Three concentric spherical metallic shells A, | vedclass

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(D) The charges on the shells are $q_A = \sigma(4\pi a^2)$,$q_B = -\sigma(4\pi b^2)$,and $q_C = \sigma(4\pi c^2)$.The potential at the surface of shel

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$24$

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(D) The charges on the shells are $q_A = \sigma(4\pi a^2)$,$q_B = -\sigma(4\pi b^2)$,and $q_C = \sigma(4\pi c^2)$.The potential at the surface of shell $A$ is $V_A = \frac{1}{4\pi\epsilon_0} [\frac{q_A}{a} + \frac{q_B}{b} + \frac{q_C}{c}] = \frac{\sigma}{\epsilon_0} [a - b + c]$.The potential at the surface of shell $C$ is $V_C = \frac{1}{4\pi\epsilon_0} [\frac{q_A}{c} + \frac{q_B}{c} + \frac{q_C}{c}] = \frac{\sigma}{\epsilon_0 c} [a^2 - b^2 + c^2]$.Given $V_A = V_C$,we equate the expressions: $a - b + c = \frac{a^2 - b^2 + c^2}{c}$.$c(a - b + c) = a^2 - b^2 + c^2 \implies ac - bc + c^2 = a^2 - b^2 + c^2$.$ac - bc = a^2 - b^2 \implies c(a - b) = (a - b)(a + b)$.Since $a 
eq b$,we divide by $(a - b)$ to get $c = a + b$.Given $a = 7 \ cm$ and $b = 17 \ cm$,$c = 7 + 17 = 24 \ cm$.

Three concentric spherical metallic shells $A$,$B$,and $C$ of radii $a=7 \ cm$,$b=17 \ cm$,and $c$ $(a < b < c)$ have surface charge densities $\sigma, -\sigma$,and $\sigma$ respectively. If $A$ and $C$ are at the same potential,then the value of $c$ is: (in $cm$)

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