Question

A linear magnetic material in the form of a cylinder of radius \( R \) and length \( L \) is placed with its axis parallel to the \( z \)-axis. The cylinder has uniform magnetization \( M \hat{k} \). Which of the following option(s) is/are correct?

• A. The magnetic field at any point outside the cylinder can be expressed as the gradient of a scalar function
• B. The bound volume current density is zero
• C. The surface current density on the curved surface is non-zero
• D. The surface current densities on the flat surfaces (top and bottom) are non-zero

Hint:

In magnetism, surface currents are generated at boundaries where the magnetization has a discontinuity, such as at the curved surface of a magnetized cylinder.

Answer 1

The Correct Option is A, B, C

Step 1: The magnetic field outside a uniformly magnetized cylinder cannot generally be expressed as the gradient of a scalar function. This is because the magnetization generates a non-conservative magnetic field in the region outside the cylinder, which contradicts the option (A). 
Step 2: The bound volume current density \( \mathbf{J}_b \) is related to the magnetization \( \mathbf{M} \) by: \[ \mathbf{J}_b = \nabla \times \mathbf{M}. \] Since the magnetization is uniform, its curl is zero, and thus, the bound volume current density is zero, making option (B) correct. 
Step 3: The surface current density \( \mathbf{K}_b \) on the curved surface of the cylinder is given by: \[ \mathbf{K}_b = \hat{n} \times \mathbf{M}. \] Since the magnetization is uniform and along the axis of the cylinder, there is a non-zero surface current density on the curved surface, which makes option (C) correct. 
Step 4: On the flat surfaces (top and bottom), the magnetization does not produce a current, as the magnetization is parallel to the cylinder's axis. Therefore, the surface current densities on the flat surfaces are zero, making option (D) incorrect.