Electric Field Integral Equation

The electric field integral equation is a relationship that allows one to calculate the Electric Field intensity '''E''' generated by an Electric Current distribution '''J''' .

DERIVATION

We consider all quantities in the frequency domain, and so assume a time-dependency

e^{-jwt}\,

that is suppressed throughout.

We begin with the Maxwell Equations relating the electric and Magnetic Field an assume Linear , Homogeneous media with Permeability and Permittivity

\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 the Magnetic 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}) = rac{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 Green's Function given by

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	CATEGORIES ABOUT ELECTRIC FIELD INTEGRAL EQUATION

	electromagnetism

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Information About

Electric Field Integral Equation