c = 3a for example, as c > b > a,
$${V_A} = {1 \over {{\varepsilon _0}}}\left\{ {{{{a^2}\sigma } \over a} - {{4{b^2}\sigma } \over {2a}} + {{{c^2}\sigma } \over c}} ight\}$$
$${V_B} = {1 \over {{\varepsilon _0}}}\left\{ {{{{a^2}\sigma } \over {2a}} - {{4{a^2}\sigma } \over {2a}} + {{{c^2}\sigma } \over c}} ight\}$$
$${V_C} = {1 \over {{\varepsilon _0}}}\left\{ {{{{a^2}\sigma } \over {3a}} - {{4{a^2}\sigma } \over {3a}} + {{{c^2}\sigma } \over c}} ight\}$$
It can seen by taking out common factors that
V
A
= V
C
> V
B
i.e.,  V
A
= V
C
$$ 
e $$ V
B

Answer

V
C
= V
B
$$ 
e $$ V
A

Answer

V
C
$$ 
e $$ V
B
$$ 
e $$ V
A

Answer

V
C
= V
B
= V
A

Answer

V
C
= V
A
$$ 
e $$ V
B

Electrostatics & Capacitance

Three concentric spherical shells have radii a, b and c (a < b < c) anf have surface charge densities $\sigma , -\sigma and \sigma $ respectively. If V A , V B and V C denote the potentials of the three shells, then, for c = a + b, we have

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