# Specifying the Relevant Integral Equation

Specify the integral equation by decomposing the fields into two parts.

## Incident and Scattered Fields

Basic laws of physics dictate when an electromagnetic field encounters an object, currents are excited on the object. These currents will subsequently re-radiate. This behaviour is referred to as electromagnetic scattering.

Maxwell's equations are linear equations which allows them to be decomposed into the sum (superposition) of the incident field and scattered field. The total field can, therefore, be written as:
${E}_{tot}={E}_{inc}+{E}_{scat}$
The incident field is typically a plane wave or it could be a voltage source. The incident field is that field that exists in the absence of the conducting body.

## Finding the Incident Field

In RCS applications, the incident field is a plane wave. For example, a plane wave incident from the negative X axis with the electric field z-polarized gives the incident field as:

${E}_{inc}={e}^{-jkx}\stackrel{^}{z}$

In antenna problems, the incident field, also denoted the excitation, is usually a voltage source. A simple form of excitation is the delta-gap feed. For an impressed voltage V at the terminals of an antenna over a gap of length δ the incident field can be written as:

${E}_{inc}=±\frac{V}{\partial }$

If this feed is applied to a wire, the length of the gap is typically the length of a wire segment. Other types of incident fields are magnetic frills and elementary Hertzian dipoles.

## Finding the Scattered Fields

To find the scattered fields an integral equation is applied to the surface currents. This is written in a simple notation as follows: $ℒ\left\{{J}_{scat}\right\}$ where $ℒ$ represents the integral operator and $\left\{{J}_{scat}\right\}$ are the unknown currents to be found.