# Magnetic current

Magnetic current is, nominally, a current composed of fictitious moving magnetic monopoles. It has the dimensions of volts. The usual symbol for magnetic current is $k$ which is analogous to $i$ for electric current. Magnetic currents produce an electric field analogously to the production of a magnetic field by electric currents. Magnetic current density, which has the units of V/m2 (volts per square meter), is usually represented by the symbols ${\mathfrak {M}}^{t}$ and ${\mathfrak {M}}^{i}$ . The superscripts indicate total and impressed magnetic current density. The impressed currents are the energy sources. In many useful cases, a distribution of electric charge can be mathematically replaced by an equivalent distribution of magnetic current. This artifice can be used to simplify some electromagnetic field problems.[a][b] It is possible to use both electric current densities and magnetic current densities in the same analysis.: 138

The direction of the electric field produced by magnetic currents is determined by the left-hand rule (opposite direction as determined by the right-hand rule) as evidenced by the negative sign in the equation

$\nabla \times {\mathcal {E}}=-{\mathfrak {M}}^{t}$ .

## Magnetic displacement current

Magnetic displacement current or more properly the magnetic displacement current density is the familiar term B/∂t[c][d][e] It is one component of ${\mathfrak {M}}^{\mathfrak {t}}$ .

${\mathfrak {M}}^{t}={\frac {\partial B}{\partial t}}+{\mathfrak {M}}^{i}$ .

where

${\mathfrak {M}}^{\mathfrak {t}}$  = total magnetic current.
${\mathfrak {M}}^{\mathfrak {i}}$  = impressed magnetic current (energy source).

## Electric vector potential

The electric vector potential, F, is computed from the magnetic current density, ${\mathfrak {M}}^{\mathfrak {i}}$ , in the same way that the magnetic vector potential, A, is computed from the electric current density. : 100 : 138 : 468  F is used in the same way the magnetic vector potential for sources attributed to magnetic current. Examples of use include finite diameter wire antennas and transformers.

magnetic vector potential: $\mathbf {A} (\mathbf {r} ,t)={\frac {\mu _{0}}{4\pi }}\int _{\Omega }{\frac {\mathbf {J} (\mathbf {r} ',t')}{|\mathbf {r} -\mathbf {r} '|}}\,\mathrm {d} ^{3}\mathbf {r} '\,.$

electric vector potential: $\mathbf {F} (\mathbf {r} ,t)={\frac {\epsilon _{0}}{4\pi }}\int _{\Omega }{\frac {{\mathfrak {M}}^{i}(\mathbf {r} ',t')}{|\mathbf {r} -\mathbf {r} '|}}\,\mathrm {d} ^{3}\mathbf {r} '\,.$

where F at point $\mathbf {r}$  and time $t$  is calculated from magnetic currents at distant position $\mathbf {r} '$  at an earlier time $t'$ . The location $\mathbf {r} '$  is a source point within volume Ω that contains the magnetic current distribution. The integration variable, $\mathrm {d} ^{3}\mathbf {r} '$ , is a volume element around position $\mathbf {r} '$ . The earlier time $t'$  is called the retarded time, and calculated as

$t'=t-{\frac {|\mathbf {r} -\mathbf {r} '|}{c}}$ .

Retarded time accounts for the accounts for the time required for electromagnetic effects to propagate from point $\mathbf {r} '$  to point $\mathbf {r}$ .

### Phasor form

When all the functions of time are sinusoids of the same frequency, the time domain equation can be replaced with a frequency domain equation. Retarded time is replaced with a phase term.

$\mathbf {F} (\mathbf {r} )={\frac {\epsilon _{0}}{4\pi }}\int _{\Omega }{\frac {{\mathfrak {M}}^{i}(\mathbf {r} )e^{(-jk|\mathbf {r} -\mathbf {r} '|)}}{|\mathbf {r} -\mathbf {r} '|}}\,\mathrm {d} ^{3}\mathbf {r} '\,.$

where $\mathbf {F}$  and ${\mathfrak {M}}^{i}$  are phasor quantities and $k$  is the wave number.

## Magnetic frill generator

A dipole antenna driven by a hypothetical annular ring of magnetic current. b is chosen so that 377 x ln(b/a) is equal to the impedance of the driving transmission line (not shown).

A distribution of magnetic current, commonly called a magnetic frill generator, may be used to replace the driving source and feed line in the analysis of a finite diameter dipole antenna.: 447–450  The voltage source and feed line impedance are subsumed into the magnetic current density. In this case, the magnetic current density is concentrated in a two dimensional surface so the units of ${\mathfrak {M}}^{i}$  are volts per meter.

The inner radius of the frill is the same as the radius of the dipole. The outer radius is chosen so that

$Z_{L}=Z_{0}\ln({\frac {b}{a}}).$

where

$Z_{L}$  = impedance of the feed transmission line (not shown).
$Z_{0}$  = impedance of free space.

The equation is the same as the equation for the impedance of a coaxial cable. However, a coaxial cable feed line is not assumed and not required.

The amplitude of the magnetic current density phasor is given by:

${M}^{i}={\frac {k}{\rho }}$  with $a\leq \rho \leq b.$

where

$\rho$  = radial distance from the axis.
$k={\frac {V_{s}}{\ln({\frac {b}{a}})}}$ .
$V_{s}$  = magnitude of the source voltage phasor driving the feed line.