Magnetic current

This article is about a current of magnetic monoples. For current as an analog for magnetic flux, see Magnetic circuit.

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 $k$ 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 units of v/m² (volts per square meter), is usually represented by the symbols ${\mathfrak {M}}^{t}$ and ${\mathfrak {M}}^{i}$ . 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.^{[lower-alpha 1]}^{[lower-alpha 2]} 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]

\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 $\partial$ B/ $\partial$ t. ^{[lower-alpha 3]} ^{[lower-alpha 4]} ^{[lower-alpha 5]} It is one component of ${\mathfrak {M}}^{\mathfrak {t}}$ .^[1]^[2]

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

where

{\mathfrak {M}}^{\mathfrak {t}}

= total magnetic current.

{\mathfrak {M}}^{\mathfrak {i}}

= impressed magnetic current (energy source).

Electric vector potential

The electric vector potential, F, is computed from the magnetic current density, ${\mathfrak {M}}^{\mathfrak {i}}$ , 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: $\mathbf {A} (\mathbf {r} ,t)={\frac {\mu _{0}}{4\pi }}\int _{\Omega }{\frac {\mathbf {J} (\mathbf {r} ',t')}{|\mathbf {r} -\mathbf {r} '|}}\,\mathrm {d} ^{3}\mathbf {r} '\,.$

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

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

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 ${\mathbf r}'$ to point $\mathbf {r}$ .

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.

\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 $\mathbf {F}$ and ${\mathfrak {M}}^{i}$ 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 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 ${\mathfrak {M}}^{i}$ 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

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

where

Z_{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}^{i}={\frac {k}{\rho }}

with

a\leq \rho \leq b.

where

\rho

= radial distance from the axis.

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

V_{s}

= magnitude of the source voltage phasor driving the feed line.

Notes

↑ "For some electromagnetic problems, their solution can often be aided by the introduction of equivalent impressed electric and magnetic current densities." ^[2]
↑ "there are many other problems where the use of fictitious magnetic currents and charges is very helpful." ^[3]
↑ "Because of the symmetry of Maxwell's equations, the ɚB/ɚt term ... has been designated as a magnetic displacement current density." ^[2]
↑ "interpreted as ... magnetic displacement current ..." ^[3]
↑ "it also is convenient to consider the term ɚB/ɚt as a magnetic displacement current density." ^[1]

References

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

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