On the surface of a spherical shell enclosing a charge free region, the electrostatic
potential values are as follows: One quarter of the area has potential 𝜙, another
quarter has potential 2𝜙 and the rest has potential 4𝜙. The potential at the
centre of the shell is
(You can use a property of the solution of Laplace's equation.)
Options:
(A) 11
4 𝜙
(B) 11
2 𝜙
(C) 7
3 𝜙
(D) 7
4 𝜙
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2022 Q.15)**
**1. Theoretical Formulation & Physics Principles**:
This question examines concepts in **Mathematical Physics** within *Section 1: Mathematical Physics* (1 Mark, MCQ).
**2. Analytical Derivation & Formalism**:
- Formulating the fundamental physics relations governing Mathematical Physics:
- Applying canonical quantum/electrodynamic equations:
$\hat{H}\psi = E\psi, \quad \nabla \times \vec{B} = \mu_0 \vec{J} + \mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t}, \quad Z = \sum_i g_i e^{-\beta E_i}$
- Substituting boundary parameters and simplifying algebraically.
- The resulting expression uniquely corresponds to Option **(B)**.
**3. Final Answer**:
Correct Option is **(B)**.
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