Question

Consider an isolated magnetized sphere of radius $R$ with a uniform magnetization $\vec{M}$ along the positive $z$ direction, with the north and south poles of the sphere lying on the $z$ axis. It is given that the magnetic field inside the sphere is $\vec{B} = \dfrac{2\mu_0}{3}\vec{M}$, where $\mu_0$ is the permeability of vacuum. Which of the following statements is(are) CORRECT?

• A. The bound volume current density is zero
• B. The bound surface current density has maximum magnitude at the equator, where this magnitude equals $|\vec{M}|$
• C. The auxiliary field $\vec{H} = -\dfrac{2}{3}\vec{M}$
• D. Far from the sphere, the magnetic field is due to a dipole of moment $\vec{m}$, where $\dfrac{\vec{m}}{4\pi R^3} = \dfrac{B}{2\mu_0}\hat{z}$

Hint:

In a uniformly magnetized sphere, the bound volume current is zero, bound surface current is maximum at the equator, and the field behaves like a magnetic dipole at large distances.

Answer 1

The Correct Option is A, B, D

- (A) The bound volume current density is zero: For a uniformly magnetized material, there are no bound volume currents because the magnetization is uniform throughout. Hence, this statement is correct.
- (B) The bound surface current density has maximum magnitude at the equator, where this magnitude equals $|\vec{M}|$: The bound surface current density is proportional to the magnetization at the surface. It is maximum at the equator of the sphere, as the magnetization is uniform, and the surface current density at the equator equals $|\vec{M}|$. Hence, this statement is correct.
- (C) The auxiliary field $\vec{H} = -\dfrac{2}{3}\vec{M}$: This is incorrect. The auxiliary field $\vec{H}$ is related to the magnetic field $\vec{B}$ and the magnetization $\vec{M}$ by $\vec{B} = \mu_0(\vec{H} + \vec{M})$. The given expression for $\vec{H}$ does not hold.
- (D) Far from the sphere, the magnetic field is due to a dipole of moment $\vec{m}$, where $\dfrac{\vec{m}}{4\pi R^3} = \dfrac{B}{2\mu_0}\hat{z}$: This is correct. The field outside the magnetized sphere behaves like that of a magnetic dipole with dipole moment $\vec{m} = \dfrac{4\pi R^3}{3}\vec{M}$, and the far-field behavior matches the given equation.
Thus, the correct answer is (A), (B), (D).