User:Constant314/Magnetic vector potential in the frequency domain

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Magnetic vector potential

Time domain

The time domain potentials are defined as

$${\begin{aligned}\mathbf {A} \!\left(\mathbf {r} ,t\right)&={\frac {\mu _{0}}{\ 4\pi \ }}\int _{\Omega }{\frac {\mathbf {J} \left(\mathbf {r} ',t'\right)}{R}}d^{3}\mathbf {r} '\\\phi \!\left(\mathbf {r} ,t\right)&={\frac {1}{4\pi \epsilon _{0}}}\int _{\Omega }{\frac {\rho \left(\mathbf {r} ',t'\right)}{R}}\ d^{3}\mathbf {r} '\end{aligned}}$$

where $R={\bigl \|}r'-r{\bigr \|}$

The potential, at position vector $\ \mathbf {r} \$ and time $\ t\$ are calculated from sources, $\mathbf {J}$ and $\phi$ at distant position $\ \mathbf {r} '\$ and at an earlier time $\ t'~.$ The location $\ \mathbf {r} '\$ is a source point in the charge or current distribution, within volume $\ \Omega \$. The earlier time $\ t'\$ is called the retarded time, and calculated as $$t'=t-{\frac {\ R\ }{c}}~.$$

Potentials calculted in this way satisfy the Lorenz gauge $$\ \nabla \cdot \mathbf {A} +{\frac {1}{\ c^{2}}}{\frac {\partial \phi }{\partial t}}=0$$.

The time domain wave equations for the potentials are given by: $${\begin{aligned}\nabla ^{2}\phi -{\frac {1}{c^{2}}}{\frac {\partial ^{2}\phi }{\ \partial t^{2}}}&=-{\frac {\rho }{\epsilon _{0}}}\\[2.734ex]\nabla ^{2}\mathbf {A} -{\frac {1}{\ c^{2}}}{\frac {\partial ^{2}\mathbf {A} }{\ \partial t^{2}}}&=-\mu _{0}\ \mathbf {J} \end{aligned}}$$

The time domain electromagnetic field equations in terms of the potentials are given by:

$$\mathbf {B} =\nabla \times \mathbf {A} \ ,\quad \mathbf {E} =-\nabla \phi -{\frac {\partial \mathbf {A} }{\partial t}}\ ,$$ where $\ \mathbf {B} \$ is the magnetic field and $\ \mathbf {E} \$ is the electric field.

Frequency domain

The solutions of Maxwell's equations in the Lorenz gauge (see Feynman and Jackson) with the boundary condition that both potentials go to zero sufficiently fast as they approach infinity are called the retarded potentials, which are the magnetic vector potential $\ \mathbf {A} (\mathbf {r} ,t)\$ and the electric scalar potential $\ \phi (\mathbf {r} ,t)\$ due to a current distribution of current density $\ \mathbf {J} (\mathbf {r} ,t)\ ,$ charge density $\ \rho (\mathbf {r} ,t)\ ,$ and volume $\ \Omega \ ,$ within which $\ \rho \$ and $\ \mathbf {J} \$ are non-zero at least sometimes and some places):

The frequency domain potentials are defined by:

$${\begin{aligned}\mathbf {A} \!\left(\mathbf {r} ,\omega \right)&={\frac {\mu _{0}}{\ 4\pi \ }}\int _{\Omega }{\frac {\mathbf {J} \left(\mathbf {r} ',\omega \right)}{R}}\ e^{-jkR}\operatorname {d} ^{3}\mathbf {r} '\\\phi \!\left(\mathbf {r} ,\omega \right)&={\frac {1}{4\pi \epsilon _{0}}}\int _{\Omega }{\frac {\rho \left(\mathbf {r} ',\omega \right)}{R}}\ e^{-jkR}\operatorname {d} ^{3}\mathbf {r} '\\R={\bigl \|}\mathbf {r} -\mathbf {r} '{\bigr \|}\end{aligned}}$$

The potential, at position vector $\ \mathbf {r} \$ and frequency $\ \omega \$ are calculated from sources, $\mathbf {J}$ and $\phi$ at distant position $\ \mathbf {r} '\$. The location $\ \mathbf {r} '\$ is a source point in the charge or current distribution, within volume $\ \Omega \$. Propagation from location $\ \mathbf {r} '\$ to location $\ \mathbf {r} \$ is accounted for by the term $e^{-jkR}$ which plays a role equvalent to retarded time.

Potentials calculated in this way satisfy the Lorenz gauge $\ \nabla \cdot \mathbf {A} +{\frac {j\omega }{\ c^{2}}}\phi =0.$ .

The wave equations in terms of potentials are given by the following:

$\nabla ^{2}\phi +k^{2}\phi =-{\frac {\rho }{\epsilon _{0}}},$

$\nabla ^{2}\mathbf {A} +k^{2}\mathbf {A} =-\mu _{0}\ \mathbf {J} .$

The Lorenz gauge is given by: $\ \nabla \cdot \mathbf {A} +{\frac {j\omega }{\ c^{2}}}\phi =0.$

$$\mathbf {B} =\nabla \times \mathbf {A} \$$

$$\mathbf {E} =-\nabla \phi -j\omega \mathbf {A}$$

$$\mathbf {E} =-\nabla \phi -j\omega \mathbf {A} =-j\omega \mathbf {A} -j{\frac {c^{2}}{\omega }}\nabla (\nabla \cdot \mathbf {A} )=-j\omega \mathbf {A} -j{\frac {\omega }{k^{2}}}\nabla (\nabla \cdot \mathbf {A} )$$

where

$\phi ,\rho ,\mathbf {A} ,$ and $\mathbf {J}$ are phasors

$k={\frac {\omega }{c}}$

Spherical coordinate

When the source current is an electrically short, straight, current filament at the origin and parallel to the Z axis, A can be written as:

Frequency domain notes

There are a few notable things about $\ \mathbf {A} \$ and $\ \phi \$ calculated in this way:

• The equation for $\ \mathbf {A} \$ is a vector equation. In Cartesian coordinates, the equation separates into three scalar equations:

$${\begin{aligned}\mathbf {A} _{x}\!\left(\mathbf {r} ,\omega \right)&={\frac {\mu _{0}}{4\pi }}\int _{\Omega }{\frac {\mathbf {J} _{x}\left(\mathbf {r} ',\omega \right)}{R}}\ e^{-jkR}\ d^{3}\mathbf {r} ',\qquad \mathbf {A} _{y}\!\left(\mathbf {r} ,\omega \right)&={\frac {\mu _{0}}{4\pi }}\int _{\Omega }{\frac {\mathbf {J} _{y}\left(\mathbf {r} ',\omega \right)}{R}}\ e^{-jkR}\ d^{3}\mathbf {r} ',\qquad \mathbf {A} _{z}\!\left(\mathbf {r} ,\omega \right)&={\frac {\mu _{0}}{4\pi }}\int _{\Omega }{\frac {\mathbf {J} _{z}\left(\mathbf {r} ',\omega \right)}{R}}\ e^{-jkR}\ d^{3}\mathbf {r} '\end{aligned}}$$

In this form it is apparent that the component of $\ \mathbf {A} \$ in a given direction depends only on the components of $\ \mathbf {J} \$ that are in the same direction. If the current is carried in a straight wire, $\ \mathbf {A} \$ points in the same direction as the wire.

• The integrand uses the phase shift term $e^{-jkR}$ which plays a role equivalent to retarded time. This reflects the fact that changes in the sources propagate at the speed of light; propagation delay in the time domain is equivalent to a phase shift in the frequency domain.

• The Lorenz gauge condition is satisfied: $\phi =-{\frac {c^{2}}{j\omega }}\nabla \cdot \mathbf {A} .$ This implies that the electric potential, $\phi$, can be computed entirely from the current density distribution, $\mathbf {J}$.

• The position of $\ \mathbf {r} \ ,$ the point at which values for $\ \phi \$ and $\ \mathbf {A} \$ are found, only enters the equation as part of the scalar distance from $\ \mathbf {r} '\$ to $\ \mathbf {r} ~.$ The direction from $\ \mathbf {r} '\$ to $\ \mathbf {r} \$ does not enter into the equation. The only thing that matters about a source point is how far away it is.

• Balanis, Advanced Engineering

• Jordan_Balman

• Balanis, Antenna Theory