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 ${\displaystyle k}$ which is analogous to ${\displaystyle i}$ 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 ${\displaystyle {\mathfrak {M}}^{t}}$ and ${\displaystyle {\mathfrak {M}}^{i}}$. The superscripts indicate total and impressed magnetic current density.^[1] 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.^[4]:138

Magnetic current (flowing magnetic monopoles), M, creates an electric field, E, in accordance with the left-hand rule.

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^[1]

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

Magnetic displacement currentEdit

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 ${\displaystyle {\mathfrak {M}}^{\mathfrak {t}}}$.^[1][2]

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

where

${\displaystyle {\mathfrak {M}}^{\mathfrak {t}}}$ = total magnetic current.

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

Electric vector potentialEdit

The electric vector potential, F, is computed from the magnetic current density, ${\displaystyle {\mathfrak {M}}^{\mathfrak {i}}}$, in the same way that the magnetic vector potential, A, is computed from the electric current density. ^{[1]:100[4]:138[3]: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.^[5]

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

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

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

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

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

Phasor formEdit

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.

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

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

Magnetic frill generatorEdit

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.^{[4]: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 ${\displaystyle {\mathfrak {M}}^{i}}$ 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

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

where

${\displaystyle Z_{L}}$ = impedance of the feed transmission line (not shown).

${\displaystyle Z_{0}}$ = 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:

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

where

${\displaystyle \rho }$ = radial distance from the axis.

${\displaystyle k={\frac {V_{s}}{\ln({\frac {b}{a}})}}}$.

${\displaystyle V_{s}}$ = magnitude of the source voltage phasor driving the feed line.

This article uses material from the Wikipedia article  Metasyntactic variable, which is released under the  Creative Commons Attribution-ShareAlike 3.0 Unported License.