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Sum of Product of Binomial Coefficients

Two concentri...

Question

Two concentric, thin metallic spherical shells of radii $R_{1}$ and $R_{2} (R_{1} > R_{2})$ bear charges $Q_{1}$ and $Q_{2}$ respectively. Then the potential at radius ‘r’ between $R_{1}$ and $R_{2}$ will be $\frac{1}{4 π ϵ_{0}}$ times

A

\frac{Q_{1} + Q_{2}}{r}

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B

\frac{Q_{1}}{R_{1}} + \frac{Q_{2}}{r}

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C

\frac{Q_{1}}{R_{1}} + \frac{Q_{2}}{R_{2}}

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D

\frac{Q_{1}}{R_{2}} + \frac{Q_{2}}{R_{2}}

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Solution

The correct option is B $\frac{Q_{1}}{R_{1}} + \frac{Q_{2}}{r}$
The potential at radius r between $R_{1}$ and $R_{2} =$ potential inside the shell of radius $R_{1} +$ potential outside the shell of radius $R_{2}$
or $V_{r} = \frac{Q_{1}}{4 π ϵ_{0} R_{1}} + \frac{Q_{2}}{4 π ϵ_{0} r}$ as $(V_{i n} = V_{s u r f a c e})$ for shell
or $V_{r} = \frac{1}{4 π ϵ_{0}} [\frac{Q_{1}}{R_{1}} + \frac{Q_{2}}{r}]$

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