91Ó°ÊÓ

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.

The equation of the dipole magnetic field is expressed as:

$$\begin{gathered} B=\frac{{\mathrm{μ}}_0\mathrm{m}}{4{\mathrm{Ï€°ù}}^3}\left[2\cos \mathrm{θ}\hat{r}+\sin \mathrm{θ}\widehat{\mathrm{θ}}\right] \\ =\frac{{\mathrm{μ}}_0\mathrm{m}}{4{\mathrm{Ï€°ù}}^3}\hat{z} \end{gathered}$$ ...... (i)

Here, ${\mathrm{μ}}_0$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 $\mathrm{θ}$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:

$\stackrel{→}{m}=m\hat{z}$

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

Substitute $\mathrm{³¦´Ç²õθ}\hat{\mathrm{r}}-\mathrm{²õ¾\pm ²Ôθ}\widehat{\mathrm{θ}}$for$\hat{z}$in the above equation.

$$\begin{gathered} \stackrel{→}{m}=m\left(\mathrm{³¦´Ç²õθ}\hat{\mathrm{r}}-\mathrm{²õ¾\pm ²Ôθ}\widehat{\mathrm{θ}}\right) \\ =m\mathrm{³¦´Ç²õθ}\hat{\mathrm{r}}-m\mathrm{²õ¾\pm ²Ôθ}\widehat{\mathrm{θ}} \\ =\left(\stackrel{→}{\mathrm{m}}.\hat{\mathrm{r}}\right)\hat{\mathrm{r}}-\mathrm{m³¦´Ç²õθ}\hat{\mathrm{r}}-\stackrel{→}{\mathrm{m}} \\ =\left(2\stackrel{→}{\mathrm{m}}.\hat{\mathrm{r}}\right)\hat{\mathrm{r}}+\left(\stackrel{→}{\mathrm{m}}.\hat{\mathrm{r}}\right)\hat{\mathrm{r}}-\stackrel{→}{\mathrm{m}} \end{gathered}$$

Hence, further as:

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

Substitute the above value in equation (i).

localid="1657530579098" $B=\frac{{\mathrm{μ}}_0}{4{\mathrm{Ï€°ù}}^3}\left[\right(3\mathrm{m}⃗â‹\dots \hat{\mathrm{r}})\hat{\mathrm{r}}-\mathrm{m}⃗]$

Thus, the magnetic field of a dipolelocalid="1657528216508" $\frac{{\mathrm{μ}}_0}{4{\mathrm{Ï€°ù}}^3}\left[\right(3\mathrm{m}⃗â‹\dots \hat{\mathrm{r}})\hat{\mathrm{r}}-\mathrm{m}⃗]$has been proved.