Subtopic: Biot-Savart Law & Ampere's Circuital Law
Question Statement:
The x-y plane is the boundary between free space and a magnetic material with relative permeability \mu_r. The magnetic field in the free space is B_x \hat{i} + B_z \hat{k}. The magnetic field in the magnetic material is
Options:
(A) \frac{B_x}{\mu_r} \hat{i} + B_z \hat{k}
(B) B_x \hat{i} + \mu_r B_z \hat{k}
(C) \mu_r B_x \hat{i} + B_z \hat{k}
(D) B_x \hat{i} + \frac{B_z}{\mu_r} \hat{k}
Official Answer Key:
D
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2016 Q.59 / PH Q.49)**
**1. Electromagnetic Boundary Conditions at Interface**:
- The boundary is the $xy$-plane ($z = 0$).
- Normal direction: $\hat{n} = \hat{k}$ (along $z$-axis).
- Tangential direction: $\hat{i}$ (along $x$-axis).
**2. Boundary Continuity Conditions**:
- The normal component of magnetic induction $\vec{B}$ is continuous:
$B_{2z} = B_{1z} = B_z$
- In the absence of free surface currents ($K_f = 0$), the tangential component of magnetic field intensity $\vec{H}$ is continuous:
$H_{2x} = H_{1x} \implies \frac{B_{2x}}{\mu_r \mu_0} = \frac{B_{1x}}{\mu_0} \implies B_{2x} = \mu_r B_{1x} = \mu_r B_x$
**3. Total Magnetic Field in Material**:
- Combining the components:
$\vec{B}_2 = \mu_r B_x \hat{i} + B_z \hat{k}$
- **Official Master Answer**: **C**
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