Magnetic current

Magnetic displacement current or more properly the magnetic displacement current density is the familiar term ∂B/∂t[lower-alpha 3][lower-alpha 4][lower-alpha 5] It is one component of ${\displaystyle {\mathfrak {M}}^{\text{t}}}$.[1][2]

$${\displaystyle {\mathfrak {M}}^{\text{t}}={\frac {\partial B}{\partial t}}+{\mathfrak {M}}^{\text{i}}.}$$

where

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

The electric vector potential, F, is computed from the magnetic current density, ${\displaystyle {\mathfrak {M}}^{\text{i}}}$, in the same way that the magnetic vector potential, A, is computed from the electric current density.[1]^: 100 [4]^: 138 [3]^: 468 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:

$${\displaystyle \mathbf {F} (\mathbf {r} ,t)={\frac {\varepsilon _{0}}{4\pi }}\int _{\Omega }{\frac {{\mathfrak {M}}^{\text{i}}(\mathbf {r} ',t')}{|\mathbf {r} -\mathbf {r} '|}}\,\mathrm {d} ^{3}\mathbf {r} '\,,}$$

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.

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

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

A dipole antenna driven by a hypothetical annular ring of magnetic current. b is chosen so that 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 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 ${\displaystyle {\mathfrak {M}}^{\text{i}}}$ 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

$${\displaystyle Z_{\text{L}}=Z_{0}\ln \left({\frac {b}{a}}\right),}$$

where

• ${\displaystyle Z_{\text{L}}}$ = 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:

$${\displaystyle {\mathfrak {M}}^{\text{i}}={\frac {k}{\rho }}}$$

with ${\displaystyle a\leq \rho \leq b.}$

where

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