Three concentric spherical shells have radii a, b and c (a < b < c) and have surface charge densities σ, −σ and σ respectively. If VA, VB and VC denotes the potentials of the three shells, then for c = a + b, we have ?

Three concentric spherical shells have radii \( a \), \( b \), and \( c \) such that \( a < b < c \), and 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 ?

(a) \( V_C = V_B \ne V_A \)

(b) \( V_C \ne V_B \ne V_A \)

(c) \( V_C = V_B = V_A \)

(d) \( V_C = V_A \ne V_B \)

Answer :  (d) \( V_C = V_A \ne V_B \)

\( V_A = k \frac{q_A}{r_A} + k \frac{q_B}{r_B} + k \frac{q_C}{r_C} \)

\( V_B = k \frac{q_A}{r_B} + k \frac{q_B}{r_B} + k \frac{q_C}{r_C} \)

\( V_C = k \frac{q_A}{r_C} + k \frac{q_B}{r_C} + k \frac{q_C}{r_C} \)

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\( V_A = k \frac{q_A}{r_A} - k \frac{q_B}{r_B} + k \frac{q_C}{r_C} \)

\( = k \sigma 4\pi a^2 - k \sigma 4\pi b^2 + k \sigma 4\pi c^2 \)

\( = k \sigma 4\pi (a^2 - b^2 + c^2) \)

\( \text{Given } c = a + b \Rightarrow c^2 = (a + b)^2 = a^2 + 2ab + b^2 \)

\( V_A = k \sigma 4\pi (a^2 - b^2 + a^2 + 2ab + b^2) = k \sigma 4\pi (2a^2 + 2ab) \)

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\( V_B = k \frac{q_A}{b} - k \frac{q_B}{b} + k \frac{q_C}{c} \)

\( = k \sigma 4\pi a^2 - k \sigma 4\pi b^2 + k \sigma 4\pi c^2 \)

\( = k \sigma 4\pi \left( \frac{a^2 - b^2}{b} + \frac{c^2}{c} \right) \)

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\( V_C = k \frac{q_A}{c} - k \frac{q_B}{c} + k \frac{q_C}{c} \)

\( = k \sigma 4\pi a^2 - k \sigma 4\pi b^2 + k \sigma 4\pi c^2 \)

\( = k \sigma 4\pi \left( \frac{a^2 - b^2 + c^2}{c} \right) \)

\( = V_A \)