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GATE 2024 IN – Question 20

Electricity and Magnetism · Electrostatics and capacitance · 1 mark · Multiple choice

Two concentric spherical metal shells have radii $x,y$ with $y>x$. If the intervening medium has permittivity $\epsilon$, their capacitance is

  1. $4\pi\epsilon\dfrac{xy}{y-x}$
  2. $4\pi\epsilon\dfrac{x^2}{y-x}$
  3. $4\pi\epsilon\dfrac{y^2}{y-x}$
  4. $4\pi\epsilon\dfrac{y^2-xy}{x}$

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Correct answer: (A) $4\pi\epsilon\dfrac{xy}{y-x}$

Explanation

Gauss’s law gives $E(r)=Q/(4\pi\epsilon r^2)$. The shell-to-shell potential difference is $V=\int_x^yE(r)\,dr=Q(1/x-1/y)/(4\pi\epsilon)$.

Thus $C=Q/V=4\pi\epsilon xy/(y-x)$, **A**.