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GATE 2025 PH – Question 58

Electromagnetic theory · magnetic materials · 2 marks · Multiple select

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 Mk̂. Which of the following option(s) is/are correct?

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

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Correct answer: (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

Explanation

The cylinder has uniform magnetisation $\mathbf M=M\hat k$ (along its axis).

**Bound currents.**
- **Volume current:** $\mathbf J_b=\nabla\times\mathbf M=0$, since $\mathbf M$ is uniform. So B is true. ✓
- **Surface current:** $\mathbf K_b=\mathbf M\times\hat n$. On the **curved** surface $\hat n=\hat s$ (radial), so $\mathbf K_b=M\,\hat z\times\hat s=M\hat\phi\neq0$. So C is true (the cylinder acts like a solenoid). ✓ On the flat top and bottom, $\hat n=\pm\hat z$ is parallel to $\mathbf M$, so $\mathbf M\times\hat n=0$. D is false.

**Field outside.** Outside the cylinder there is no free or bound current, so $\nabla\times\mathbf B=0$ there. A curl-free field can be written as a gradient of a scalar potential. So A is true. ✓

Answer **A, B and C**.