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GATE 2023 PH – Question 59

Electromagnetic theory · Solutions of electrostatic and magnetostatic problems including boundary value problems · 2 marks · Numerical answer

The two source configurations contain q=1 C, isolated or centered inside a neutral conducting shell with $R_1=1$ m, $R_2=2$ m. $W_I=\epsilon_0\int E_I^2dV/2$ and $W_{II}=\epsilon_0\int E_{II}^2dV/2$ over all space. If $(8\pi/\epsilon_0)|W_I-W_{II}|=1/n$, find integer n.

Isolated charge and identical charge inside a conducting shell.

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Correct answer: 2

Explanation

The shell removes the field energy $\Delta W=q^2(1/R_1-1/R_2)/(8\pi\epsilon_0)=1/(16\pi\epsilon_0)$. With the dimensionally intended multiplier $8\pi\epsilon_0$, this gives 1/n=1/2.