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GATE 2022 EE – Question 53

Electromagnetic Fields · Coulomb's Law, Electric Field Intensity, Electric Flux Density, Gauss's Law, Divergence · 2 marks · Multiple choice

As shown in the figure below, two concentric conducting spherical shells, centered at $r=0$ and having radii $r=c$ and $r=d$ are maintained at potentials such that the potential $V(r)$ at $r=c$ is $V_1$ and $V(r)$ at $r=d$ is $V_2$. Assume that $V(r)$ depends only on $r$, where $r$ is the radial distance. The expression for $V(r)$ in the region between $r=c$ and $r=d$ is

Diagram for GATE 2022 EE question 53
  1. $V(r)=\dfrac{cd(V_2-V_1)}{(d-c)r}-\dfrac{V_1c+V_2d-2V_1d}{d-c}$
  2. $V(r)=\dfrac{cd(V_1-V_2)}{(d-c)r}+\dfrac{V_2d-V_1c}{d-c}$
  3. $V(r)=\dfrac{cd(V_1-V_2)}{(d-c)r}-\dfrac{V_1c-V_2c}{d-c}$
  4. $V(r)=\dfrac{cd(V_2-V_1)}{(d-c)r}-\dfrac{V_2c-V_1c}{d-c}$

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Correct answer: (B) $V(r)=\dfrac{cd(V_1-V_2)}{(d-c)r}+\dfrac{V_2d-V_1c}{d-c}$

Explanation

Laplace's equation with spherical symmetry gives $V(r)=\frac Ar+B$. The conditions $V(c)=V_1$ and $V(d)=V_2$ give $A=\frac{cd(V_1-V_2)}{d-c}$ and $B=\frac{V_2d-V_1c}{d-c}$, which is option B.