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7.5: A Point Magnetic Dipole

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  • Consider an oscillating magnetic dipole moment, mz, oriented along the z-axis and located at the origin of co-ordinates similar to the case of the oscillating electric dipole of Figure (7.4.3). If the dipole were static it would generate a vector potential having only a \(\phi\)-component:

    \[\text{A}_{\phi}=\frac{\mu_{0}}{4 \pi} \frac{\text{m}_{\text{z}} \sin \theta}{\text{R}^{2}}. \label{7.34}\]

    This follows from the general expression for the vector potential generated by a point dipole, Equation (4.3.4)

    \[\vec{\text{A}}=\frac{\mu_{0}}{4 \pi} \frac{(\vec{\text{m}} \times \vec{\text{R}})}{\text{R}^{3}}. \nonumber \]

    However, it can be shown that due to the effects of time retardation the equation for the vector potential,(\ref{7.34}) must be modified to read

    \[\text{A}_{\phi}=\frac{\mu_{0}}{4 \pi} \sin \theta\left[\frac{\text{m}_{\text{z}}}{\text{R}^{2}}+\frac{\text{m}_{\text{z}}}{\text{cR}}\right]. \label{7.35}\]

    The fields derived from this expression for the vector potential, \(\vec{\text{B}}=\operatorname{curl}(\vec{\text{A}})\), are

    \[\text{B}_{\text{R}}=\frac{\mu_{0}}{4 \pi} 2 \cos \theta\left[\frac{\text{m}_{\text{z}}}{\text{R}^{3}}+\frac{\dot{\text{m}}_{\text{z}}}{\text{cR}^{2}}\right]_{\text{t}_{\text{R}}}, \label{7.36}\]

    \[\text{B}_{\theta}=\frac{\mu_{0}}{4 \pi} \sin \theta\left[\frac{\text{m}_{z}}{\text{R}^{3}}+\frac{\dot{\text{m}}_{z}}{\text{c} \text{R}^{2}}+\frac{\ddot{\text{m}}_{z}}{\text{c}^{2} \text{R}}\right]_{\text{t}_{\text{R}}}, \nonumber \]

    \[ \text{B}_{\phi}=0=\text{E}_{\text{R}}=\text{E}_{\theta}, \nonumber \]

    \[\text{E}_{\phi}=-\frac{\mu_{0}}{4 \pi} \sin \theta\left[\frac{\dot{\text{m}}_{\text{z}}}{\text{R}^{2}}+\frac{\ddot{\text{m}}_{\text{z}}}{\text{cR}}\right]_{\text{t}_{\text{R}}}. \nonumber \]

    where tR = t − R/c. Far from the dipole the radiation fields that decrease with distance like (1/R) are given by

    \[\text{B}_{\theta}=\frac{\mu_{0}}{4 \pi} \frac{\ddot{\text{m}}_{z}}{\text{c}^{2} \text{R}} \sin \theta, \label{7.37}\]
    \[\text{E}_{\phi}=-\frac{\mu_{0}}{4 \pi} \frac{\ddot{\text{m}}_{z}}{\text{cR}} \sin \theta=\text{cB}_{\theta}, \nonumber\]

    both evaluated at the retarded time tR. Just as for the electric dipole far fields \(|\vec{\text{E}}|=\text{c}|\vec{\text{B}}|\), and \(\vec E\) and \(\vec B\) are orthogonal to each other and to the line joining the position of the observer to the dipole.