View source for Physics:Magnetic current

{{Short description|Flow of magnetic monopole charge}}
[[File:Magnetic Current Produces Electric Field By Left Hand Rule.svg|thumb|Magnetic current (flowing magnetic monopoles), '''''M''''', creates an electric field, '''''E''''', in accordance with the left-hand rule.]]
{{Electromagnetism}}
'''Magnetic current''' is, nominally, a current composed of fictitious moving [[Physics:Magnetic monopole|magnetic monopole]]s.  It has the unit [[Volt|volt]].  The usual symbol for magnetic current is <math>k</math>, which is analogous to <math>i</math> for [[Physics:Electric current|electric current]].  Magnetic currents produce an [[Physics:Electric field|electric field]] analogously to the production of a magnetic field by electric currents.  '''Magnetic current density''', which has the unit V/m<sup>2</sup> (volt per square meter), is usually represented by the symbols <math> \mathfrak{M}^\text{t}</math> and <math> \mathfrak{M}^\text{i}</math>.  The superscripts indicate total and impressed magnetic current density.<ref name="Harrington">{{Citation |last=Harrington |first= Roger F. |year= 1961 |title= Time-Harmonic Electromagnetic Fields |publisher= McGraw-Hill |isbn=0-07-026745-6 |pages=7–8|hdl=2027/mdp.39015002091489 |hdl-access=free }}</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.{{efn| "For some electromagnetic problems, their solution can often be aided by the introduction of equivalent impressed electric and magnetic current densities."<ref name="Balanis_EE">{{Citation |last=Balanis |first= Constantine A. |author-link=Constantine A. Balanis |year= 2012 |title= Advanced Engineering Electromagnetics |publisher= John Wiley |isbn= 978-0-470-58948-9 |pages=2–3}}</ref>}}{{efn|"there are many other problems where the use of fictitious magnetic currents and charges is very helpful."<ref name="Jordan_Balman">{{Citation |last1=Jordan|first1= Edward |last2=Balmain|first2=Keith G.|year= 1968 |title=Electromagnetic Waves and Radiating Systems|edition= 2nd |publisher= Prentice-Hall | lccn=68-16319 | page=466}}</ref>}}  It is possible to use both electric current densities and magnetic current densities in the same analysis.<ref name="Balanis_Ant">{{Citation |last=Balanis |first= Constantine A. |year= 2005 |title= Antenna Theory |edition=third |publisher= John Wiley |isbn= 047166782X |author-link=Constantine A. Balanis}}</ref>{{rp|138}}

The direction of the electric field produced by magnetic currents is determined by the left-hand rule (opposite direction as determined by the [[Physics:Right-hand rule|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 {{anchor|Displacement}} ==

'''Magnetic displacement current''' or more properly the '''magnetic displacement current density''' is the familiar term {{math|∂'''''B'''''/∂''t''}}{{efn| "Because of the symmetry of Maxwell's equations, the ∂B/∂t term ... has been designated as a magnetic displacement current density."<ref name="Balanis_EE"/>}}{{efn| "interpreted as ... magnetic displacement current ..."<ref name="Jordan_Balman"/>}}{{efn| "it also is convenient to consider the term ∂B/∂t as a magnetic displacement current density."<ref name="Harrington"/>}}  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 [[Physics:Magnetic vector potential|magnetic vector potential]], '''A''', is computed from the electric current density.<ref name="Harrington"/>{{rp|100}} <ref name="Balanis_Ant"/>{{rp|138}} <ref name="Jordan_Balman"/>{{rp|468}} Examples of use include finite diameter wire [[Engineering:Antenna (radio)|antennas]] and [[Engineering:Transformer|transformer]]s.<ref name="Kulkarni">{{Citation |last1=Kulkarni |first1= S. V. |last2=Khaparde | first2= S. A. |year= 2004 |title=Transformer Engineering: Design and Practice  |edition=third |publisher= CRC Press |isbn= 0824756533 |pages=179–180}}</ref>

<!-- This time domain version intentionally parallels the style of the vector magnetic potential in the [[Physics:Magnetic potential|Magnetic potential]] article that appears to come from Jackson.  The sources that are cited give frequency (phasor) domain equations where retarded time is replaced by a term that provides phase shift that is proportional to separation.  Harrington omits the epsilon_0 constant. Jordan_Balman is the only source that is explicit that only the impressed currents are used in the calculation. -->
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 <big>Ω</big> 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 ''[[Physics:Retarded time|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|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|phasor]] quantities and <math>k</math> is the wave number.

== Magnetic frill generator ==

[[File:Magnetic Frill Driving Dipole Antenna.png|thumb|300px|A dipole antenna driven by a hypothetical annular ring of magnetic current.  ''b'' is chosen so that {{nowrap|377 Ω × 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 [[Engineering:Feed line|feed line]] in the analysis of a finite diameter [[Physics:Dipole antenna|dipole antenna]].<ref name="Balanis_Ant"/>{{rp|447–450}}  The voltage source and feed line [[Physics:Electrical impedance|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 [[Engineering:Coaxial cable|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> = [[Cylindrical coordinate system|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.
{{Clear}}

== See also ==
[[Physics:Surface equivalence principle|Surface equivalence principle]]

== Notes ==
{{Notelist}}

== References ==
{{reflist}}

[[Category:Electromagnetism]]
[[Category:Antennas]]

{{Sourceattribution|Magnetic current}}