Magnetic current
| Articles about |
| Electromagnetism |
|---|
 |
Magnetic current is, nominally, a current composed of moving magnetic monopoles. It has the unit volt. The usual symbol for magnetic current is <math>k</math>, which is analogous to <math>i</math> 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 <math> \mathfrak{M}^\text{t}</math> and <math> \mathfrak{M}^\text{i}</math>.<ref group="lower-alpha">Not to be confused with magnetization M</ref> The superscripts indicate total and impressed magnetic current density.<ref name="Harrington">Harrington, Roger F. (1961), Time-Harmonic Electromagnetic Fields, McGraw-Hill, pp. 7–8, hdl:2027/mdp.39015002091489, ISBN 0-07-026745-6</ref> 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.<ref group="lower-alpha"></ref><ref group="lower-alpha"></ref> It is possible to use both electric current densities and magnetic current densities in the same analysis.<ref name="Balanis_Ant">Balanis, Constantine A. (2005), Antenna Theory (third ed.), John Wiley, ISBN 047166782X</ref>: 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<ref name="Harrington"/> <math display="block"> \nabla \times \mathcal{E} = -\mathfrak{M}^\text{t} .</math>
Magnetic displacement current
Magnetic displacement current or more properly the magnetic displacement current density is the familiar term ∂B/∂t<ref group="lower-alpha"></ref><ref group="lower-alpha"></ref><ref group="lower-alpha"></ref> It is one component of <math> \mathfrak{M}^\text{t}</math>.<ref name="Harrington"/><ref name="Balanis_EE"/> <math display="block"> \mathfrak{M}^\text{t} = \frac {\partial B} {\partial t} + \mathfrak{M}^\text{i} .</math> where
- <math> \mathfrak{M}^\text{t}</math> is the total magnetic current.
- <math> \mathfrak{M}^\text{i}</math> is the impressed magnetic current (energy source).
Electric vector potential
The electric vector potential, F, is computed from the magnetic current density, <math> \mathfrak{M}^\text{i}</math>, in the same way that the magnetic vector potential, A, is computed from the electric current density.<ref name="Harrington"/>: 100 <ref name="Balanis_Ant"/>: 138 <ref name="Jordan_Balman"/>: 468 Examples of use include finite diameter wire antennas and transformers.<ref name="Kulkarni">Kulkarni, S. V.; Khaparde, S. A. (2004), Transformer Engineering: Design and Practice (third ed.), CRC Press, pp. 179–180, ISBN 0824756533</ref>
magnetic vector potential: <math display="block">\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'\,.</math>
electric vector potential: <math display="block">\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'\,,</math> where F at point <math>\mathbf r</math> and time <math> t</math> is calculated from magnetic currents at distant position <math>\mathbf r'</math> at an earlier time <math> t'</math>. The location <math>\mathbf r'</math> is a source point within volume Ω that contains the magnetic current distribution. The integration variable, <math>\mathrm{d}^3\mathbf r'</math>, is a volume element around position <math>\mathbf r'</math>. The earlier time <math> t'</math> is called the retarded time, and calculated as <math display="block">t' = t - \frac{|\mathbf r - \mathbf r'|}{c}.</math>
Retarded time accounts for the accounts for the time required for electromagnetic effects to propagate from point <math>\mathbf r'</math> to point <math>\mathbf r</math>.
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. <math display="block">\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'\,,</math> where <math>\mathbf F</math> and <math>\mathfrak{M}^\text{i}</math> are phasor quantities and <math>k</math> is the wave number.
Magnetic frill generator
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.<ref name="Balanis_Ant"/>: 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 <math> \mathfrak{M}^\text{i}</math> 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 <math display="block"> Z_\text{L} = Z_0 \ln\left( \frac b a\right),</math> where
- <math> Z_\text{L} </math> = impedance of the feed transmission line (not shown).
- <math> Z_0 </math> = 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: <math display="block"> \mathfrak{M}^\text{i} = \frac k \rho</math> with <math> a \le \rho \le b.</math> where
- <math> \rho </math> = radial distance from the axis.
- <math> k = \frac {V_\text{s}} {\ln\left( \frac b a\right)} </math>.
- <math> V_\text{s} </math> = magnitude of the source voltage phasor driving the feed line.
