Consider an isolated magnetized sphere of radius R with a uniform magnetization
M⃗⃗ along the positive z direction, with the north and south poles of the sphere
lying on the z axis. It is given that the magnetic field inside the sphere is
B⃗ =
2𝜇0
3 M⃗⃗ , where 𝜇0 is the permeability of vacuum. Which of the following
statements is(are) CORRECT?
Options:
(A) The bound volume current density is zero
(B) The bound surface current density has maximum magnitude at the equator, where
this magnitude equals |M⃗⃗ |
(C) The auxiliary field H⃗⃗ = -
2
3 M⃗⃗
(D) Far from the sphere, the magnetic field is due to a dipole of moment m⃗⃗ , where
m⃗⃗⃗
4𝜋R3 =
B
2𝜇0
ẑ
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2023 Q.31)**
**1. Theoretical Formulation & Physics Principles**:
This question examines concepts in **Electromagnetic Theory** within *Section 3: Electromagnetic Theory* (1 Mark, 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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