Question
Three long concentric conducting cylindrical shells have radii $R, 2R$ and $2\sqrt 2 $ $ R$ . Inner and outer shells are connected to each other. The capacitance across middle and inner shells per unit length is:
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Answer
The equivalent structure is shown in figure. Here the capacitance between inner and middle cylinder is equal to equivalent capacitance when $C_{1}$ and $C_{2}$ are in parallel.
$C_{1}=\frac{2 \pi \epsilon_{0}}{\ln \left(\frac{2 R}{R}\right)}=\frac{2 \pi \epsilon_{0}}{\ln 2}$
$C_{2}=\frac{2 \pi \epsilon_{0}}{\ln \left(\frac{2 \sqrt{2 R}}{2 R}\right)}=\frac{4 \pi \epsilon_{0}}{\ln 2}$
$C_{e q}=C_{1}+C_{2}=\frac{2 \pi \epsilon_{0}}{l n 2}+\frac{4 \pi \epsilon_{0}}{l n 2}=\frac{6 \pi \epsilon_{0}}{l n 2}$
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