A point charge q is placed at a distance d above an infinite, grou nded conducting
plate placed on the xy plane at z = 0. The electrostatic potential in z > 0 region is
given by 𝜙 = 𝜙1 + 𝜙2, where 𝜙1 =
1
4𝜋𝜖0
q
√x2+y2+(z-d)2 and
𝜙2 = -
1
4𝜋𝜖0
q
√x2+y2+(z+d)2. Which of the following option(s) is/are correct?
Options:
(A) The magnitude of the force experienced by the point charge q is
1
16𝜋𝜖0
q
d2
(B) The electrostatic energy of the system is
1
8𝜋𝜖0
q2
d
(C) The induced surface charge density on the plate is proportional to
1
√x2+y2+d2
(D) The electrostatic potential 𝜙1 satisfies Poisson's equation for z > 0
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2025 Q.49)**
**1. Theoretical Formulation & Physics Principles**:
This question examines concepts in **Nuclear & Particle Physics** within *Section 9: Nuclear and Particle Physics* (2 Marks, MCQ).
**2. Analytical Derivation & Formalism**:
- Formulating the fundamental physics relations governing Nuclear & Particle 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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