Magnetic current

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Magnetic current is, nominally, a current composed of moving magnetic monopoles. It has the unit volt. 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 unit V/m² (volt per square meter), is usually represented by the symbols ${\displaystyle {\mathfrak {M}}^{\text{t}}}$ and ${\displaystyle {\mathfrak {M}}^{\text{i}}}$.^[a] 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.^[b][c] It is possible to use both electric current densities and magnetic current densities in the same analysis.^[4]: 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^[1] $${\displaystyle \nabla \times {\mathcal {E}}=-{\mathfrak {M}}^{\text{t}}.}$$

Magnetic displacement current or more properly the magnetic displacement current density is the familiar term ∂B/∂t^[d][e][f] It is one component of ${\displaystyle {\mathfrak {M}}^{\text{t}}}$.^[1][2] $${\displaystyle {\mathfrak {M}}^{\text{t}}={\frac {\partial B}{\partial t}}+{\mathfrak {M}}^{\text{i}}.}$$ where

• ${\displaystyle {\mathfrak {M}}^{\text{t}}}$ is the total magnetic current.
• ${\displaystyle {\mathfrak {M}}^{\text{i}}}$ is the impressed magnetic current (energy source).

The electric vector potential, F, is computed from the magnetic current density, ${\displaystyle {\mathfrak {M}}^{\text{i}}}$, in the same way that the magnetic vector potential, A, is computed from the electric current density.^{[1]: 100  [4]: 138  [3]: 468} 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: $${\displaystyle \mathbf {F} (\mathbf {r} ,t)={\frac {\varepsilon _{0}}{4\pi }}\int _{\Omega }{\frac {{\mathfrak {M}}^{\text{i}}(\mathbf {r} ',t')}{|\mathbf {r} -\mathbf {r} '|}}\,\mathrm {d} ^{3}\mathbf {r} '\,,}$$ 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 ${\displaystyle \mathbf {r} }$.

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. $${\displaystyle \mathbf {F} (\mathbf {r} )={\frac {\varepsilon _{0}}{4\pi }}\int _{\Omega }{\frac {{\mathfrak {M}}^{\text{i}}(\mathbf {r} )e^{-jk|\mathbf {r} -\mathbf {r} '|}}{|\mathbf {r} -\mathbf {r} '|}}\,\mathrm {d} ^{3}\mathbf {r} '\,,}$$ where ${\displaystyle \mathbf {F} }$ and ${\displaystyle {\mathfrak {M}}^{\text{i}}}$ are phasor quantities and ${\displaystyle k}$ is the wave number.

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 ${\displaystyle {\mathfrak {M}}^{\text{i}}}$ 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 $${\displaystyle Z_{\text{L}}=Z_{0}\ln \left({\frac {b}{a}}\right),}$$ where

• ${\displaystyle Z_{\text{L}}}$ = 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: $${\displaystyle {\mathfrak {M}}^{\text{i}}={\frac {k}{\rho }}}$$ with ${\displaystyle a\leq \rho \leq b.}$ where

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

Surface equivalence principle