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JEE MAIN - Physics (2018 (Offline) - No. 9)

Three concentric metal shells A, B and C of respective radii a, b and c (a < b < c) have surface charge densities $$ + \sigma $$, $$ - \sigma $$ and $$ + \sigma $$ respectively. The potential of shell B is :
$${\sigma \over { \in {}_0}}\left[ {{{{b^2} - {c^2}} \over c} + a} \right]$$
$${\sigma \over { \in {}_0}}\left[ {{{{a^2} - {b^2}} \over a} + c} \right]$$
$${\sigma \over { \in {}_0}}\left[ {{{{a^2} - {b^2}} \over b} + c} \right]$$
$${\sigma \over { \in {}_0}}\left[ {{{{b^2} - {c^2}} \over b} + a} \right]$$

Explanation

JEE Main 2018 (Offline) Physics - Electrostatics Question 198 English Explanation

Let charge of shell A, B and C are QA, QB and QC respectively.

Potential of B shell will be due to charge QA, QB and QC.

Here charge QA is inside of the shell B and QB is on the surface of the shell B in both cases you have to take the radius of the shell B, while calculating potential of shell B.

Charge QC is outside of the shell B so take radius of shell C for calculating potential of shell B.

$$\therefore$$ VB = V$$_{{Q_a}}$$ + V$$_{{Q_b}}$$ + V$$_{{Q_c}}$$

= $${1 \over {4\pi {\varepsilon _0}}}$$ $$\left[ {{{4\pi {a^2}\left( { + \sigma } \right)} \over b} + {{4\pi {b^2}\left( { - \sigma } \right)} \over b} + {{4\pi {c^2}\left( { + \sigma } \right)} \over c}} \right]$$

= $${1 \over {4\pi {\varepsilon _0}}}$$ $$ \times $$ 4$$\pi $$$$\sigma $$ $$\left[ {{{{a^2}} \over b} - {{{b^2}} \over b} + c} \right]$$

= $${\sigma \over {{\varepsilon _0}}}\left[ {{{{a^2} - {b^2}} \over b} + c} \right]$$

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