A long conducting cylinder of length $l$ and radius $a (a < < l)$ carrying a total charge $q$ is surrounded by a conducting cylindrical shell of length $l$ and radius $2 a$ having a total charge $- 2 q$ . Assume electric potential at $r = 2 a$ to be zero, mark the CORRECT option(s) : $[given K = \frac{1}{4 π \in_{0}}]$

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The electric field at $r = \frac{3 a}{2} is \frac{4 Kq}{3 al}$

The electric potential at $r = \frac{3 a}{2} is \frac{2 Kq}{l} \ln (\frac{4}{3})$

The electric potential at $r = \frac{5 a}{2} is - \frac{2 Kq}{l} \ln (\frac{5}{4})$

The electric field at $r = \frac{5 a}{2} is - \frac{4 Kq}{5 al}$

answer is A, B, C.

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Detailed Solution

E=0 for r < a
$= \frac{2 k (\frac{q}{l})}{r} for a < r < 2 a = \frac{- 2 k (\frac{q}{l})}{r} for r > 2 a$
$r = \frac{3 a}{2}, E = \frac{2 k (\frac{q}{l})}{\frac{3 a}{2}} = \frac{4 ka}{3 al}$
For, $r = \frac{5 a}{2}, E = \frac{2 k (\frac{q}{l})}{\frac{5 a}{2}} = \frac{- 4 kq}{5 al}$
For, $(\frac{3 a}{2}), v_{1} - 0 = - \int_{2 a}^{\frac{3 a}{2}} \frac{2 k (\frac{q}{l})}{r} dr$
$V_{1} = - 2 k (\frac{q}{l}) \int_{2 a}^{\frac{3 a}{2}} \frac{dr}{r} = \frac{- 2 k (q)}{l} \ln (\frac{4}{3})$
For, $(\frac{5 a}{2}), V_{2} - 0 = 2 k (\frac{q}{l}) \ln (\frac{5}{4})$