Magnetic current#Magnetic displacement current

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."}}{{efn| "interpreted as ... magnetic displacement current ..."}}{{efn| "it also is convenient to consider the term ∂B/∂t as a magnetic displacement current density."}} It is one component of $\backslash mathfrak\{M\}^\backslash text\{t\}$.

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

where

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

The electric vector potential, F, is computed from the magnetic current density, $\backslash mathfrak\{M\}^\backslash text\{i\}$, in the same way that the magnetic vector potential, A, is computed from the electric current density.{{rp|100}} {{rp|138}} {{rp|468}} Examples of use include finite diameter wire antennas and transformers.{{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}}

magnetic vector potential:

$$\backslash mathbf\; A\; (\backslash mathbf\; r\; ,\; t)\; =\; \backslash frac\{\backslash mu\_0\}\{4\backslash pi\}\backslash int\_\backslash Omega\; \backslash frac\{\backslash mathbf\; J\; (\backslash mathbf\; r\text{'}\; ,\; t\text{'})\}$$

electric vector potential:

$$\backslash mathbf\; F\; (\backslash mathbf\; r\; ,\; t)\; =\; \backslash frac\{\backslash varepsilon\_0\}\{4\backslash pi\}\backslash int\_\backslash Omega\; \backslash frac\{\backslash mathfrak\{M\}^\backslash text\{i\}\; (\backslash mathbf\; r\text{'}\; ,\; t\text{'})\}$$

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

$$t\text{'}\; =\; t\; -\; \backslash frac$$

Retarded time accounts for the accounts for the time required for electromagnetic effects to propagate from point $\backslash mathbf\; r\text{'}$ to point $\backslash mathbf\; r$.

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.

$$\backslash mathbf\; F\; (\backslash mathbf\; r\; )\; =\; \backslash frac\{\backslash varepsilon\_0\}\{4\backslash pi\}\; \backslash int\_\backslash Omega\; \backslash frac\{\backslash mathfrak\{M\}^\backslash text\{i\}\; (\backslash mathbf\{r\})\; e^\{-jk\; |\backslash mathbf\{r\}\; -\; \backslash mathbf\; r\text{'}|\}\}$$

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

File:Magnetic Frill Driving Dipole Antenna.png

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.{{rp|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 $\backslash mathfrak\{M\}^\backslash text\{i\}$ 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

$$Z\_\backslash text\{L\}\; =\; Z\_0\; \backslash ln\backslash left(\; \backslash frac\; b\; a\backslash right),$$

where

• $Z\_\backslash text\{L\}$ = 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:

$$\backslash mathfrak\{M\}^\backslash text\{i\}\; =\; \backslash frac\; k\; \backslash rho$$ with $a\; \backslash le\; \backslash rho\; \backslash le\; b.$

where

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

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Surface equivalence principle

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Category:Electromagnetism

Category:Antennas