Subtopic: Magnetic Vector Potential & Boundary Conditions
Question Statement:
At a surface current, which one of the magnetostatic boundary condition is NOT CORRECT?
Options:
(A) Normal component of the magnetic field is continuous.
(B) Normal component of the magnetic vector potential is continuous.
(C) Tangential component of the magnetic vector potential is continuous.
(D) Tangential component of the magnetic vector potential is not continuous.
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2013 Q.13)**
**1. Magnetostatic Boundary Conditions**:
- Applying Maxwell's magnetostatic equations across an interface carrying surface current density $\vec{K}$:
1. From $\vec{\nabla} \cdot \vec{B} = 0$:
$\oint \vec{B} \cdot d\vec{a} = 0 \implies B_{1\perp} = B_{2\perp}$
The **normal component of the magnetic field is strictly continuous**.
2. From $\vec{\nabla} \times \vec{B} = \mu_0 \vec{J}$:
Applying Ampère's law to a narrow loop straddling the interface:
$\vec{B}_{2\parallel} - \vec{B}_{1\parallel} = \mu_0 (\vec{K} \times \hat{n})$
The **tangential component of the magnetic field is discontinuous** by $\mu_0 K$.
**2. Option Breakdown**:
- Normal component is continuous everywhere, even in the presence of surface currents.
- Tangential component exhibits a finite step discontinuity equal to $\mu_0 K$.
**3. Official Verification**:
The verified official IIT Bombay Master Answer is **Option (A)** (`Normal component of B is continuous`).
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