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This is a small addition to Phasor form of (extended) Maxwell’s equations in Geometric Algebra.

Relative to the observer frame implicitly specified by \( \gamma_0 \), here’s an expansion of the curl of the electric four potential

\begin{equation}\label{eqn:phasorMaxwellsWithElectricAndMagneticCharges:720}

\begin{aligned}

\grad \wedge A_{\textrm{e}}

&=

\inv{2}\lr{

\grad A_{\textrm{e}}

–

A_{\textrm{e}} \grad

} \\

&=

\inv{2}\lr{

\gamma_0 \lr{ \spacegrad + j k } \gamma_0 \lr{ A_{\textrm{e}}^0 – \BA_{\textrm{e}} }

–

\gamma_0 \lr{ A_{\textrm{e}}^0 – \BA_{\textrm{e}} } \gamma_0 \lr{ \spacegrad + j k }

} \\

&=

\inv{2}\lr{

\lr{ -\spacegrad + j k } \lr{ A_{\textrm{e}}^0 – \BA_{\textrm{e}} }

–

\lr{ A_{\textrm{e}}^0 + \BA_{\textrm{e}} } \lr{ \spacegrad + j k }

} \\

&=

\inv{2}\lr{

– 2 \spacegrad A_{\textrm{e}}^0 + j k A_{\textrm{e}}^0 – j k A_{\textrm{e}}^0

+ \spacegrad \BA_{\textrm{e}} – \BA_{\textrm{e}} \spacegrad

– 2 j k \BA_{\textrm{e}}

} \\

&=

– \lr{ \spacegrad A_{\textrm{e}}^0 + j k \BA_{\textrm{e}} }

+ \spacegrad \wedge \BA_{\textrm{e}}

\end{aligned}

\end{equation}

In the above expansion when the gradients appeared on the right of the field components, they are acting from the right (i.e. implicitly using the Hestenes dot convention.)

The electric and magnetic fields can be picked off directly from above, and in the units implied by this choice of four-potential are

\begin{equation}\label{eqn:phasorMaxwellsWithElectricAndMagneticCharges:760}

\BE_{\textrm{e}} = – \lr{ \spacegrad A_{\textrm{e}}^0 + j k \BA_{\textrm{e}} } = -j \lr{ \inv{k}\spacegrad \spacegrad \cdot \BA_{\textrm{e}} + k \BA_{\textrm{e}} }

\end{equation}

\begin{equation}\label{eqn:phasorMaxwellsWithElectricAndMagneticCharges:780}

c \BB_{\textrm{e}} = \spacegrad \cross \BA_{\textrm{e}}.

\end{equation}

For the fields due to the magnetic potentials

\begin{equation}\label{eqn:phasorMaxwellsWithElectricAndMagneticCharges:800}

\lr{ \grad \wedge A_{\textrm{e}} } I

=

– \lr{ \spacegrad A_{\textrm{e}}^0 + j k \BA_{\textrm{e}} } I

– \spacegrad \cross \BA_{\textrm{e}},

\end{equation}

so the fields are

\begin{equation}\label{eqn:phasorMaxwellsWithElectricAndMagneticCharges:840}

c \BB_{\textrm{m}} = – \lr{ \spacegrad A_{\textrm{m}}^0 + j k \BA_{\textrm{m}} } = -j \lr{ \inv{k}\spacegrad \spacegrad \cdot \BA_{\textrm{m}} + k \BA_{\textrm{m}} }

\end{equation}

\begin{equation}\label{eqn:phasorMaxwellsWithElectricAndMagneticCharges:860}

\BE_{\textrm{m}} = -\spacegrad \cross \BA_{\textrm{m}}.

\end{equation}

Including both electric and magnetic sources the fields are

\begin{equation}\label{eqn:phasorMaxwellsWithElectricAndMagneticCharges:900}

\BE = -\spacegrad \cross \BA_{\textrm{m}} -j \lr{ \inv{k}\spacegrad \spacegrad \cdot \BA_{\textrm{e}} + k \BA_{\textrm{e}} }

\end{equation}

\begin{equation}\label{eqn:phasorMaxwellsWithElectricAndMagneticCharges:920}

c \BB = \spacegrad \cross \BA_{\textrm{e}} -j \lr{ \inv{k}\spacegrad \spacegrad \cdot \BA_{\textrm{m}} + k \BA_{\textrm{m}} }

\end{equation}