Three concentric spherical metallic shells , and of radius , and respectively have surface charge densities , and , respectively. The shells and are at the same potential. If the radii of and are 2 cm and 3 cm, respectively, the radius of shell

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Solution

Q.
Three concentric spherical metallic shells $X$ , $Y$ and $Z$ of radius $a$ , $b$ and $c$ respectively $(a < b < c)$ have surface charge densities $σ$ , $- σ$ and $σ$ , respectively. The shells $X$ and $Z$ are at the same potential. If the radii of $X$ and $Y$ are 2 cm and 3 cm, respectively, the radius of shell $Z$ is ________ cm. [2023]

Ans.
(5)

$q_{x} = σ 4 π a^{2}, q_{y} = - σ 4 π b^{2}, q_{z} = σ 4 π c^{2}$

$Potential at x = potential at z$

$V_{x} = V_{z}$

$\frac{q_{x}}{4 π ε_{0} a} + \frac{q_{y}}{4 π ε_{0} b} + \frac{q_{z}}{4 π ε_{0} c} = \frac{q_{x}}{4 π ε_{0} c} + \frac{q_{y}}{4 π ε_{0} c} + \frac{q_{z}}{4 π ε_{0} c}$

$\frac{σ 4 π a^{2}}{a} - \frac{σ 4 π b^{2}}{b} + \frac{σ 4 π c^{2}}{c} = \frac{4 π σ (a^{2} - b^{2} + c^{2})}{c}$

$c (a - b + c) = a^{2} - b^{2} + c^{2}$

$c (a - b) = a^{2} - b^{2}$

$\Rightarrow c = a + b \Rightarrow c = 5 cm$

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