A neutral conducting sphere of radius R is placed in a uniform electric field of
magnitude E0, that points along the z axis. The electrostatic potential at any point r
outside the sphere is given by V(r, ð) = V0 - E0r (1 -
R3
r3) cos ð, where V0 is the
constant potential of the sphere. Which of the following option(s) is/are correct?
Options:
(A) The induced surface charge density on the sphere is proportional to sin ð
(B) As r â â, Eâ = E0 cos ð rĖ
(C) The electric field at any point is curl free for r > R
(D) The electric field at any point is divergence free for r > R
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2025 Q.56)**
**1. Theoretical Formulation & Physics Principles**:
This question examines concepts in **Electromagnetic Theory** within *Section 3: Electromagnetic Theory* (2 Marks, MCQ).
**2. Analytical Derivation & Formalism**:
- Formulating the fundamental physics relations governing Electromagnetic Theory:
- 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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