Magnetic Materials

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) –
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) –
This equation is a restatement of Gauss's law for magnetism, which states that the magnetic flux density ( ) 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) –
This is not true in general. The divergence of the magnetic field intensity is not zero; instead, it is related to the presence of free charges. The correct relation involves the displacement current and is given by . Hence, Option (C) is incorrect. Step 4: Option (D) –
This equation is valid because the bound current density is related to the magnetization of the material. The curl of 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)

The magnetic field in the air-gap can be calculated using Ampere's law: where:
- ,
- ,
- is the length of the air-gap.
First, calculate , the permeability of free space, which is . Now, substituting the values: Thus, the magnitude of the magnetic field is approximately .