The exchange coupling mentioned in Section 32-11 as being responsible for ferromagnetism is not the mutual magnetic interaction between two elementary magnetic dipoles. To show this, (a) Calculate the magnitude of the magnetic field a distance of $\mathbf{10}\mathbf{}\mathbf{nm}$away, along the dipole axis, from an atom with magnetic dipole moment $\mathbf{1}\mathbf{.}\mathbf{5}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{23}}\mathbf{}\mathbf{J}\mathbf{/}\mathbf{T}$(cobalt), and (b) Calculate the minimum energy required to turn a second identical dipole end for end in this field. (c) By comparing the latter with the mean translational kinetic energy of $\mathbf{0}\mathbf{.}\mathbf{040}\mathbf{}\mathbf{eV}$, what can you conclude?

${\mathrm{\mu }}_{\mathrm{B}}=1.5\times {10}^{-23}\mathrm{J}/\mathrm{T}$

$\begin{array}{c}\mathrm{r}=10\mathrm{nm}\\ =10\times {10}^{-9}\mathrm{m}\end{array}$

Here, we need to use the equation of magnetic field due to magnetic dipole moment. The minimum energy can be calculated using the equation of energy related to magnetic dipole moment and magnetic field.

Formulae:

Magnetic field due to the dipole moment ${\text{μ}}_{\text{B}}$at a distance r is

$\mathrm{B}=\frac{{\mathrm{\mu }}_0}{2\mathrm{\pi }}\left(\frac{{\mathrm{\mu }}_{\mathrm{B}}}{{\mathrm{r}}^3}\right)$

The required field along the dipole axis:

$\begin{array}{c}\mathrm{B}=\frac{{\mathrm{\mu }}_0}{2\mathrm{\pi }}\left(\frac{{\mathrm{\mu }}_{\mathrm{B}}}{{\mathrm{r}}^3}\right)\\ =\frac{4\mathrm{\pi }\times {10}^{-7}}{2\mathrm{\pi }}\left(\frac{1.5\times {10}^{-23}}{{\left({10}^{-8}\right)}^3}\right)\\ =3.0\times {10}^{-6}\text{T}\end{array}$

The magnitude of the magnetic field is,$\mathrm{B}=3.0\times {10}^{-6}\mathrm{T}$

$\begin{array}{c}{\mathrm{U}}_{\mathrm{minimum}}={\mathrm{\mu }}_{\mathrm{B}}\mathrm{B}(\cos \left({\mathrm{\varphi }}_2\right)-\cos \left({\mathrm{\varphi }}_1\right))\\ =1.5\times {10}^{-23}\times 3.0\times {10}^{-6}\times (\cos \left(0\right)-\cos \left(180\right))\\ =9.0\times {10}^{-29}\mathrm{J}\\ =\frac{9.0\times {10}^{-29}}{1.6\times {10}^{-19}}\\ =5.6\times {10}^{-10}\mathrm{eV}\end{array}$

The minimum energy required is,${\mathrm{U}}_{\mathrm{minimum}}=5.6\times {10}^{-10}\mathrm{eV}$

As the mean translational kinetic energy (0.04 eV) is much larger than the required energy of the aligning dipoles, if dipole – dipole interactions were responsible for aligning dipoles, the collision would easily disturb the direction of moments, and they would not remain aligned.