Magnetic current is, nominally, a current composed of movingmagnetic monopoles. It has the unitvolt. The usual symbol for magnetic current is${\displaystyle k}$, which is analogous to${\displaystyle i}$ forelectric current. Magnetic currents produce anelectric 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 theright-hand rule) as evidenced by the negative sign in the equation^[1]$${\displaystyle \mathrm{\nabla }\times \mathcal{E}=-{\mathfrak{M}}^{\text{t}}.}$$

Magnetic displacement current or more properly themagnetic displacement current density is the familiar term∂B/∂t^[d][e][f] It is one component of${\displaystyle {\mathfrak{M}}^{\text{t}}}$ .^[1][2]$${\displaystyle {\mathfrak{M}}^{\text{t}}=\frac{\mathrm{\partial }B}{\mathrm{\partial }t}+{\mathfrak{M}}^{\text{i}}.}$$ 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 themagnetic vector potential,A, is computed from the electric current density.^{[1]: 100 [4]: 138 [3]: 468} Examples of use include finite diameter wireantennas andtransformers.^[5]

magnetic vector potential:$${\displaystyle \mathbf{A}(\mathbf{r},t)=\frac{{\mu }_0}{4\pi }{\int }_{\mathrm{\Omega }}\frac{\mathbf{J}({\mathbf{r}}^{\prime },t^{\prime })}{|\mathbf{r}-{\mathbf{r}}^{\prime }|}{\mathrm{d}}^3{\mathbf{r}}^{\prime }.}$$ 

electric vector potential:$${\displaystyle \mathbf{F}(\mathbf{r},t)=\frac{{\epsilon }_0}{4\pi }{\int }_{\mathrm{\Omega }}\frac{{\mathfrak{M}}^{\text{i}}({\mathbf{r}}^{\prime },t^{\prime })}{|\mathbf{r}-{\mathbf{r}}^{\prime }|}{\mathrm{d}}^3{\mathbf{r}}^{\prime },}$$ whereF atpoint${\displaystyle \mathbf{r}}$  and time${\displaystyle t}$  is calculated from magnetic currents at distant position${\displaystyle {\mathbf{r}}^{\prime }}$  at an earlier time${\displaystyle t^{\prime }}$ . The location${\displaystyle {\mathbf{r}}^{\prime }}$  is a source point within volumeΩ that contains the magnetic current distribution. The integration variable,${\displaystyle {\mathrm{d}}^3{\mathbf{r}}^{\prime }}$ , is a volume element around position${\displaystyle {\mathbf{r}}^{\prime }}$ . The earlier time${\displaystyle t^{\prime }}$  is called theretarded time, and calculated as$${\displaystyle t^{\prime }=t-\frac{|\mathbf{r}-{\mathbf{r}}^{\prime }|}{c}.}$$ 

Retarded time accounts for the accounts for the time required for electromagnetic effects to propagate from point${\displaystyle {\mathbf{r}}^{\prime }}$  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 afrequency domain equation. Retarded time is replaced with a phase term.$${\displaystyle \mathbf{F}(\mathbf{r})=\frac{{\epsilon }_0}{4\pi }{\int }_{\mathrm{\Omega }}\frac{{\mathfrak{M}}^{\text{i}}(\mathbf{r})e^{-jk|\mathbf{r}-{\mathbf{r}}^{\prime }|}}{|\mathbf{r}-{\mathbf{r}}^{\prime }|}{\mathrm{d}}^3{\mathbf{r}}^{\prime },}$$ where${\displaystyle \mathbf{F}}$  and${\displaystyle {\mathfrak{M}}^{\text{i}}}$  arephasor quantities and${\displaystyle k}$  is the wave number.

A distribution of magnetic current, commonly called amagnetic frill generator, may be used to replace the driving source andfeed line in the analysis of a finite diameterdipole antenna.^{[4]: 447–450} The voltage source and feed lineimpedance 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 acoaxial 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${\displaystyle a\le \rho \le b.}$ 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