Potentials of concentric conducting spherical shells
Q. There are three co-centric conducting spherical shells A, B and C of radii \(a, b\) and \(c\) respectively \((c > b > a)\) and they are charged with charge \(q_1, q_2\) and \(q_3\) respectively. The potentials of the spheres A, B and C respectively, are :

(A) \( \dfrac{1}{4\pi\varepsilon_0}\left(\dfrac{q_1}{a}+\dfrac{q_2}{b}+\dfrac{q_3}{c}\right), \dfrac{1}{4\pi\varepsilon_0}\left(\dfrac{q_1+q_2+q_3}{b}\right), \dfrac{1}{4\pi\varepsilon_0}\left(\dfrac{q_1+q_2+q_3}{c}\right) \)

(B) \( \dfrac{1}{4\pi\varepsilon_0}\left(\dfrac{q_1+q_2+q_3}{a}\right), \dfrac{1}{4\pi\varepsilon_0}\left(\dfrac{q_1+q_2+q_3}{b}\right), \dfrac{1}{4\pi\varepsilon_0}\left(\dfrac{q_1+q_2+q_3}{c}\right) \)

(C) \( \dfrac{1}{4\pi\varepsilon_0}\left(\dfrac{q_1+q_2+q_3}{a}\right), \dfrac{1}{4\pi\varepsilon_0}\left(\dfrac{q_1+q_2}{b}+\dfrac{q_3}{c}\right), \dfrac{1}{4\pi\varepsilon_0}\left(\dfrac{q_1+q_2+q_3}{c}\right) \)

(D) \( \dfrac{1}{4\pi\varepsilon_0}\left(\dfrac{q_1}{a}+\dfrac{q_2}{b}+\dfrac{q_3}{c}\right), \dfrac{1}{4\pi\varepsilon_0}\left(\dfrac{q_1+q_2}{b}+\dfrac{q_3}{c}\right), \dfrac{1}{4\pi\varepsilon_0}\left(\dfrac{q_1+q_2+q_3}{c}\right) \)

Correct Answer: \( \dfrac{1}{4\pi\varepsilon_0}\left(\dfrac{q_1}{a}+\dfrac{q_2}{b}+\dfrac{q_3}{c}\right), \dfrac{1}{4\pi\varepsilon_0}\left(\dfrac{q_1+q_2}{b}+\dfrac{q_3}{c}\right), \dfrac{1}{4\pi\varepsilon_0}\left(\dfrac{q_1+q_2+q_3}{c}\right) \)

Step-by-Step Explanation

Important concept: Potential at any conducting shell is the algebraic sum of potentials due to all charges present, evaluated at the radius of that shell.

Potential of shell A (radius a):

\[ V_A = \frac{1}{4\pi\varepsilon_0} \left(\frac{q_1}{a} + \frac{q_2}{b} + \frac{q_3}{c}\right) \]

Potential of shell B (radius b):

\[ V_B = \frac{1}{4\pi\varepsilon_0} \left(\frac{q_1+q_2}{b} + \frac{q_3}{c}\right) \]

Potential of shell C (radius c):

\[ V_C = \frac{1}{4\pi\varepsilon_0} \left(\frac{q_1+q_2+q_3}{c}\right) \]

Hence, option (D) is correct.

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