- Electric field integral equation
The

**electric field integral equation**is a relationship that allows one to calculate theelectric field intensity**E**generated by anelectric current distribution**J**.**Derivation**We consider all quantities in the frequency domain, and so assume a time-dependency $e^\{-jwt\},$ that is suppressed throughout.

Begin with the

Maxwell equations relating the electric andmagnetic field an assume linear, media with permeability andpermittivity $epsilon,$ and $mu,$, respectively::$abla\; imes\; extbf\{E\}\; =\; -j\; omega\; mu\; extbf\{H\},$:$abla\; imes\; extbf\{H\}\; =\; j\; omega\; epsilon\; extbf\{E\}\; +\; extbf\{J\},$

Following the third equation involving the

divergence of**H**:$abla\; cdot\; extbf\{H\}\; =\; 0,$

by

vector calculus we can write any divergenceless vector as the curl of another vector, hence:$abla\; imes\; extbf\{A\}\; =\; extbf\{H\},$

where

**A**is called themagnetic vector potential . Substituting this into the above we get:$abla\; imes\; (\; extbf\{E\}\; +\; j\; omega\; mu\; extbf\{A\})\; =\; 0,$

and any curl-free vector can be written as the

gradient of a scalar, hence:$extbf\{E\}\; +\; j\; omega\; mu\; extbf\{A\}\; =\; -\; abla\; Phi$

where $Phi$ is the

electric scalar potential . These relationships now allow us to write:$abla\; imes\; abla\; imes\; extbf\{A\}\; -\; k^\{2\}\; extbf\{A\}\; =\; extbf\{J\}\; -\; j\; omega\; epsilon\; abla\; Phi\; ,$

which can be rewritten by vector identity as

:$abla\; (\; abla\; cdot\; extbf\{A\})\; -\; abla^\{2\}\; extbf\{A\}\; -\; k^\{2\}\; extbf\{A\}\; =\; extbf\{J\}\; -\; j\; omega\; epsilon\; abla\; Phi\; ,$

As we have only specified the curl of

**A**, we are free to define the divergence, and choose the following::$abla\; cdot\; extbf\{A\}\; =\; -\; j\; omega\; epsilon\; Phi\; ,$

which is called the

Lorenz gauge condition . The previous expression for**A**now reduces to:$abla^\{2\}\; extbf\{A\}\; +\; k^\{2\}\; extbf\{A\}\; =\; -\; extbf\{J\},$

which is the vector

Helmholtz equation . The solution of this equation for**A**is:$extbf\{A\}(\; extbf\{r\})\; =\; frac\{1\}\{4\; pi\}\; iiint\; extbf\{J\}(\; extbf\{r\}^\{prime\})\; G(\; extbf\{r\},\; extbf\{r\}^\{prime\})\; d\; extbf\{r\}^\{prime\}\; ,$

where $G(\; extbf\{r\},\; extbf\{r\}^\{prime\}),$ is the three-dimensional homogeneous

Green's function given by:$G(\; extbf\{r\},\; extbf\{r\}^\{prime\})\; =\; frac\{e^\{-j\; k\; |\; extbf\{r\}\; -\; extbf\{r\}^\{prime\}|,$

We can now write what is called the electric field integral equation (EFIE), relating the electric field

**E**to the vector potential**A**:$extbf\{E\}\; =\; -j\; omega\; mu\; extbf\{A\}\; +\; frac\{1\}\{j\; omega\; epsilon\}\; abla\; (\; abla\; cdot\; extbf\{A\}),$

We can further represent the EFIE in the dyadic form as

:$extbf\{E\}\; =\; -j\; omega\; mu\; int\_V\; d\; extbf\{r\}^\{prime\}\; extbf\{G\}(\; extbf\{r\},\; extbf\{r\}^\{prime\})\; cdot\; extbf\{J\}(\; extbf\{r\}^\{prime\})\; ,$

where $extbf\{G\}(\; extbf\{r\},\; extbf\{r\}^\{prime\}),$ here is the dyadic homogeneous Green's Function given by

:$extbf\{G\}(\; extbf\{r\},\; extbf\{r\}^\{prime\})\; =\; frac\{1\}\{4\; pi\}\; left\; [\; extbf\{I\}+frac\{\; abla\; abla\}\{k^2\}\; ight]\; G(\; extbf\{r\},\; extbf\{r\}^\{prime\})\; ,$

**Interpretation**The EFIE describes a radiated field

**E**given a set of sources**J**, and as such it is the fundamental equation used in antenna analysis and design. It is a very general relationship that can be used to compute the radiated field of any sort of antenna once the current distribution on it is known. The most important aspect of the EFIE is that it allows us to solve the radiation/scattering problem in anunbounded region, or one whose boundary is located atinfinity . For closed surfaces it is possible to use theMagnetic Field Integral Equation or theCombined Field Integral Equation , both of which result in a set of equations with improved condition number compared to the EFIE. However, the MFIE and CFIE can still contain resonances.In scattering problems, it is desirable to determine an unknown scattered field $E\_\{s\}$ that is due to a known incident field $E\_\{i\}$. Unfortunately, the EFIE relates the "scattered" field to

**J**, not the incident field, so we do not know what**J**is. This sort of problem can be solved by imposing theboundary conditions on the incident and scattered field, allowing one to write the EFIE in terms of $E\_\{i\}$ and**J**alone. Once this has been done, the integral equation can then be solved by a numerical technique appropriate to integral equations such as the method of moments.**Notes**By the Helmholtz theorem a vector is described completely by its divergence and curl. As the divergence was not defined, we are justified by choosing the Lorenz Gauge condition above provided that we consistently use this definition of the divergence of

**A**in all subsequent analysis.This vector

**A**should not be interpreted as a real physical quantity, it is just a mathematical tool to help us solve electromagnetic problems.**References***Gibson, Walton C. "The Method of Moments in Electromagnetics". Chapman & Hall/CRC, 2008. ISBN 978-1-4200-6145-1

*Harrington, Roger F. "Time-Harmonic Electromagnetic Fields". McGraw-Hill, Inc., 1961. ISBN 0-07-026745-6.

*Balanis, Constantine A. "Advanced Engineering Electromagnetics". Wiley, 1989. ISBN 0-471-62194-3.

*Chew, Weng C. "Waves and Fields in Inhomogeneous Media". IEEE Press, 1995. ISBN 0-7803-4749-8.

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