Question

A magnetic dipole $\stackrel{\rightharpoonup }{m}=m_0\hat{z}$ is situated at the origin, in an otherwiseuniform magnetic field $\stackrel{\rightharpoonup }{B}=B_0\hat{z}$ . Show that there exists a spherical surface, centered at the origin, through which no magnetic field lines pass. Find the radius of this sphere, and sketch the field lines, inside and out.

Short answer

There exists a spherical surface of radius $\sqrt[3]{\frac{{\mathrm{\mu }}_0{\mathrm{m}}_0}{2{\mathrm{\pi B}}_0}}$, centered at the origin, through which no magnetic field lines pass.

Step-by-step solution

Given data

There is an magnetic dipole $\stackrel{\rightharpoonup }{m}=-m_0\hat{z}$ is situated at the origin, in an uniform magnetic field $\stackrel{\rightharpoonup }{B}=B_0\hat{z}$.

Magnetic field due to a dipole

The magnetic field due to a magnetic dipole $\stackrel{\rightharpoonup }{m}$ is

localid="1658559878707" ${\mathit{B}}_{\mathbf{dip}}\mathbf{=}\frac{{\mathbf{\mu }}_{\mathbf{0}}\mathbf{m}}{\mathbf{4}{\mathbf{\pi }}^{\mathbf{3}\mathbf{r}}}\mathbf{}\mathbf{[}\mathbf{2}\mathbf{cos\theta }\hat{\mathbf{r}}\mathbf{+}\mathbf{sin\theta }\hat{\mathbf{\theta }}\mathbf{]}$

Here, ${\mu }_0$ is the permeability of free space.

Net magnetic field near origin

From equation (1), the net magnetic field near the origin is,

$$\begin{gathered} \stackrel{\rightharpoonup }{B}=B_0\hat{z}+\overrightarrow{B_{dip}} \\ =B_0\hat{z}-\frac{{\mu }_0{m}_0}{4\pi r^3}\left[2c0s\theta \hat{r}+\sin \theta \hat{\theta }\right] \end{gathered}$$

The radial component of this field is,

$$\begin{gathered} \stackrel{\rightharpoonup }{B}.\hat{r}=B_0\cos \theta -\frac{{\mu }_0{m}_0}{4\pi r^3}2\cos \theta \\ =\left(B_{0-}\frac{{\mu }_0{m}_0}{2\pi r^3}\right)\cos \theta \end{gathered}$$

Thus, the net field is zero for any value of $\theta$ at radius R where

localid="1657777042303" $$\begin{gathered} B_{0-}\frac{{\mu }_0{m}_0}{2\pi R^3}=0 \\ R=\sqrt[3]{\frac{{\mu }_0{m}_0}{2\pi B^3}} \end{gathered}$$

The field lines are shown in the following figure

Thus, the magnetic field lines are absent in the sphere of radius$\sqrt[3]{\frac{{\mu }_0{m}_0}{2\pi B^3}}$ .