Magnetic current

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

Template:Electromagnetism Magnetic current is, nominally, a current composed of moving magnetic monopoles. It has the unit volt. 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 unit V/m² (volt per square meter), is usually represented by the symbols ${𝔐}^{\text{t}}$ and ${𝔐}^{\text{i}}$.Template:Efn 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.Template:EfnTemplate:Efn It is possible to use both electric current densities and magnetic current densities in the same analysis.Template:Rp

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 $$\mathrm{\nabla }\times ℰ=-{𝔐}^{\text{t}}.$$

Magnetic displacement current Template:Anchor

Magnetic displacement current or more properly the magnetic displacement current density is the familiar term Template:MathTemplate:EfnTemplate:EfnTemplate:Efn It is one component of ${𝔐}^{\text{t}}$. $${𝔐}^{\text{t}}=\frac{\partial B}{\partial t}+{𝔐}^{\text{i}}.$$ where

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

Electric vector potential

The electric vector potential, F, is computed from the magnetic current density, ${𝔐}^{\text{i}}$, in the same way that the magnetic vector potential, A, is computed from the electric current density.Template:Rp Template:Rp Template:Rp Examples of use include finite diameter wire antennas and transformers.

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

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

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

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. $$𝐅(𝐫)=\frac{{\epsilon }_0}{4\pi }\underset{\mathrm{\Omega }}{\int }\frac{{𝔐}^{\text{i}}(𝐫)e^{-jk|𝐫-{𝐫}^{\prime }|}}{|𝐫-{𝐫}^{\prime }|}{\mathrm{d}}^3{𝐫}^{\prime },$$ where $𝐅$ and ${𝔐}^{\text{i}}$ 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 Template:Nowrap 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.Template:Rp 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 ${𝔐}^{\text{i}}$ 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 $$Z_{\text{L}}=Z_0\ln \left(\frac{b}{a}\right),$$ where

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

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

See also

Surface equivalence principle