Electric Field Integrals

The formula for determining a set of charges’ induced electric field is very similar to the Coloumb-Equation. The influence of a region of charge on a field point $P=r$ is modelled by

$$F=E(r)=\frac{1}{4\pi ϵ\_0}\int \frac{\lambda }{{\gamma }^2}\hat{\gamma }d{\tau }^{\prime }$$

• $\gamma =r-r^{\prime }$ the distance vector from source to field point $P=r$ (pointing from $r^{\prime }$ to $r$)
• $\hat{\gamma }=\frac{\gamma }{|\gamma |}$
• $\lambda (r^{\prime })$ the charge distribution along the region
• $d\tau$ the differential according to the region of integration

The word “region” encompasses here all possible spaces of integration (line, surface, volume, …) in any chosen coordinate system