Question

A material is placed in a magnetic field intensity \(H\). As a result, bound current density \(J_b\) is induced and magnetization of the material is \(M\). The magnetic flux density is \(B\). Choose the correct option(s) valid at the surface of the material.

• A. \( \nabla \cdot M = 0 \)
• B. \( \nabla \cdot B = 0 \)
• C. \( \nabla \cdot H = 0 \)
• D. \( \nabla \times J_b = 0 \)

Hint:

The curl of the bound current density is zero at the surface of the material, and the divergence of the magnetic flux density is always zero, as per Gauss's law for magnetism.

Answer 1

The Correct Option is B, D

This question involves the concepts of magnetic fields, magnetization, and bound current in a material. Let's break down each option and discuss why the correct answers are (B) and (D). Step 1: Option (A) – \( \nabla \cdot M = 0 \)
This equation indicates that the divergence of the magnetization is zero. While magnetization is a vector field, it is not always true that its divergence is zero. In fact, the divergence of the magnetization relates to bound charges in the material, so this option is incorrect. Step 2: Option (B) – \( \nabla \cdot B = 0 \)
This equation is a restatement of Gauss's law for magnetism, which states that the magnetic flux density (\(B\)) has no net divergence. It is always valid because there are no magnetic monopoles in classical electromagnetism. Therefore, Option (B) is correct. Step 3: Option (C) – \( \nabla \cdot H = 0 \)
This is not true in general. The divergence of the magnetic field intensity \(H\) is not zero; instead, it is related to the presence of free charges. The correct relation involves the displacement current and is given by \( \nabla \cdot H = \rho_{\text{free}} \). Hence, Option (C) is incorrect. Step 4: Option (D) – \( \nabla \times J_b = 0 \)
This equation is valid because the bound current density \(J_b\) is related to the magnetization of the material. The curl of \(J_b\) is zero at the surface because the bound current does not generate any net circulating field at the boundary, making this option correct. Final Answer: (B), (D)