Question

# Three concentric metal shells A, B and C of respectively radii a, b and c $$(a < b < c)$$ have surface charge densities $$+\sigma$$, $$-\sigma$$ and $$+\sigma$$ respectively. The potential of shell B is?

A
σϵ0[b2c2b+a]
B
σϵ0[b2c2c+a]
C
σϵ0[a2b2a+c]
D
σϵ0[a2b2b+c]

Solution

## The correct option is D $$\displaystyle\frac{\sigma}{\epsilon_0}\left[\displaystyle\frac{a^2-b^2}{b}+c\right]$$$$V_{outer} = \dfrac{KQ}{r}$$where r is distance of point from the centre of shell$$V_{inside} = \dfrac{KQ}{R}$$where R is the radius of shell$$V_B = \dfrac{Kq_A}{r_b} - \dfrac{Kq_B}{r_b} + \dfrac{Kq_C}{r_c}$$$$V_B =\dfrac{1}{4 \pi \epsilon_o} [\dfrac{\sigma 4 \pi a^2}{b} - \dfrac{\sigma 4 \pi b^2}{b} + \dfrac{\sigma 4 \pi c^2}{c}]$$$$V_B = \dfrac{\sigma}{\epsilon_o} [ \dfrac{a^2 - b^2}{b} + c]$$Physics

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