Use the “duality” transformation of Prob. 7.64, together with the fields of an oscillating electric dipole (Eqs. 11.18 and 11.19), to determine the fields that would be produced by an oscillating “Gilbert” magnetic dipole (composed of equal and opposite magnetic charges, instead of an electric current loop). Compare Eqs. 11.36 and 11.37, and comment on the result.

Write the expression for the duality transformation equations.

$$\begin{gathered} \mathit{E}\mathbf{\text{'}}\mathbf{=}\mathit{E}\mathit{c}\mathit{o}\mathit{s}\mathit{\alpha }\mathbf{+}\mathit{c}\mathit{B}\mathit{s}\mathit{i}\mathit{n}\mathit{\alpha } \\ \mathit{c}\mathit{B}\mathbf{\text{'}}\mathbf{=}\mathit{c}\mathit{B}\mathit{c}\mathit{o}\mathit{s}\mathit{\alpha }\mathbf{-}\mathit{E}\mathit{s}\mathit{i}\mathit{n}\mathit{\alpha } \\ \mathit{c}{\mathit{q}}_{\mathbf{e}}^{\mathbf{\text{'}}}\mathbf{=}\mathit{c}{\mathit{q}}_{\mathbf{e}}\mathit{c}\mathit{o}\mathit{s}\mathit{\alpha }\mathbf{+}{\mathit{q}}_{\mathbf{m}}\mathit{s}\mathit{i}\mathit{n}\mathit{\alpha } \\ {\mathit{q}}_{\mathbf{m}}^{\mathbf{\text{'}}}\mathbf{=}{\mathit{q}}_{\mathbf{m}}\mathit{c}\mathit{o}\mathit{s}\mathit{\alpha }\mathbf{-}\mathit{c}{\mathit{q}}_{\mathbf{e}}\mathit{s}\mathit{i}\mathit{n}\mathit{\alpha } \end{gathered}$$

Substitute $\alpha =90°$in all the duality transformation equations.

$$\begin{gathered} E\text{'}=E\cos \left(90°\right)+cB\sin \left(90°\right) \\ E\text{'}=cB........\left(1\right) \end{gathered}$$

$$\begin{gathered} cB\text{'}=cB\cos \left(90°\right)-E\sin \left(90°\right) \\ cB\text{'}=-E.......\left(2\right) \end{gathered}$$

$$\begin{gathered} c{q}_e^{\text{'}}=c{q}_e\cos \left(90°\right)+q_m\sin \left(90°\right) \\ c{q}_e^{\text{'}}=q_m \end{gathered}$$

$$\begin{gathered} q_m^{\text{'}}=q_m\cos \left(90°\right)-c{q}_e\sin \left(90°\right) \\ {q}_n^{\text{'}}=-c{q}_e \end{gathered}$$

Write the expression for the magnetic field using the equation .

$B=-\frac{{\mu }_0{p}_0{\omega }^2}{4\pi c}\left(\frac{\sin \theta }{r}\right)\cos \left[\omega \left(t-\frac{r}{c}\right)\right]\hat{\varphi }$

Here,$p_0=-\frac{m_0}{c}$ .

Hence, the above equation becomes,

$B=-\frac{{\mu }_0{m}_0{\omega }^2}{4\pi c^2}\left(\frac{\sin \theta }{r}\right)\cos \left[\omega \left(t-\frac{r}{c}\right)\right]\hat{\varphi }$

Substitute $B=-\frac{{\mu }_0{m}_0{\omega }^2}{4\pi c^2}\left(\frac{\sin \theta }{r}\right)\cos \left[\omega \left(t-\frac{r}{c}\right)\right]\hat{\varphi }$in equation (1).

$$\begin{gathered} E\text{'}=c\left[-\frac{{\mu }_0{m}_0{\omega }^2}{4\pi c^2}\left(\frac{\sin \theta }{r}\right)\cos \left[\omega \left(t-\frac{r}{c}\right)\right]\hat{\varphi }\right] \\ E\text{'}=\frac{{\mu }_0{m}_0{\omega }^2}{4\pi c}\left(\frac{\sin \theta }{r}\right)\cos \left[\omega \left(t-\frac{r}{c}\right)\right]\hat{\varphi } \end{gathered}$$

Write the expression for an electric field using the equation .

$E=\frac{{\mu }_0{m}_0{\omega }^2}{4\pi c}\left(\frac{\sin \theta }{r}\right)\cos \left[\omega \left(t-\frac{r}{c}\right)\right]\hat{\varphi }$

Substitute $E=\frac{{\mu }_0{m}_0{\omega }^2}{4\pi c}\left(\frac{\sin \theta }{r}\right)\cos \left[\omega \left(t-\frac{r}{c}\right)\right]\hat{\varphi }$in equation (2).

$$\begin{gathered} B\text{'}=\frac{1}{c}\left[-\frac{{\mu }_0{m}_0{\omega }^2}{4\pi c}\left(\frac{\sin \theta }{r}\right)\cos \left[\omega \left(t-\frac{r}{c}\right)\right]\hat{\varphi }\right] \\ {B}^{\text{'}}=\frac{-{\mu }_0{m}_0{\omega }^2}{4\pi c^2}\left[\left(\frac{\sin \theta }{r}\right)\cos \left[\omega ct-\frac{r}{c}\right]\hat{\varphi }\right] \end{gathered}$$

Hence, these are identical to the fields of an ampere dipole.

Therefore, the fields that would be produced by an oscillating “Gilbert” magnetic dipole is$E\text{'}=\frac{{\mu }_0{m}_0{\omega }^2}{4\pi c}\left(\frac{\sin \theta }{r}\right)\cos \left[\omega \left(t-\frac{r}{c}\right)\right]\hat{\varphi }$ and$B\text{'}=\frac{-{\mu }_0{m}_0{\omega }^2}{4\pi c^2}\left[\left(\frac{\sin \theta }{r}\right)\cos \left[\omega ct-\frac{r}{c}\right]\hat{\varphi }\right]$ and the equations 11.36 and 11.37 are identical to the fields of an ampere dipole.