Question

Show that the magnetic field of a dipole can be written in coordinate-free form:

$\boxed{{\mathbf{B}}_{\mathbf{dip}}\mathbf{\left(}\mathbf{r}\mathbf{\right)}\mathbf{=}\frac{{\mathbf{\mu }}_{\mathbf{0}}}{\mathbf{4}\mathbf{\pi }}\frac{\mathbf{1}}{{\mathbf{r}}^{\mathbf{3}}}\mathbf{\left[}\mathbf{3}\mathbf{\right(}\mathbf{m}\mathbf{\cdot }\hat{\mathbf{r}}\mathbf{)}\hat{\mathbf{r}}\mathbf{-}\mathbf{m}\mathbf{]}}$

Short answer

The magnetic field of a dipole$\frac{{\mathrm{\mu }}_0}{4{\mathrm{\pi r}}^3}\left[\right(3\mathrm{m}⃗\cdot \hat{\mathrm{r}})\hat{\mathrm{r}}-\mathrm{m}⃗]$ has been proved.

Step-by-step solution

Significance of the magnetism

Magnetism is a type of physical phenomenon produced by a motion of electric charges. Magnetism significantly results in repulsive or attractive force amongst the objects.

Determination of the magnetic field of a dipole

The equation of the dipole magnetic field is expressed as:

$$\begin{gathered} B=\frac{{\mathrm{\mu }}_0\mathrm{m}}{4{\mathrm{\pi r}}^3}\left[2\cos \theta \hat{r}+\sin \theta \hat{\theta }\right] \\ =\frac{{\mathrm{\mu }}_0\mathrm{m}}{4{\mathrm{\pi r}}^3}\hat{z} \end{gathered}$$ ...... (i)

Here, ${\mu }_0$is the permeability, m is the magnetic dipole moment ,r is the distance between the dipole charges, $\hat{z}$is the position vector in the z direction, and $\theta$is the angle between the dipoles.

If the dipole orients towards the z axis, then the equation of the magnetic dipole moment can be expressed as:

$\overrightarrow{m}=m\hat{z}$

Here, $\overrightarrow{m}$is the magnetic dipole moment vector and $\hat{z}$is the position vector in the z direction.

Substitute $\mathrm{cos\theta }\hat{\mathrm{r}}-\mathrm{sin\theta }\hat{\mathrm{\theta }}$for$\hat{z}$in the above equation.

$$\begin{gathered} \overrightarrow{m}=m\left(\mathrm{cos\theta }\hat{\mathrm{r}}-\mathrm{sin\theta }\hat{\mathrm{\theta }}\right) \\ =m\mathrm{cos\theta }\hat{\mathrm{r}}-m\mathrm{sin\theta }\hat{\mathrm{\theta }} \\ =\left(\overrightarrow{\mathrm{m}}.\hat{\mathrm{r}}\right)\hat{\mathrm{r}}-\mathrm{mcos\theta }\hat{\mathrm{r}}-\overrightarrow{\mathrm{m}} \\ =\left(2\overrightarrow{\mathrm{m}}.\hat{\mathrm{r}}\right)\hat{\mathrm{r}}+\left(\overrightarrow{\mathrm{m}}.\hat{\mathrm{r}}\right)\hat{\mathrm{r}}-\overrightarrow{\mathrm{m}} \end{gathered}$$

Hence, further as:

$$\begin{gathered} \overrightarrow{m}=\left(2\overrightarrow{m}.\hat{r}\right)\hat{r}+\left(\overrightarrow{m}.\hat{r}\right)\hat{r}-\overrightarrow{m} \\ =\left(3\overrightarrow{m}.\hat{r}\right)\hat{r}-\overrightarrow{m} \end{gathered}$$

Substitute the above value in equation (i).

localid="1657530579098" $B=\frac{{\mathrm{\mu }}_0}{4{\mathrm{\pi r}}^3}\left[\right(3\mathrm{m}⃗\cdot \hat{\mathrm{r}})\hat{\mathrm{r}}-\mathrm{m}⃗]$

Thus, the magnetic field of a dipolelocalid="1657528216508" $\frac{{\mathrm{\mu }}_0}{4{\mathrm{\pi r}}^3}\left[\right(3\mathrm{m}⃗\cdot \hat{\mathrm{r}})\hat{\mathrm{r}}-\mathrm{m}⃗]$has been proved.