magnetic vector potential

A comparison of Geometric Algebra electrodynamic potential methods

January 7, 2017 math and physics play , , , , , , , , , , , , , , , , , , ,

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Motivation

Geometric algebra (GA) allows for a compact description of Maxwell’s equations in either an explicit 3D representation or a STA (SpaceTime Algebra [2]) representation. The 3D GA and STA representations Maxwell’s equation both the form

\begin{equation}\label{eqn:potentialMethods:1280}
L \boldsymbol{\mathcal{F}} = J,
\end{equation}

where \( J \) represents the sources, \( L \) is a multivector gradient operator that includes partial derivative operator components for each of the space and time coordinates, and

\begin{equation}\label{eqn:potentialMethods:1020}
\boldsymbol{\mathcal{F}} = \boldsymbol{\mathcal{E}} + \eta I \boldsymbol{\mathcal{H}},
\end{equation}

is an electromagnetic field multivector, \( I = \Be_1 \Be_2 \Be_3 \) is the \R{3} pseudoscalar, and \( \eta = \sqrt{\mu/\epsilon} \) is the impedance of the media.

When Maxwell’s equations are extended to include magnetic sources in addition to conventional electric sources (as used in antenna-theory [1] and microwave engineering [3]), they take the form

\begin{equation}\label{eqn:chapter3Notes:20}
\spacegrad \cross \boldsymbol{\mathcal{E}} = – \boldsymbol{\mathcal{M}} – \PD{t}{\boldsymbol{\mathcal{B}}}
\end{equation}
\begin{equation}\label{eqn:chapter3Notes:40}
\spacegrad \cross \boldsymbol{\mathcal{H}} = \boldsymbol{\mathcal{J}} + \PD{t}{\boldsymbol{\mathcal{D}}}
\end{equation}
\begin{equation}\label{eqn:chapter3Notes:60}
\spacegrad \cdot \boldsymbol{\mathcal{D}} = q_{\textrm{e}}
\end{equation}
\begin{equation}\label{eqn:chapter3Notes:80}
\spacegrad \cdot \boldsymbol{\mathcal{B}} = q_{\textrm{m}}.
\end{equation}

The corresponding GA Maxwell equations in their respective 3D and STA forms are

\begin{equation}\label{eqn:potentialMethods:300}
\lr{ \spacegrad + \inv{v} \PD{t}{} } \boldsymbol{\mathcal{F}}
=
\eta
\lr{ v q_{\textrm{e}} – \boldsymbol{\mathcal{J}} }
+ I \lr{ v q_{\textrm{m}} – \boldsymbol{\mathcal{M}} }
\end{equation}
\begin{equation}\label{eqn:potentialMethods:320}
\grad \boldsymbol{\mathcal{F}} = \eta J – I M,
\end{equation}

where the wave group velocity in the medium is \( v = 1/\sqrt{\epsilon\mu} \), and the medium is isotropic with
\( \boldsymbol{\mathcal{B}} = \mu \boldsymbol{\mathcal{H}} \), and \( \boldsymbol{\mathcal{D}} = \epsilon \boldsymbol{\mathcal{E}} \). In the STA representation, \( \grad, J, M \) are all four-vectors, the specific meanings of which will be spelled out below.

How to determine the potential equations and the field representation using the conventional distinct Maxwell’s \ref{eqn:chapter3Notes:20}, … is well known. The basic procedure is to consider the electric and magnetic sources in turn, and observe that in each case one of the electric or magnetic fields must have a curl representation. The STA approach is similar, except that it can be observed that the field must have a four-curl representation for each type of source. In the explicit 3D GA formalism
\ref{eqn:potentialMethods:300} how to formulate a natural potential representation is not as obvious. There is no longer an reason to set any component of the field equal to a curl, and the representation of the four curl from the STA approach is awkward. Additionally, it is not obvious what form gauge invariance takes in the 3D GA representation.

Ideas explored in these notes

  • GA representation of Maxwell’s equations including magnetic sources.
  • STA GA formalism for Maxwell’s equations including magnetic sources.
  • Explicit form of the GA potential representation including both electric and magnetic sources.
  • Demonstration of exactly how the 3D and STA potentials are related.
  • Explore the structure of gauge transformations when magnetic sources are included.
  • Explore the structure of gauge transformations in the 3D GA formalism.
  • Specify the form of the Lorentz gauge in the 3D GA formalism.

Traditional vector algebra

No magnetic sources

When magnetic sources are omitted, it follows from \ref{eqn:chapter3Notes:80} that there is some \( \boldsymbol{\mathcal{A}}^{\mathrm{e}} \) for which

\begin{equation}\label{eqn:potentialMethods:20}
\boxed{
\boldsymbol{\mathcal{B}} = \spacegrad \cross \boldsymbol{\mathcal{A}}^{\mathrm{e}},
}
\end{equation}

Substitution into Faraday’s law \ref{eqn:chapter3Notes:20} gives

\begin{equation}\label{eqn:potentialMethods:40}
\spacegrad \cross \boldsymbol{\mathcal{E}} = – \PD{t}{}\lr{ \spacegrad \cross \boldsymbol{\mathcal{A}}^{\mathrm{e}} },
\end{equation}

or
\begin{equation}\label{eqn:potentialMethods:60}
\spacegrad \cross \lr{ \boldsymbol{\mathcal{E}} + \PD{t}{ \boldsymbol{\mathcal{A}}^{\mathrm{e}} } } = 0.
\end{equation}

A gradient representation of this curled quantity, say \( -\spacegrad \phi \), will provide the required zero

\begin{equation}\label{eqn:potentialMethods:80}
\boxed{
\boldsymbol{\mathcal{E}} = -\spacegrad \phi -\PD{t}{ \boldsymbol{\mathcal{A}}^{\mathrm{e}} }.
}
\end{equation}

The final two Maxwell equations yield

\begin{equation}\label{eqn:potentialMethods:100}
\begin{aligned}
-\spacegrad^2 \boldsymbol{\mathcal{A}}^{\mathrm{e}} + \spacegrad \lr{ \spacegrad \cdot \boldsymbol{\mathcal{A}}^{\mathrm{e}} } &= \mu \lr{ \boldsymbol{\mathcal{J}} + \epsilon \PD{t}{} \lr{ -\spacegrad \phi -\PD{t}{ \boldsymbol{\mathcal{A}}^{\mathrm{e}} } } } \\
\spacegrad \cdot \lr{ -\spacegrad \phi -\PD{t}{ \boldsymbol{\mathcal{A}}^{\mathrm{e}} } } &= q_e/\epsilon,
\end{aligned}
\end{equation}

or
\begin{equation}\label{eqn:potentialMethods:120}
\boxed{
\begin{aligned}
\spacegrad^2 \boldsymbol{\mathcal{A}}^{\mathrm{e}} – \inv{v^2} \PDSq{t}{ \boldsymbol{\mathcal{A}}^{\mathrm{e}} }
– \spacegrad \lr{
\inv{v^2} \PD{t}{\phi}
+\spacegrad \cdot \boldsymbol{\mathcal{A}}^{\mathrm{e}}
}
&= -\mu \boldsymbol{\mathcal{J}} \\
\spacegrad^2 \phi + \PD{t}{} \lr{ \spacegrad \cdot \boldsymbol{\mathcal{A}}^{\mathrm{e}} } &= -q_e/\epsilon.
\end{aligned}
}
\end{equation}

Note that the Lorentz condition \( \PDi{t}{(\phi/v^2)} + \spacegrad \cdot \boldsymbol{\mathcal{A}}^{\mathrm{e}} = 0 \) can be imposed to decouple these, leaving non-homogeneous wave equations for the vector and scalar potentials respectively.

No electric sources

Without electric sources, a curl representation of the electric field can be assumed, satisfying Gauss’s law

\begin{equation}\label{eqn:potentialMethods:140}
\boxed{
\boldsymbol{\mathcal{D}} = – \spacegrad \cross \boldsymbol{\mathcal{A}}^{\mathrm{m}}.
}
\end{equation}

Substitution into the Maxwell-Faraday law gives
\begin{equation}\label{eqn:potentialMethods:160}
\spacegrad \cross \lr{ \boldsymbol{\mathcal{H}} + \PD{t}{\boldsymbol{\mathcal{A}}^{\mathrm{m}}} } = 0.
\end{equation}

This is satisfied with any gradient, say, \( -\spacegrad \phi_m \), providing a potential representation for the magnetic field

\begin{equation}\label{eqn:potentialMethods:180}
\boxed{
\boldsymbol{\mathcal{H}} = -\spacegrad \phi_m – \PD{t}{\boldsymbol{\mathcal{A}}^{\mathrm{m}}}.
}
\end{equation}

The remaining Maxwell equations provide the required constraints on the potentials

\begin{equation}\label{eqn:potentialMethods:220}
-\spacegrad^2 \boldsymbol{\mathcal{A}}^{\mathrm{m}} + \spacegrad \lr{ \spacegrad \cdot \boldsymbol{\mathcal{A}}^{\mathrm{m}} } = -\epsilon
\lr{
-\boldsymbol{\mathcal{M}} – \mu \PD{t}{}
\lr{
-\spacegrad \phi_m – \PD{t}{\boldsymbol{\mathcal{A}}^{\mathrm{m}}}
}
}
\end{equation}
\begin{equation}\label{eqn:potentialMethods:240}
\spacegrad \cdot
\lr{
-\spacegrad \phi_m – \PD{t}{\boldsymbol{\mathcal{A}}^{\mathrm{m}}}
}
= \inv{\mu} q_m,
\end{equation}

or
\begin{equation}\label{eqn:potentialMethods:260}
\boxed{
\begin{aligned}
\spacegrad^2 \boldsymbol{\mathcal{A}}^{\mathrm{m}} – \inv{v^2} \PDSq{t}{\boldsymbol{\mathcal{A}}^{\mathrm{m}}} – \spacegrad \lr{ \inv{v^2} \PD{t}{\phi_m} + \spacegrad \cdot \boldsymbol{\mathcal{A}}^{\mathrm{m}} } &= -\epsilon \boldsymbol{\mathcal{M}} \\
\spacegrad^2 \phi_m + \PD{t}{}\lr{ \spacegrad \cdot \boldsymbol{\mathcal{A}}^{\mathrm{m}} } &= -\inv{\mu} q_m.
\end{aligned}
}
\end{equation}

The general solution to Maxwell’s equations is therefore
\begin{equation}\label{eqn:potentialMethods:280}
\begin{aligned}
\boldsymbol{\mathcal{E}} &=
-\spacegrad \phi -\PD{t}{ \boldsymbol{\mathcal{A}}^{\mathrm{e}} }
– \inv{\epsilon} \spacegrad \cross \boldsymbol{\mathcal{A}}^{\mathrm{m}} \\
\boldsymbol{\mathcal{H}} &=
\inv{\mu} \spacegrad \cross \boldsymbol{\mathcal{A}}^{\mathrm{e}}
-\spacegrad \phi_m – \PD{t}{\boldsymbol{\mathcal{A}}^{\mathrm{m}}},
\end{aligned}
\end{equation}

subject to the constraints \ref{eqn:potentialMethods:120} and \ref{eqn:potentialMethods:260}.

Potential operator structure

Knowing that there is a simple underlying structure to the potential representation of the electromagnetic field in the STA formalism inspires the question of whether that structure can be found directly using the scalar and vector potentials determined above.

Specifically, what is the multivector representation \ref{eqn:potentialMethods:1020} of the electromagnetic field in terms of all the individual potential variables, and can an underlying structure for that field representation be found? The composite field is

\begin{equation}\label{eqn:potentialMethods:280b}
\boldsymbol{\mathcal{F}}
=
-\spacegrad \phi -\PD{t}{ \boldsymbol{\mathcal{A}}^{\mathrm{e}} }
– \inv{\epsilon} \spacegrad \cross \boldsymbol{\mathcal{A}}^{\mathrm{m}} \\
+ I \eta
\lr{
\inv{\mu} \spacegrad \cross \boldsymbol{\mathcal{A}}^{\mathrm{e}}
-\spacegrad \phi_m – \PD{t}{\boldsymbol{\mathcal{A}}^{\mathrm{m}}}
}.
\end{equation}

Can this be factored into into multivector operator and multivector potentials? Expanding the cross products provides some direction

\begin{equation}\label{eqn:potentialMethods:1040}
\begin{aligned}
\boldsymbol{\mathcal{F}}
&=
– \PD{t}{ \boldsymbol{\mathcal{A}}^{\mathrm{e}} }
– \eta \PD{t}{I \boldsymbol{\mathcal{A}}^{\mathrm{m}}}
– \spacegrad \lr{ \phi – \eta I \phi_m } \\
&\quad + \frac{\eta}{2 \mu} \lr{ \rspacegrad \boldsymbol{\mathcal{A}}^{\mathrm{e}} – \boldsymbol{\mathcal{A}}^{\mathrm{e}} \lspacegrad }
+ \frac{1}{2 \epsilon} \lr{ \rspacegrad I \boldsymbol{\mathcal{A}}^{\mathrm{m}} – I \boldsymbol{\mathcal{A}}^{\mathrm{m}} \lspacegrad }.
\end{aligned}
\end{equation}

Observe that the
gradient and the time partials can be grouped together

\begin{equation}\label{eqn:potentialMethods:1060}
\begin{aligned}
\boldsymbol{\mathcal{F}}
&=
– \PD{t}{ } \lr{\boldsymbol{\mathcal{A}}^{\mathrm{e}} + \eta I \boldsymbol{\mathcal{A}}^{\mathrm{m}}}
– \spacegrad \lr{ \phi + \eta I \phi_m }
+ \frac{v}{2} \lr{ \rspacegrad (\boldsymbol{\mathcal{A}}^{\mathrm{e}} + I \eta \boldsymbol{\mathcal{A}}^{\mathrm{m}}) – (\boldsymbol{\mathcal{A}}^{\mathrm{e}} + I \eta \boldsymbol{\mathcal{A}}^{\mathrm{m}}) \lspacegrad } \\
&=
\inv{2} \lr{
\lr{ \rspacegrad – \inv{v} {\stackrel{ \rightarrow }{\partial_t}} } \lr{ v \boldsymbol{\mathcal{A}}^{\mathrm{e}} + \eta v I \boldsymbol{\mathcal{A}}^{\mathrm{m}} }
–
\lr{ v \boldsymbol{\mathcal{A}}^{\mathrm{e}} + \eta v I \boldsymbol{\mathcal{A}}^{\mathrm{m}}} \lr{ \lspacegrad + \inv{v} {\stackrel{ \leftarrow }{\partial_t}} }
} \\
&+\quad \inv{2} \lr{
\lr{ \rspacegrad – \inv{v} {\stackrel{ \rightarrow }{\partial_t}} } \lr{ -\phi – \eta I \phi_m }
– \lr{ \phi + \eta I \phi_m } \lr{ \lspacegrad + \inv{v} {\stackrel{ \leftarrow }{\partial_t}} }
}
,
\end{aligned}
\end{equation}

or

\begin{equation}\label{eqn:potentialMethods:1080}
\boxed{
\boldsymbol{\mathcal{F}}
=
\inv{2} \Biglr{
\lr{ \rspacegrad – \inv{v} {\stackrel{ \rightarrow }{\partial_t}} }
\lr{
– \phi
+ v \boldsymbol{\mathcal{A}}^{\mathrm{e}}
+ \eta I v \boldsymbol{\mathcal{A}}^{\mathrm{m}}
– \eta I \phi_m
}
–
\lr{
\phi
+ v \boldsymbol{\mathcal{A}}^{\mathrm{e}}
+ \eta I v \boldsymbol{\mathcal{A}}^{\mathrm{m}}
+ \eta I \phi_m
}
\lr{ \lspacegrad + \inv{v} {\stackrel{ \leftarrow }{\partial_t}} }
}
.
}
\end{equation}

There’s a conjugate structure to the potential on each side of the curl operation where we see a sign change for the scalar and pseudoscalar elements only. The reason for this becomes more clear in the STA formalism.

Potentials in the STA formalism.

Maxwell’s equation in its explicit 3D form \ref{eqn:potentialMethods:300} can be
converted to STA form, by introducing a four-vector basis \( \setlr{ \gamma_\mu } \), where the spatial basis
\( \setlr{ \Be_k = \gamma_k \gamma_0 } \)
is expressed in terms of the Dirac basis \( \setlr{ \gamma_\mu } \).
By multiplying from the left with \( \gamma_0 \) a STA form of Maxwell’s equation
\ref{eqn:potentialMethods:320}
is obtained,
where
\begin{equation}\label{eqn:potentialMethods:340}
\begin{aligned}
J &= \gamma^\mu J_\mu = ( v q_e, \boldsymbol{\mathcal{J}} ) \\
M &= \gamma^\mu M_\mu = ( v q_m, \boldsymbol{\mathcal{M}} ) \\
\grad &= \gamma^\mu \partial_\mu = ( (1/v) \partial_t, \spacegrad ) \\
I &= \gamma_0 \gamma_1 \gamma_2 \gamma_3,
\end{aligned}
\end{equation}

Here the metric choice is \( \gamma_0^2 = 1 = -\gamma_k^2 \). Note that in this representation the electromagnetic field \( \boldsymbol{\mathcal{F}} = \boldsymbol{\mathcal{E}} + \eta I \boldsymbol{\mathcal{H}} \) is a bivector, not a multivector as it is explicit (frame dependent) 3D representation of \ref{eqn:potentialMethods:300}.

A potential representation can be obtained as before by considering electric and magnetic sources in sequence and using superposition to assemble a complete potential.

No magnetic sources

Without magnetic sources, Maxwell’s equation splits into vector and trivector terms of the form

\begin{equation}\label{eqn:potentialMethods:380}
\grad \cdot \boldsymbol{\mathcal{F}} = \eta J
\end{equation}
\begin{equation}\label{eqn:potentialMethods:400}
\grad \wedge \boldsymbol{\mathcal{F}} = 0.
\end{equation}

A four-vector curl representation of the field will satisfy \ref{eqn:potentialMethods:400} allowing an immediate potential solution

\begin{equation}\label{eqn:potentialMethods:560}
\boxed{
\begin{aligned}
&\boldsymbol{\mathcal{F}} = \grad \wedge {A^{\mathrm{e}}} \\
&\grad^2 {A^{\mathrm{e}}} – \grad \lr{ \grad \cdot {A^{\mathrm{e}}} } = \eta J.
\end{aligned}
}
\end{equation}

This can be put into correspondence with \ref{eqn:potentialMethods:120} by noting that

\begin{equation}\label{eqn:potentialMethods:460}
\begin{aligned}
\grad^2 &= (\gamma^\mu \partial_\mu) \cdot (\gamma^\nu \partial_\nu) = \inv{v^2} \partial_{tt} – \spacegrad^2 \\
\gamma_0 {A^{\mathrm{e}}} &= \gamma_0 \gamma^\mu {A^{\mathrm{e}}}_\mu = {A^{\mathrm{e}}}_0 + \Be_k {A^{\mathrm{e}}}_k = {A^{\mathrm{e}}}_0 + \BA^{\mathrm{e}} \\
\gamma_0 \grad &= \gamma_0 \gamma^\mu \partial_\mu = \inv{v} \partial_t + \spacegrad \\
\grad \cdot {A^{\mathrm{e}}} &= \partial_\mu {A^{\mathrm{e}}}^\mu = \inv{v} \partial_t {A^{\mathrm{e}}}_0 – \spacegrad \cdot \BA^{\mathrm{e}},
\end{aligned}
\end{equation}

so multiplying from the left with \( \gamma_0 \) gives

\begin{equation}\label{eqn:potentialMethods:480}
\lr{ \inv{v^2} \partial_{tt} – \spacegrad^2 } \lr{ {A^{\mathrm{e}}}_0 + \BA^{\mathrm{e}} } – \lr{ \inv{v} \partial_t + \spacegrad }\lr{ \inv{v} \partial_t {A^{\mathrm{e}}}_0 – \spacegrad \cdot \BA^{\mathrm{e}} } = \eta( v q_e – \boldsymbol{\mathcal{J}} ),
\end{equation}

or

\begin{equation}\label{eqn:potentialMethods:520}
\lr{ \inv{v^2} \partial_{tt} – \spacegrad^2 } \BA^{\mathrm{e}} – \spacegrad \lr{ \inv{v} \partial_t {A^{\mathrm{e}}}_0 – \spacegrad \cdot \BA^{\mathrm{e}} } = -\eta \boldsymbol{\mathcal{J}}
\end{equation}
\begin{equation}\label{eqn:potentialMethods:540}
\spacegrad^2 {A^{\mathrm{e}}}_0 – \inv{v} \partial_t \lr{ \spacegrad \cdot \BA^{\mathrm{e}} } = -q_e/\epsilon.
\end{equation}

So \( {A^{\mathrm{e}}}_0 = \phi \) and \( -\ifrac{\BA^{\mathrm{e}}}{v} = \boldsymbol{\mathcal{A}}^{\mathrm{e}} \), or

\begin{equation}\label{eqn:potentialMethods:600}
\boxed{
{A^{\mathrm{e}}} = \gamma_0\lr{ \phi – v \boldsymbol{\mathcal{A}}^{\mathrm{e}} }.
}
\end{equation}

No electric sources

Without electric sources, Maxwell’s equation now splits into

\begin{equation}\label{eqn:potentialMethods:640}
\grad \cdot \boldsymbol{\mathcal{F}} = 0
\end{equation}
\begin{equation}\label{eqn:potentialMethods:660}
\grad \wedge \boldsymbol{\mathcal{F}} = -I M.
\end{equation}

Here the dual of an STA curl yields a solution

\begin{equation}\label{eqn:potentialMethods:680}
\boxed{
\boldsymbol{\mathcal{F}} = I ( \grad \wedge {A^{\mathrm{m}}} ).
}
\end{equation}

Substituting this gives

\begin{equation}\label{eqn:potentialMethods:720}
\begin{aligned}
0
&=
\grad \cdot (I ( \grad \wedge {A^{\mathrm{m}}} ) ) \\
&=
\gpgradeone{ \grad I ( \grad \wedge {A^{\mathrm{m}}} ) } \\
&=
-I \grad \wedge ( \grad \wedge {A^{\mathrm{m}}} ).
\end{aligned}
\end{equation}
\begin{equation}\label{eqn:potentialMethods:740}
\begin{aligned}
-I M
&=
\grad \wedge (I ( \grad \wedge {A^{\mathrm{m}}} ) ) \\
&=
\gpgradethree{ \grad I ( \grad \wedge {A^{\mathrm{m}}} ) } \\
&=
-I \grad \cdot ( \grad \wedge {A^{\mathrm{m}}} ).
\end{aligned}
\end{equation}

The \( \grad \cdot \boldsymbol{\mathcal{F}} \) relation \ref{eqn:potentialMethods:720} is identically zero as desired, leaving

\begin{equation}\label{eqn:potentialMethods:760}
\boxed{
\grad^2 {A^{\mathrm{m}}} – \grad \lr{ \grad \cdot {A^{\mathrm{m}}} }
=
M.
}
\end{equation}

So the general solution with both electric and magnetic sources is

\begin{equation}\label{eqn:potentialMethods:800}
\boxed{
\boldsymbol{\mathcal{F}} = \grad \wedge {A^{\mathrm{e}}} + I (\grad \wedge {A^{\mathrm{m}}}),
}
\end{equation}

subject to the constraints of \ref{eqn:potentialMethods:560} and \ref{eqn:potentialMethods:760}. As before the four-potential \( {A^{\mathrm{m}}} \) can be put into correspondence with the conventional scalar and vector potentials by left multiplying with \( \gamma_0 \), which gives

\begin{equation}\label{eqn:potentialMethods:820}
\lr{ \inv{v^2} \partial_{tt} – \spacegrad^2 } \lr{ {A^{\mathrm{m}}}_0 + \BA^{\mathrm{m}} } – \lr{ \inv{v} \partial_t + \spacegrad }\lr{ \inv{v} \partial_t {A^{\mathrm{m}}}_0 – \spacegrad \cdot \BA^{\mathrm{m}} } = v q_m – \boldsymbol{\mathcal{M}},
\end{equation}

or
\begin{equation}\label{eqn:potentialMethods:860}
\lr{ \inv{v^2} \partial_{tt} – \spacegrad^2 } \BA^{\mathrm{m}} – \spacegrad \lr{ \inv{v} \partial_t {A^{\mathrm{m}}}_0 – \spacegrad \cdot \BA^{\mathrm{m}} } = – \boldsymbol{\mathcal{M}}
\end{equation}
\begin{equation}\label{eqn:potentialMethods:880}
\spacegrad^2 {A^{\mathrm{m}}}_0 – \inv{v} \partial_t \spacegrad \cdot \BA^{\mathrm{m}} = -v q_m.
\end{equation}

Comparing with \ref{eqn:potentialMethods:260} shows that \( {A^{\mathrm{m}}}_0/v = \mu \phi_m \) and \( -\ifrac{\BA^{\mathrm{m}}}{v^2} = \mu \boldsymbol{\mathcal{A}}^{\mathrm{m}} \), or

\begin{equation}\label{eqn:potentialMethods:900}
\boxed{
{A^{\mathrm{m}}} = \gamma_0 \eta \lr{ \phi_m – v \boldsymbol{\mathcal{A}}^{\mathrm{m}} }.
}
\end{equation}

Potential operator structure

Observe that there is an underlying uniform structure of the differential operator that acts on the potential to produce the electromagnetic field. Expressed as a linear operator of the
gradient and the potentials, that is

\( \boldsymbol{\mathcal{F}} = L(\lrgrad, {A^{\mathrm{e}}}, {A^{\mathrm{m}}}) \)

\begin{equation}\label{eqn:potentialMethods:980}
\begin{aligned}
\boldsymbol{\mathcal{F}}
&=
L(\grad, {A^{\mathrm{e}}}, {A^{\mathrm{m}}}) \\
&= \grad \wedge {A^{\mathrm{e}}} + I (\grad \wedge {A^{\mathrm{m}}}) \\
&=
\inv{2} \lr{ \rgrad {A^{\mathrm{e}}} – {A^{\mathrm{e}}} \lgrad }
+ \frac{I}{2} \lr{ \rgrad {A^{\mathrm{m}}} – {A^{\mathrm{m}}} \lgrad } \\
&=
\inv{2} \lr{ \rgrad {A^{\mathrm{e}}} – {A^{\mathrm{e}}} \lgrad }
+ \frac{1}{2} \lr{ -\rgrad I {A^{\mathrm{m}}} – I {A^{\mathrm{m}}} \lgrad } \\
&=
\inv{2} \lr{ \rgrad ({A^{\mathrm{e}}} -I {A^{\mathrm{m}}}) – ({A^{\mathrm{e}}} + I {A^{\mathrm{m}}}) \lgrad }
,
\end{aligned}
\end{equation}

or
\begin{equation}\label{eqn:potentialMethods:1000}
\boxed{
\boldsymbol{\mathcal{F}}
=
\inv{2} \lr{ \rgrad ({A^{\mathrm{e}}} -I {A^{\mathrm{m}}}) – ({A^{\mathrm{e}}} – I {A^{\mathrm{m}}})^\dagger \lgrad }
.
}
\end{equation}

Observe that \ref{eqn:potentialMethods:1000} can be
put into correspondence with \ref{eqn:potentialMethods:1080} using a factoring of unity \( 1 = \gamma_0 \gamma_0 \)

\begin{equation}\label{eqn:potentialMethods:1100}
\boldsymbol{\mathcal{F}}
=
\inv{2} \lr{ (-\rgrad \gamma_0) (-\gamma_0 ({A^{\mathrm{e}}} -I {A^{\mathrm{m}}})) – (({A^{\mathrm{e}}} + I {A^{\mathrm{m}}}) \gamma_0)(\gamma_0 \lgrad) },
\end{equation}

where

\begin{equation}\label{eqn:potentialMethods:1140}
\begin{aligned}
-\grad \gamma_0
&=
-(\gamma^0 \partial_0 + \gamma^k \partial_k) \gamma_0 \\
&=
-\partial_0 – \gamma^k \gamma_0 \partial_k \\
&=
\spacegrad
-\inv{v} \partial_t
,
\end{aligned}
\end{equation}
\begin{equation}\label{eqn:potentialMethods:1160}
\begin{aligned}
\gamma_0 \grad
&=
\gamma_0 (\gamma^0 \partial_0 + \gamma^k \partial_k) \\
&=
\partial_0 – \gamma^k \gamma_0 \partial_k \\
&=
\spacegrad
+ \inv{v} \partial_t
,
\end{aligned}
\end{equation}

and
\begin{equation}\label{eqn:potentialMethods:1200}
\begin{aligned}
-\gamma_0 ( {A^{\mathrm{e}}} – I {A^{\mathrm{m}}} )
&=
-\gamma_0 \gamma_0 \lr{ \phi -v \boldsymbol{\mathcal{A}}^{\mathrm{e}} + \eta I \lr{ \phi_m – v \boldsymbol{\mathcal{A}}^{\mathrm{m}} } } \\
&=
-\lr{ \phi -v \boldsymbol{\mathcal{A}}^{\mathrm{e}} + \eta I \phi_m – \eta v I \boldsymbol{\mathcal{A}}^{\mathrm{m}} } \\
&=
– \phi
+ v \boldsymbol{\mathcal{A}}^{\mathrm{e}}
+ \eta v I \boldsymbol{\mathcal{A}}^{\mathrm{m}}
– \eta I \phi_m
\end{aligned}
\end{equation}
\begin{equation}\label{eqn:potentialMethods:1220}
\begin{aligned}
( {A^{\mathrm{e}}} + I {A^{\mathrm{m}}} )\gamma_0
&=
\lr{ \gamma_0 \lr{ \phi -v \boldsymbol{\mathcal{A}}^{\mathrm{e}} } + I \gamma_0 \eta \lr{ \phi_m – v \boldsymbol{\mathcal{A}}^{\mathrm{m}} } } \gamma_0 \\
&=
\phi + v \boldsymbol{\mathcal{A}}^{\mathrm{e}} + I \eta \phi_m + I \eta v \boldsymbol{\mathcal{A}}^{\mathrm{m}} \\
&=
\phi
+ v \boldsymbol{\mathcal{A}}^{\mathrm{e}}
+ \eta v I \boldsymbol{\mathcal{A}}^{\mathrm{m}}
+ \eta I \phi_m
,
\end{aligned}
\end{equation}

This recovers \ref{eqn:potentialMethods:1080} as desired.

Potentials in the 3D Euclidean formalism

In the conventional scalar plus vector differential representation of Maxwell’s equations \ref{eqn:chapter3Notes:20}…, given electric(magnetic) sources the structure of the electric(magnetic) potential follows from first setting the magnetic(electric) field equal to the curl of a vector potential. The procedure for the STA GA form of Maxwell’s equation was similar, where it was immediately evident that the field could be set to the four-curl of a four-vector potential (or the dual of such a curl for magnetic sources).

In the 3D GA representation, there is no immediate rationale for introducing a curl or the equivalent to a four-curl representation of the field. Reconciliation of this is possible by recognizing that the fact that the field (or a component of it) may be represented by a curl is not actually fundamental. Instead, observe that the two sided gradient action on a potential to generate the electromagnetic field in the STA representation of \ref{eqn:potentialMethods:1000} serves to select the grade two component product of the gradient and the multivector potential \( {A^{\mathrm{e}}} – I {A^{\mathrm{m}}} \), and that this can in fact be written as
a single sided gradient operation on a potential, provided the multivector product is filtered with a four-bivector grade selection operation

\begin{equation}\label{eqn:potentialMethods:1240}
\boxed{
\boldsymbol{\mathcal{F}} = \gpgradetwo{ \grad \lr{ {A^{\mathrm{e}}} – I {A^{\mathrm{m}}} } }.
}
\end{equation}

Similarly, it can be observed that the
specific function of the conjugate structure in the two sided potential representation of
\ref{eqn:potentialMethods:1080}
is to discard all the scalar and pseudoscalar grades in the multivector product. This means that a single sided potential can also be used, provided it is wrapped in a grade selection operation

\begin{equation}\label{eqn:potentialMethods:1260}
\boxed{
\boldsymbol{\mathcal{F}} =
\gpgrade{ \lr{ \spacegrad – \inv{v} \PD{t}{} }
\lr{
– \phi
+ v \boldsymbol{\mathcal{A}}^{\mathrm{e}}
+ \eta I v \boldsymbol{\mathcal{A}}^{\mathrm{m}}
– \eta I \phi_m
} }{1,2}.
}
\end{equation}

It is this grade selection operation that is really the fundamental defining action in the potential of the STA and conventional 3D representations of Maxwell’s equations. So, given Maxwell’s equation in the 3D GA representation, defining a potential representation for the field is really just a demand that the field have the structure

\begin{equation}\label{eqn:potentialMethods:1320}
\boldsymbol{\mathcal{F}} = \gpgrade{ (\alpha \spacegrad + \beta \partial_t)( A_0 + A_1 + I( A_0′ + A_1′ ) }{1,2}.
\end{equation}

This is a mandate that the electromagnetic field is the grades 1 and 2 components of the vector product of space and time derivative operators on a multivector field \( A = \sum_{k=0}^3 A_k = A_0 + A_1 + I( A_0′ + A_1′ ) \) that can potentially have any grade components. There are more degrees of freedom in this specification than required, since the multivector can absorb one of the \( \alpha \) or \( \beta \) coefficients, so without loss of generality, one of these (say \( \alpha\)) can be set to 1.

Expanding \ref{eqn:potentialMethods:1320} gives

\begin{equation}\label{eqn:potentialMethods:1340}
\begin{aligned}
\boldsymbol{\mathcal{F}}
&=
\spacegrad A_0
+ \beta \partial_t A_1
– \spacegrad \cross A_1′
+ I (\spacegrad \cross A_1
+ \beta \partial_t A_1′
+ \spacegrad A_0′) \\
&=
\boldsymbol{\mathcal{E}} + I \eta \boldsymbol{\mathcal{H}}.
\end{aligned}
\end{equation}

This naturally has all the right mixes of curls, gradients and time derivatives, all following as direct consequences of applying a grade selection operation to the action of a “spacetime gradient” on a general multivector potential.

The conclusion is that the potential representation of the field is

\begin{equation}\label{eqn:potentialMethods:1360}
\boldsymbol{\mathcal{F}} =
\gpgrade{ \lr{ \spacegrad – \inv{v} \PD{t}{} } A }{1,2},
\end{equation}

where \( A \) is a multivector potentially containing all grades, where grades 0,1 are required for electric sources, and grades 2,3 are required for magnetic sources. When it is desirable to refer back to the conventional scalar and vector potentials this multivector potential can be written as \( A = -\phi + v \boldsymbol{\mathcal{A}}^{\mathrm{e}} + \eta I \lr{ -\phi_m + v \boldsymbol{\mathcal{A}}^{\mathrm{m}} } \).

Gauge transformations

Recall that for electric sources the magnetic field is of the form

\begin{equation}\label{eqn:potentialMethods:1380}
\boldsymbol{\mathcal{B}} = \spacegrad \cross \boldsymbol{\mathcal{A}},
\end{equation}

so adding the gradient of any scalar field to the potential \( \boldsymbol{\mathcal{A}}’ = \boldsymbol{\mathcal{A}} + \spacegrad \psi \)
does not change the magnetic field

\begin{equation}\label{eqn:potentialMethods:1400}
\begin{aligned}
\boldsymbol{\mathcal{B}}’
&= \spacegrad \cross \lr{ \boldsymbol{\mathcal{A}} + \spacegrad \psi } \\
&= \spacegrad \cross \boldsymbol{\mathcal{A}} \\
&= \boldsymbol{\mathcal{B}}.
\end{aligned}
\end{equation}

The electric field with this changed potential is

\begin{equation}\label{eqn:potentialMethods:1420}
\begin{aligned}
\boldsymbol{\mathcal{E}}’
&= -\spacegrad \phi – \partial_t \lr{ \BA + \spacegrad \psi} \\
&= -\spacegrad \lr{ \phi + \partial_t \psi } – \partial_t \BA,
\end{aligned}
\end{equation}

so if
\begin{equation}\label{eqn:potentialMethods:1440}
\phi = \phi’ – \partial_t \psi,
\end{equation}

the electric field will also be unaltered by this transformation.

In the STA representation, the field can similarly be altered by adding any (four)gradient to the potential. For example with only electric sources

\begin{equation}\label{eqn:potentialMethods:1460}
\boldsymbol{\mathcal{F}} = \grad \wedge (A + \grad \psi) = \grad \wedge A
\end{equation}

and for electric or magnetic sources

\begin{equation}\label{eqn:potentialMethods:1480}
\boldsymbol{\mathcal{F}} = \gpgradetwo{ \grad (A + \grad \psi) } = \gpgradetwo{ \grad A }.
\end{equation}

In the 3D GA representation, where the field is given by \ref{eqn:potentialMethods:1360}, there is no field that is being curled to add a gradient to. However, if the scalar and vector potentials transform as

\begin{equation}\label{eqn:potentialMethods:1500}
\begin{aligned}
\boldsymbol{\mathcal{A}} &\rightarrow \boldsymbol{\mathcal{A}} + \spacegrad \psi \\
\phi &\rightarrow \phi – \partial_t \psi,
\end{aligned}
\end{equation}

then the multivector potential transforms as
\begin{equation}\label{eqn:potentialMethods:1520}
-\phi + v \boldsymbol{\mathcal{A}}
\rightarrow -\phi + v \boldsymbol{\mathcal{A}} + \partial_t \psi + v \spacegrad \psi,
\end{equation}

so the electromagnetic field is unchanged when the multivector potential is transformed as

\begin{equation}\label{eqn:potentialMethods:1540}
A \rightarrow A + \lr{ \spacegrad + \inv{v} \partial_t } \psi,
\end{equation}

where \( \psi \) is any field that has scalar or pseudoscalar grades. Viewed in terms of grade selection, this makes perfect sense, since the transformed field is

\begin{equation}\label{eqn:potentialMethods:1560}
\begin{aligned}
\boldsymbol{\mathcal{F}}
&\rightarrow
\gpgrade{ \lr{ \spacegrad – \inv{v} \PD{t}{} } \lr{ A + \lr{ \spacegrad + \inv{v} \partial_t } \psi } }{1,2} \\
&=
\gpgrade{ \lr{ \spacegrad – \inv{v} \PD{t}{} } A + \lr{ \spacegrad^2 – \inv{v^2} \partial_{tt} } \psi }{1,2} \\
&=
\gpgrade{ \lr{ \spacegrad – \inv{v} \PD{t}{} } A }{1,2}.
\end{aligned}
\end{equation}

The \( \psi \) contribution to the grade selection operator is killed because it has scalar or pseudoscalar grades.

Lorenz gauge

Maxwell’s equations are completely decoupled if the potential can be found such that

\begin{equation}\label{eqn:potentialMethods:1580}
\begin{aligned}
\boldsymbol{\mathcal{F}}
&=
\gpgrade{ \lr{ \spacegrad – \inv{v} \PD{t}{} } A }{1,2} \\
&=
\lr{ \spacegrad – \inv{v} \PD{t}{} } A.
\end{aligned}
\end{equation}

When this is the case, Maxwell’s equations are reduced to four non-homogeneous potential wave equations

\begin{equation}\label{eqn:potentialMethods:1620}
\lr{ \spacegrad^2 – \inv{v^2} \PDSq{t}{} } A = J,
\end{equation}

that is

\begin{equation}\label{eqn:potentialMethods:1600}
\begin{aligned}
\lr{ \spacegrad^2 – \inv{v^2} \PDSq{t}{} } \phi &= – \inv{\epsilon} q_e \\
\lr{ \spacegrad^2 – \inv{v^2} \PDSq{t}{} } \boldsymbol{\mathcal{A}}^{\mathrm{e}} &= – \mu \boldsymbol{\mathcal{J}} \\
\lr{ \spacegrad^2 – \inv{v^2} \PDSq{t}{} } \phi_m &= – \frac{I}{\mu} q_m \\
\lr{ \spacegrad^2 – \inv{v^2} \PDSq{t}{} } \boldsymbol{\mathcal{A}}^{\mathrm{m}} &= – I \epsilon \boldsymbol{\mathcal{M}}.
\end{aligned}
\end{equation}

There should be no a-priori assumption that such a field representation has no scalar, nor no pseudoscalar components. That explicit expansion in grades is

\begin{equation}\label{eqn:potentialMethods:1640}
\begin{aligned}
\lr{ \spacegrad – \inv{v} \PD{t}{} } A
&=
\lr{ \spacegrad – \inv{v} \PD{t}{} } \lr{ -\phi + v \boldsymbol{\mathcal{A}}^{\mathrm{e}} + \eta I \lr{ -\phi_m + v \boldsymbol{\mathcal{A}}^{\mathrm{m}} } } \\
&=
\inv{v} \partial_t \phi
+ v \spacegrad \cdot \boldsymbol{\mathcal{A}}^{\mathrm{e}} \\
&-\spacegrad \phi
+ I \eta v \spacegrad \wedge \boldsymbol{\mathcal{A}}^{\mathrm{m}}
– \partial_t \boldsymbol{\mathcal{A}}^{\mathrm{e}} \\
&+ v \spacegrad \wedge \boldsymbol{\mathcal{A}}^{\mathrm{e}}
– \eta I \spacegrad \phi_m
– I \eta \partial_t \boldsymbol{\mathcal{A}}^{\mathrm{m}} \\
&+ \eta I \inv{v} \partial_t \phi_m
+ I \eta v \spacegrad \cdot \boldsymbol{\mathcal{A}}^{\mathrm{m}},
\end{aligned}
\end{equation}

so if this potential representation has only vector and bivector grades, it must be true that

\begin{equation}\label{eqn:potentialMethods:1660}
\begin{aligned}
\inv{v} \partial_t \phi + v \spacegrad \cdot \boldsymbol{\mathcal{A}}^{\mathrm{e}} &= 0 \\
\inv{v} \partial_t \phi_m + v \spacegrad \cdot \boldsymbol{\mathcal{A}}^{\mathrm{m}} &= 0.
\end{aligned}
\end{equation}

The first is the well known Lorenz gauge condition, whereas the second is the dual of that condition for magnetic sources.

Should one of these conditions, say the Lorenz condition for the electric source potentials, be non-zero, then it is possible to make a potential transformation for which this condition is zero

\begin{equation}\label{eqn:potentialMethods:1680}
\begin{aligned}
0
&\ne
\inv{v} \partial_t \phi + v \spacegrad \cdot \boldsymbol{\mathcal{A}}^{\mathrm{e}} \\
&=
\inv{v} \partial_t (\phi’ – \partial_t \psi) + v \spacegrad \cdot (\boldsymbol{\mathcal{A}}’ + \spacegrad \psi) \\
&=
\inv{v} \partial_t \phi’ + v \spacegrad \boldsymbol{\mathcal{A}}’
+ v \lr{ \spacegrad^2 – \inv{v^2} \partial_{tt} } \psi,
\end{aligned}
\end{equation}

so if \( \inv{v} \partial_t \phi’ + v \spacegrad \boldsymbol{\mathcal{A}}’ \) is zero, \( \psi \) must be found such that
\begin{equation}\label{eqn:potentialMethods:1700}
\inv{v} \partial_t \phi + v \spacegrad \cdot \boldsymbol{\mathcal{A}}^{\mathrm{e}}
= v \lr{ \spacegrad^2 – \inv{v^2} \partial_{tt} } \psi.
\end{equation}

References

[1] Constantine A Balanis. Antenna theory: analysis and design. John Wiley \& Sons, 3rd edition, 2005.

[2] C. Doran and A.N. Lasenby. Geometric algebra for physicists. Cambridge University Press New York, Cambridge, UK, 1st edition, 2003.

[3] David M Pozar. Microwave engineering. John Wiley \& Sons, 2009.

Updated notes for ece1229 antenna theory

March 16, 2015 ece1229 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,

I’ve now posted a first update of my notes for the antenna theory course that I am taking this term at UofT.

Unlike most of the other classes I have taken, I am not attempting to take comprehensive notes for this class. The class is taught on slides which go by faster than I can easily take notes for (and some of which match the textbook closely). In class I have annotated my copy of textbook with little details instead. This set of notes contains musings of details that were unclear, or in some cases, details that were provided in class, but are not in the text (and too long to pencil into my book), as well as some notes Geometric Algebra formalism for Maxwell’s equations with magnetic sources (something I’ve encountered for the first time in any real detail in this class).

The notes compilation linked above includes all of the following separate notes, some of which have been posted separately on this blog:

Duality transformation of the far field fields.

February 27, 2015 ece1229 , , , ,

We’ve seen that the far field electric and magnetic fields associated with a magnetic vector potential were

\begin{equation}\label{eqn:dualFarField:40}
\BE = -j \omega \textrm{Proj}_\T \BA,
\end{equation}
\begin{equation}\label{eqn:dualFarField:60}
\BH = \inv{\eta} \kcap \cross \BE.
\end{equation}

It’s worth a quick note that the duality transformation for this, referring to [1] tab. 3.2, is

\begin{equation}\label{eqn:dualFarField:100}
\BH = -j \omega \textrm{Proj}_\T \BF
\end{equation}
\begin{equation}\label{eqn:dualFarField:120}
\BE = -\eta \kcap \cross \BH.
\end{equation}

What does \( \BH \) look like in terms of \( \BA \), and \( \BE \) look like in terms of \( \BH \)?

The first is

\begin{equation}\label{eqn:dualFarField:140}
\BH
= -\frac{j \omega}{\eta} \kcap \cross \lr{ \BA – \lr{\BA \cdot \kcap} \kcap },
\end{equation}

in which the \( \kcap \) crossed terms are killed, leaving

\begin{equation}\label{eqn:dualFarField:160}
\BH
= -\frac{j \omega}{\eta} \kcap \cross \BA.
\end{equation}

The electric field follows again using a duality transformation, so in terms of the electric vector potential, is

\begin{equation}\label{eqn:dualFarField:180}
\BE = j \omega \eta \kcap \cross \BF.
\end{equation}

These show explicitly that neither the electric or magnetic far field have any radial component, matching with intuition for transverse propagation of the fields.

References

[1] Constantine A Balanis. Antenna theory: analysis and design. John Wiley & Sons, 3rd edition, 2005.

Notes for Balantis chapter 4: linear wire antennas.

February 16, 2015 ece1229 , , , , , , , , , , , , , ,

[Click here for a PDF of this post with nicer formatting]

These are notes for the UofT course ECE1229, Advanced Antenna Theory, taught by Prof. Eleftheriades, covering ch. 4 [1] content.

Unlike most of the other classes I have taken, I am not attempting to take comprehensive notes for this class. The class is taught on slides that match the textbook so closely, there is little value to me taking notes that just replicate the text. Instead, I am annotating my copy of textbook with little details instead. My usual notes collection for the class will contain musings of details that were unclear, or in some cases, details that were provided in class, but are not in the text (and too long to pencil into my book.)

Magnetic Vector Potential.

In class and in the problem set \( \BA \) was referred to as the Magnetic Vector Potential.  I only recalled this referred to as the Vector Potential.  Prefixing this with magnetic seemed counter intuitive to me since it is generated by electric sources (charges and currents).
This terminology can be justified due to the fact that \( \BA \) generates the magnetic field by its curl. Some mention of this can be found in [4], which also points out that the Electric Potential refers to the scalar \( \phi \). Prof. Eleftheriades points out that Electric Vector Potential refers to the vector potential \( \BF \) generated by magnetic sources (because in that case the electric field is generated by the curl of \( \BF \).)

Plots of infinitesimal dipole radial dependence.

In section 4.2 of [1] are some discussions of the \( kr < 1 \), \( kr = 1 \), and \( kr > 1 \) radial dependence of the fields and power of a solution to an infinitesimal dipole system. Here are some plots of those \( k r \) dependence, along with the \( k r = 1 \) contour as a reference. All the \( \theta \) dependence and any scaling is left out.

The CDF notebook visualizeDipoleFields.cdf is available to interactively plot these, rotate the plots and change the ranges of what is plotted.

A plot of the real and imaginary parts of \( H_\phi = \frac{j k}{r} e^{-j k r} \lr{ 1-\frac{j}{k r} } \) can be found in fig. 1 and fig. 2.

infinitesimalDipoleHphiRealFig3pn

fig 1. Radial dependence of Re H_phi

infinitesimalDipoleHphiImagFig4pn

fig 2. Radial dependence of Im H_phi

 

A plot of the real and imaginary parts of \( E_r = \inv{r^2} \lr{1-\frac{j}{k r}} e^{-j k r} \) can be found in fig. 3 and fig. 4.

infinitesimalDipoleErRealFig1pn

fig 3. Radial dependence of Re E_r

infinitesimalDipoleErImagFig2pn

fig 4. Radial dependence of Im E_r

 

Finally, a plot of the real and imaginary parts of \( E_\theta = \frac{ j k }{r} \lr{1 -\frac{j}{k r} -\frac{1}{k^2 r^2} } e^{-j k r} \) can be found in fig. 5 and fig. 6.

infinitesimalDipoleEthetaRealFig5pn

fig. 5. Radial dependence of Re E_theta

infinitesimalDipoleEthetaImagFig6pn

fig. 6. Radial dependence of Im E_theta

 

Electric Far field for a spherical potential.

It is interesting to look at the far electric field associated with an arbitrary spherical magnetic vector potential, assuming all of the radial dependence is in the spherical envelope. That is

\begin{equation}\label{eqn:chapter4Notes:20}
\BA = \frac{e^{-j k r}}{r} \lr{
\rcap a_r\lr{ \theta, \phi }
+\thetacap a_\theta\lr{ \theta, \phi }
+\phicap a_\phi\lr{ \theta, \phi }
}.
\end{equation}

The electric field is

\begin{equation}\label{eqn:chapter4Notes:40}
\BE = – j \omega \BA – j \frac{1}{\omega \mu_0 \epsilon_0 } \spacegrad \lr{\spacegrad \cdot \BA }.
\end{equation}

The divergence and gradient in spherical coordinates are

\begin{equation}\label{eqn:chapter4Notes:80}
\begin{aligned}
\spacegrad \cdot \BA
&=
\inv{r^2} \PD{r}{} \lr{ r^2 A_r }
+ \inv{r \sin\theta } \PD{\theta}{} \lr{A_\theta \sin\theta}
+ \inv{r \sin\theta } \PD{\phi}{A_\phi}
\end{aligned}
\end{equation}

\begin{equation}\label{eqn:chapter4Notes:100}
\begin{aligned}
\spacegrad \psi \\
&=
\rcap \PD{r}{\psi}
+\frac{\thetacap}{r} \PD{\theta}{\psi}
+ \frac{\phicap}{r \sin\theta} \PD{\phi}{\psi}.
\end{aligned}
\end{equation}

For the assumed potential, the divergence is

\begin{equation}\label{eqn:chapter4Notes:120}
\begin{aligned}
\spacegrad \cdot \BA
&=
\frac{a_r}{r^2} \PD{r}{} \lr{ r^2 \frac{e^{-j k r}}{r} }
+ \inv{r \sin\theta } \frac{e^{-j k r}}{r} \PD{\theta}{} \lr{\sin\theta a_\theta}
+ \inv{r \sin\theta } \frac{e^{-j k r}}{r} \PD{\phi}{a_\phi} \\
&=
a_r
e^{-j k r}
\lr{
\inv{r^2}
-j k \inv{r}
}
+ \inv{r^2 \sin\theta } e^{-j k r} \PD{\theta}{} \lr{\sin\theta a_\theta}
+ \inv{r^2 \sin\theta } e^{-j k r} \PD{\phi}{a_\phi} \\
&\approx
-j k \frac{a_r}{r}
e^{-j k r}.
\end{aligned}
\end{equation}

The last approximation dropped all the \( 1/r^2 \) terms that will be small compared to \( 1/r \) contribution that dominates when \( r \rightarrow \infty \), the far field.

The gradient can now be computed

\begin{equation}\label{eqn:chapter4Notes:140}
\begin{aligned}
\spacegrad \lr{\spacegrad \cdot \BA }
&\approx
-j k
\spacegrad
\lr{
\frac{a_r}{r}
e^{-j k r}
} \\
&=
-j k \lr{
\rcap \PD{r}{}
+\frac{\thetacap}{r} \PD{\theta}{}
+ \frac{\phicap}{r \sin\theta} \PD{\phi}{}
}
\frac{a_r}{r}
e^{-j k r} \\
&=
-j k \lr{
\rcap a_r \PD{r}{} \lr{
\frac{1}{r}
e^{-j k r}
}
+\frac{\thetacap}{r^2}
e^{-j k r}
\PD{\theta}{a_r}
+
e^{-j k r}
\frac{\phicap}{r^2 \sin\theta}
\PD{\phi}{a_r}
} \\
&=
-j k \lr{
-\rcap \frac{a_r}{r^2} \lr{
1
+ j k r
}
+\frac{\thetacap}{r^2}
\PD{\theta}{a_r}
+
\frac{\phicap}{r^2 \sin\theta}
\PD{\phi}{a_r}
}
e^{-j k r} \\
&\approx
– k^2 \rcap \frac{a_r}{r}
e^{-j k r}.
\end{aligned}
\end{equation}

Again, a far field approximation has been used to kill all the \( 1/r^2 \) terms.

The far field approximation of the electric field is now possible

\begin{equation}\label{eqn:chapter4Notes:160}
\begin{aligned}
\BE
&= – j \omega \BA – j \frac{1}{\omega \mu_0 \epsilon_0 } \spacegrad \lr{\spacegrad \cdot \BA } \\
&=
– j \omega
\frac{e^{-j k r}}{r} \lr{
\rcap a_r\lr{ \theta, \phi }
+\thetacap a_\theta\lr{ \theta, \phi }
+\phicap a_\phi\lr{ \theta, \phi }
}
+ j \frac{1}{\omega \mu_0 \epsilon_0 }
k^2 \rcap \frac{a_r}{r}
e^{-j k r} \\
&=
– j \omega
\frac{e^{-j k r}}{r} \lr{
\rcap a_r\lr{ \theta, \phi }
+\thetacap a_\theta\lr{ \theta, \phi }
+\phicap a_\phi\lr{ \theta, \phi }
}
+ j \frac{c^2}{\omega }
\lr{\frac{\omega}{c}}^2 \rcap \frac{a_r}{r}
e^{-j k r}
\\
&=
– j \omega
\frac{e^{-j k r}}{r} \lr{
\thetacap a_\theta\lr{ \theta, \phi }
+\phicap a_\phi\lr{ \theta, \phi }
}.
\end{aligned}
\end{equation}

Observe the perfect, somewhat miraculous seeming, cancellation of all the radial components of the field. If \( \BA_{\textrm{T}} \) is the non-radial projection of \( \BA \), the electric far field is just

\begin{equation}\label{eqn:chapter4Notes:180}
\boxed{
\BE_{\textrm{ff}} = -j \omega \BA_{\textrm{T}}.
}
\end{equation}

Magnetic Far field for a spherical potential.

Application of the same assumed representation for the magnetic field gives
\begin{equation}\label{eqn:chapter4Notes:220}
\begin{aligned}
\BB
&=
\spacegrad \cross \BA \\
&=
\frac{\rcap}{r \sin\theta} \partial_\theta \lr{A_\phi \sin\theta}
+ \frac{\thetacap}{r} \lr{ \inv{\sin\theta} \partial_\phi A_r – \partial_r \lr{r A_\phi}}
+ \frac{\phicap}{r} \lr{ \partial_r\lr{r A_\theta} – \partial_\theta A_r} \\
&=
\frac{\rcap}{r \sin\theta} \partial_\theta \lr{
\frac{e^{-j k r}}{r} a_\phi
\sin\theta}
+ \frac{\thetacap}{r} \lr{ \inv{\sin\theta} \partial_\phi \lr{
\frac{e^{-j k r}}{r} a_r
} – \partial_r \lr{r
\frac{e^{-j k r}}{r} a_\phi
}
}
+ \frac{\phicap}{r} \lr{ \partial_r\lr{r
\frac{e^{-j k r}}{r} a_\theta
} – \partial_\theta
\lr{
\frac{e^{-j k r}}{r} a_r
}
} \\
&=
\frac{\rcap}{r \sin\theta}
\frac{e^{-j k r}}{r}
\partial_\theta \lr{
a_\phi
\sin\theta}
+ \frac{\thetacap}{r} \lr{ \inv{\sin\theta}
\frac{e^{-j k r}}{r}
\partial_\phi
a_r
– \partial_r \lr{
e^{-j k r}
}
a_\phi
}
+ \frac{\phicap}{r} \lr{
\partial_r
\lr{
e^{-j k r}
}
a_\theta
–
\frac{e^{-j k r}}{r}
\partial_\theta
a_r
}
\approx
j k \lr{ \thetacap a_\phi
–
\phicap a_\theta
}
\frac{e^{-j k r}}{r} \\
&=
-j k \rcap \cross \lr{
\thetacap a_\theta
+\phicap a_\phi
}
\frac{e^{-j k r}}{r} \\
&=
\inv{c} \BE_{\textrm{ff}}.
\end{aligned}
\end{equation}

The approximation above drops the \( 1/r^2 \) terms. Since

\begin{equation}\label{eqn:chapter4Notes:240}
\inv{\mu_0 c} = \inv{\mu_0} \sqrt{\mu_0\epsilon_0} = \sqrt{\frac{\epsilon_0}{\mu_0}} = \inv{\eta},
\end{equation}

the magnetic far field can be expressed in terms of the electric far field as
\begin{equation}\label{eqn:chapter4Notes:260}
\boxed{
\BH = \inv{\eta} \rcap \cross \BE.
}
\end{equation}

Plane wave relations between electric and magnetic fields

I recalled an identity of the form \ref{eqn:chapter4Notes:260} in [3], but didn’t think that it required a far field approximation.
The reason for this was because the Jackson identity assumed a plane wave representation of the field, something that the far field assumptions also locally require.

Assuming a plane wave representation for both fields

\begin{equation}\label{eqn:chapter4Notes:300}
\boldsymbol{\mathcal{E}}(\Bx, t) = \BE e^{j \lr{\omega t – \Bk \cdot \Bx}}
\end{equation}
\begin{equation}\label{eqn:chapter4Notes:320}
\boldsymbol{\mathcal{B}}(\Bx, t) = \BB e^{j \lr{\omega t – \Bk \cdot \Bx}}
\end{equation}

The cross product relation between the fields follows from the Maxwell-Faraday law of induction

\begin{equation}\label{eqn:chapter4Notes:340}
0 = \spacegrad \cross \boldsymbol{\mathcal{E}} + \PD{t}{\boldsymbol{\mathcal{B}}},
\end{equation}

or

\begin{equation}\label{eqn:chapter4Notes:360}
\begin{aligned}
0
&=
\Be_r \cross \BE \partial_r e^{j\lr{ \omega t – \Bk \cdot \Bx}}
+
j \omega \BB e^{j \lr{\omega t – \Bk \cdot \Bx}} \\
&=
-j \Be_r k_r \cross \BE e^{j \lr{\omega t – \Bk \cdot \Bx}}
+
j \omega \BB e^{j \lr{\omega t – \Bk \cdot \Bx}} \\
&=
\lr{ – \Bk \cross \BE + \omega \BB } j
e^{j \lr{\omega t – \Bk \cdot \Bx}},
\end{aligned}
\end{equation}

or

\begin{equation}\label{eqn:chapter4Notes:380}
\begin{aligned}
\BH
&= \frac{ k}{k c \mu_0 } \kcap \cross \BE \\
&= \inv{ \eta } \kcap \cross \BE,
\end{aligned}
\end{equation}

which also finds \ref{eqn:chapter4Notes:260}, but with much less work and less mess.

Transverse only nature of the far-field fields

Also observe that its possible to tell that the far field fields have only transverse components using the same argument that they are locally plane waves at that distance. The plane waves must satisfy the zero divergence Maxwell’s equations

\begin{equation}\label{eqn:chapter4Notes:420}
\spacegrad \cdot \boldsymbol{\mathcal{E}} = 0
\end{equation}
\begin{equation}\label{eqn:chapter4Notes:440}
\spacegrad \cdot \boldsymbol{\mathcal{B}} = 0,
\end{equation}

so by the same logic

\begin{equation}\label{eqn:chapter4Notes:480}
\Bk \cdot \BE = 0
\end{equation}
\begin{equation}\label{eqn:chapter4Notes:500}
\Bk \cdot \BB = 0.
\end{equation}

In the far field the electric field must equal its transverse projection

\begin{equation}\label{eqn:chapter4Notes:520}
\BE = \textrm{Proj}_\T \lr{-j \omega \BA
– j \frac{1}{\omega \mu_0 \epsilon_0 } \spacegrad \lr{\spacegrad \cdot \BA } }.
\end{equation}

Since by \ref{eqn:chapter4Notes:140} the scalar potential term has only a radial component, that leaves

\begin{equation}\label{eqn:chapter4Notes:540}
\BE = -j \omega \textrm{Proj}_\T \BA,
\end{equation}

which provides \ref{eqn:chapter4Notes:180} with slightly less work.

Vertical dipole reflection coefficient

In class a ground reflection scenario was covered for a horizontal dipole. Reading the text I was surprised to see what looked like the same sort of treatment section 4.7.2, but ending up with a quite different result. It turns out the difference is because the text was treating the vertical dipole configuration, whereas Prof. Eleftheriades was treating a horizontal dipole configuration, which have different reflection coefficients. These differing reflection coefficients are due to differences in the polarization of the field.

To understand these differences in reflection coefficients, consider first the field due to a vertical dipole as sketched in fig. 7, with a wave vector directed from the transmission point downwards in the z-y plane.

verticalDipoleConfigurationFig1

fig. 7. vertical dipole configuration.

 

The wave vector has direction

\begin{equation}\label{eqn:chapter4Notes:560}
\kcap = \zcap e^{\zcap \xcap \theta} = \zcap \cos\theta + \ycap \sin\theta.
\end{equation}

Suppose that the (magnetic) vector potential is that of an infinitesimal dipole

\begin{equation}\label{eqn:chapter4Notes:580}
\BA = \zcap \frac{\mu_0 I_0 l}{4 \pi r} e^{-j k r} %= \frac{A_r}{4 \pi r} e^{-j k r}
\end{equation}

The electric field, in the far field, can be computed by computing the normal projection to the wave vector direction

\begin{equation}\label{eqn:chapter4Notes:600}
\begin{aligned}
\BE
&= -j \omega \lr{\BA \wedge \kcap} \cdot \kcap \\
&= -j \omega \frac{\mu_0 I_0 l}{4 \pi r} \lr{\zcap \wedge \lr{\zcap \cos\theta
+ \ycap \sin\theta} } \lr{\zcap \cos\theta + \ycap \sin\theta} \\
&= -j \omega \frac{\mu_0 I_0 l}{4 \pi r} \lr{ \zcap \ycap \sin\theta }
\lr{\zcap \cos\theta + \ycap \sin\theta} \\
&= -j \omega \frac{\mu_0 I_0 l}{4 \pi r} \sin\theta \lr{-\ycap \cos\theta +
\zcap \sin\theta} \\
&= j \omega \frac{\mu_0 I_0 l}{4 \pi r} \sin\theta \ycap e^{\zcap \ycap \theta}.
\end{aligned}
\end{equation}

This is directed in the z-y plane rotated an additional \( \pi/2 \) past \( \kcap \). The magnetic field must then be directed into the page, along the x axis. This is sketched in fig. 8.

verticalDipoleConfigurationFig2

fig. 8. Electric and magnetic field directions

 

Referring to [2] (\eqntext 4.40) for the coefficient of reflection component

\begin{equation}\label{eqn:chapter4Notes:620}
R
=
\frac{
n_t \cos\theta_i – n_i \cos\theta_t
}
{
n_i \cos\theta_i + n_t \cos\theta_t
}
\end{equation}

This is the Fresnel equation for the case when
that corresponds to

\( \BE \) lies in the plane of incidence, and the magnetic field is completely parallel to the plane of reflection). For the no transmission case, allowing \( v_t \rightarrow 0 \), the index of refraction is \( n_t = c/v_t \rightarrow \infty \), and the reflection coefficient is \( 1 \) as claimed in section 4.7.2 of [1]. Because of the symmetry of this dipole configuration, the azimuthal angle that the wave vector is directed along does not matter.

Horizontal dipole reflection coefficient

In the class notes, a horizontal dipole coming out of the page is indicated. With the page representing the z-y plane, this is a magnetic vector potential directed along the x-axis direction

\begin{equation}\label{eqn:chapter4Notes:640}
\BA = \xcap \frac{\mu_0 I_0 l}{4 \pi r} e^{-j k r}.

\end{equation}

For a wave vector directed in the z-y plane as in \ref{eqn:chapter4Notes:560}, the electric far field is directed along

\begin{equation}\label{eqn:chapter4Notes:660}
\begin{aligned}
\lr{ \xcap \wedge \kcap } \cdot \kcap
&=
\xcap – \lr{ \xcap \cdot \kcap } \kcap \\
&=
\xcap – \lr{ \xcap \cdot \lr{
\zcap \cos\theta + \ycap \sin\theta
} } \kcap \\
&= \xcap.
\end{aligned}
\end{equation}

The electric far field lies completely in the plane of reflection. From [2] (\eqntext 4.34), the Fresnel reflection coefficients is

\begin{equation}\label{eqn:chapter4Notes:680}
R =
\frac{
n_i \cos\theta_i – n_t \cos\theta_t
}
{
n_i \cos\theta_i + n_t \cos\theta_t
},
\end{equation}

which approaches \( -1 \) when \( n_t \rightarrow \infty \). This is consistent with the image theorem summation that Prof. Eleftheriades used in class.

Azimuthal angle dependency of the reflection coefficient

Now consider a horizontal dipole directed along the y-axis. For the same wave vector direction as avove, the electric far field is now directed along

\begin{equation}\label{eqn:chapter4Notes:700}
\begin{aligned}
\lr{ \ycap \wedge \kcap } \cdot \kcap
&=
\ycap – \lr{ \ycap \cdot \kcap } \kcap \\
&=
\ycap – \lr{ \ycap \cdot \lr{
\zcap \cos\theta + \ycap \sin\theta
} } \kcap \\
&=
\ycap – \kcap \sin\theta \\
&=
\ycap – \sin\theta \lr{
\zcap \cos\theta + \ycap \sin\theta
} \\
&=
\ycap \cos^2 \theta – \sin\theta \cos\theta \zcap \\
&= \cos\theta \lr{ \ycap \cos\theta – \sin\theta \zcap } \\
&= \cos\theta \ycap e^{ \zcap \ycap \theta }.
\end{aligned}
\end{equation}

That is

\begin{equation}\label{eqn:chapter4Notes:720}
\BE =
-j \omega \frac{\mu_0 I_0 l}{4 \pi r} e^{-j k r}
\cos\theta \ycap e^{ \zcap \ycap \theta }.
\end{equation}

This far field electric field lies in the plane of incidence (a direction of \( \thetacap \) rotated by \( \pi/2 \)), not in the plane of reflection. The corresponding magnetic field should be directed along the plane of reflection, which is easily confirmed by calculation

\begin{equation}\label{eqn:chapter4Notes:740}
\begin{aligned}
\kcap \cross
\lr{ \ycap \cos\theta – \sin\theta \zcap }
&=
\lr{ \zcap \cos\theta + \ycap \sin\theta } \cross
\lr{ \ycap \cos\theta – \sin\theta \zcap } \\
&=
-\xcap \cos^2 \theta – \xcap \sin^2\theta \\
&= -\xcap.
\end{aligned}
\end{equation}

The far field magnetic field is seen to be

\begin{equation}\label{eqn:chapter4Notes:721}
\BH =
j \omega \frac{I_0 l}{4 \pi r} e^{-j k r}
\cos\theta \xcap,
\end{equation}

so a reflection coefficient of \( 1 \) is required to calculate the power loss after a single ground reflection signal bounce for this relative orientation of antenna to the target.

I fail to see how the horizontal dipole treatment in section 4.7.5 can use a single reflection coefficient without taking into account the azimuthal dependency of that reflection coefficient.

References

[1] Constantine A Balanis. Antenna theory: analysis and design. John Wiley \& Sons, 3rd edition, 2005.

[2] E. Hecht. Optics. 1998.

[3] JD Jackson. Classical Electrodynamics. John Wiley and Sons, 2nd edition, 1975.

[4] Wikipedia. Magnetic potential — Wikipedia, The Free Encyclopedia, 2015. URL https://en.wikipedia.org/w/index.php?title=Magnetic_potential&oldid=642387563. [Online; accessed 5-February-2015].