GATE 2022 PH – Question 15
A spherical boundary encloses a charge-free region. A quarter has potential $\phi_0$, another quarter $2\phi_0$, and the rest $4\phi_0$. Find its central potential.
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Correct answer: (A) $11\phi_0/4$
Explanation
The region inside the sphere is charge-free, so the potential satisfies Laplace's equation. By the **mean value theorem**, the potential at the centre equals the average of the potential over the surface of the sphere.
**Fractions of the surface:**
- one quarter at $\phi_0$,
- one quarter at $2\phi_0$,
- the remaining half at $4\phi_0$.
$$\phi(0)=\tfrac14\phi_0+\tfrac14(2\phi_0)+\tfrac12(4\phi_0)=\left(\tfrac14+\tfrac12+2\right)\phi_0=\frac{11}{4}\phi_0\quad(\text{option A}).$$