# 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 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 ${\displaystyle {\mathfrak {M}}^{\mathfrak {t}}}$ .[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 potential

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 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.

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

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

## Magnetic frill 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.[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.

## Notes

1. ^ "For some electromagnetic problems, their solution can often be aided by the introduction of equivalent impressed electric and magnetic current densities." [2]
2. ^ "there are many other problems where the use of fictitious magnetic currents and charges is very helpful."[3]
3. ^ "Because of the symmetry of Maxwell's equations, the ∂B/∂t term ... has been designated as a magnetic displacement current density."[2]
4. ^ "interpreted as ... magnetic displacement current ..."[3]
5. ^ "it also is convenient to consider the term ∂B/∂t as a magnetic displacement current density."[1]

## References

1. Harrington, Roger F. (1961), Time-Harmonic Electromagnetic Fields, McGraw-Hill, pp. 7–8, hdl:2027/mdp.39015002091489, ISBN 0-07-026745-6
2. ^ a b c Balanis, Constantine A. (2012), Advanced Engineering Electromagnetics, John Wiley, pp. 2–3, ISBN 978-0-470-58948-9
3. ^ a b c Jordan, Edward; Balmain, Keith G. (1968), Electromagnetic Waves and Radiating Systems (2nd ed.), Prentice-Hall, p. 466, LCCN 68-16319
4. ^ a b c Balanis, Constantine A. (2005), Antenna Theory (third ed.), John Wiley, ISBN 047166782X
5. ^ Kulkarni, S. V.; Khaparde, S. A. (2004), Transformer Engineering: Design and Practice (third ed.), CRC Press, pp. 179–180, ISBN 0824756533