Three long concentric conducting cylindrical | vedclass

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(B) Let the radii of the shells be $r_1 = R$,$r_2 = 2R$,and $r_3 = 2\sqrt{2}R$. The inner shell $(r_1)$ and outer shell $(r_3)$ are connected,so they

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$\frac{6 \pi \epsilon_0}{\ln 2}$

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(B) Let the radii of the shells be $r_1 = R$,$r_2 = 2R$,and $r_3 = 2\sqrt{2}R$. The inner shell $(r_1)$ and outer shell $(r_3)$ are connected,so they are at the same potential.
The capacitance per unit length of a cylindrical capacitor is given by $C = \frac{2 \pi \epsilon_0}{\ln(r_{outer}/r_{inner})}$.
There are two capacitors formed:
$C_1$ between the inner shell $(R)$ and middle shell $(2R)$: $C_1 = \frac{2 \pi \epsilon_0}{\ln(2R/R)} = \frac{2 \pi \epsilon_0}{\ln 2}$.
$C_2$ between the middle shell $(2R)$ and outer shell $(2\sqrt{2}R)$: $C_2 = \frac{2 \pi \epsilon_0}{\ln(2\sqrt{2}R/2R)} = \frac{2 \pi \epsilon_0}{\ln(\sqrt{2})} = \frac{2 \pi \epsilon_0}{\frac{1}{2} \ln 2} = \frac{4 \pi \epsilon_0}{\ln 2}$.
Since the inner and outer shells are connected,these two capacitors are connected in parallel with respect to the middle shell.
Therefore,the equivalent capacitance is $C_{eq} = C_1 + C_2 = \frac{2 \pi \epsilon_0}{\ln 2} + \frac{4 \pi \epsilon_0}{\ln 2} = \frac{6 \pi \epsilon_0}{\ln 2}$.

Three long concentric conducting cylindrical shells have radii $R$,$2R$,and $2\sqrt{2}R$. The inner and outer shells are connected to each other. The capacitance across the middle and inner shells per unit length is:

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