Two concentric conducting spherical shells of radii a1 and a2 a2 - a1 are charged to potentials -1 and -2- respectively- What is the charge on the inner shell-

Name: JEE - Grade 12 - Physics - Electrostatics - Session16 - W09
Uploaded: 2022-07-04T10:36:42+05:30
Description: Let the charges on the inner and outer shells be q_1 and q_2 respectively. nbsp; Potential of inner spherical shell, nbsp; nbsp; nbsp; nbsp; n ...

Let the charges on the inner and outer shells be q_1 and q_2 respectively. nbsp; Potential of inner spherical shell, nbsp; nbsp; nbsp; nbsp; n ...

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Question

Two concentric conducting spherical shells of radii $a_{1}$ and $a_{2} (a_{2} > a_{1})$ are charged to potentials $ϕ_{1}$ and $ϕ_{2}$ , respectively. What is the charge on the inner shell?

4 π ϵ_{0} (\frac{ϕ_{1} - ϕ_{2}}{a_{2} - a_{1}}) a_{1} a_{2}

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4 π ϵ_{0} (\frac{ϕ_{1} + ϕ_{2}}{a_{2} + a_{1}}) a_{1} a_{2}

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4 π ϵ_{0} (\frac{ϕ_{1} - ϕ_{2}}{a_{2} + a_{1}}) a_{1} a_{2}

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2 π ϵ_{0} (\frac{ϕ_{1} - ϕ_{2}}{a_{2} + a_{1}}) a_{1} a_{2}

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Solution

The correct option is A $4 π ϵ_{0} (\frac{ϕ_{1} - ϕ_{2}}{a_{2} - a_{1}}) a_{1} a_{2}$
Let the charges on the inner and outer shells be $q_{1}$ and $q_{2}$ respectively.

Potential of inner spherical shell,
$ϕ_{1} = \frac{k q_{1}}{a_{1}} + \frac{k q_{2}}{a_{2}} . . . (1)$

Potential of outer spherical shell,
$ϕ_{2} = \frac{k q_{1}}{a_{2}} + \frac{k q_{2}}{a_{2}} . . . (2)$

From $(1)$ and $(2)$ , we get
$ϕ_{1} - ϕ_{2} = (\frac{k q_{1}}{a_{1}} + \frac{k q_{2}}{a_{2}}) - (\frac{k q_{1}}{a_{2}} + \frac{k q_{2}}{a_{2}})$

$\Rightarrow ϕ_{1} - ϕ_{2} = (\frac{k q_{1}}{a_{1}} - \frac{k q_{1}}{a_{2}})$

$\Rightarrow q_{1} = \frac{1}{k} [\frac{ϕ_{1} - ϕ_{2}}{a_{2} - a_{1}}] a_{1} a_{2}$

Substituting, $k = \frac{1}{4 π ϵ_{0}}$ , we get

$q_{1} = 4 π ϵ_{0} [\frac{ϕ_{1} - ϕ_{2}}{a_{2} - a_{1}}] a_{1} a_{2}$

Hence, option (a) is the correct answer.

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