Infinite Parallel Planes Of Charge

Two parallel plates with infinite area can be used as an approximation for a Capacitor.

Electric Field §

Given two planes with infinite area, with total charges ${\sigma }_1$ and ${\sigma }_2$ respectively, and separated by a distance of $d$, the net electric field at a point with $z$ in the Z-axis is:

Formula §

$${\vec{E}}_{net}={\vec{E}}_1+{\vec{E}}_2=\{\begin{array}{ll}\frac{{\sigma }_1-{\sigma }_2}{2{ϵ}_0}\hat{u},& -\frac{d}{2}\le z\le \frac{d}{2}(\text{i.e. within the plates})\\ \frac{{\sigma }_1+{\sigma }_2}{2{ϵ}_0},& \text{outside the plates}\end{array}$$

• $\vec{E}$ = Electric Field (in $N/C$)
• $\sigma$ = Surface Charge Density (in $C/m^2$)

Meaning §

This means the field in the gap inside like-charged plates is always zero, which also proves the property of Electrostatic Equilibrium for a Infinite Conducting Charged Slab.

For unlike charges, such as in a Capacitor, the electric field is doubled if $-q_1=q_2$ and is zero in the points outside the plates.

Derivation §

The formula can be calculated by using the superposition of electric fields, since we already have the electric field for a single Infinite Plane Of Charge:

If we define $\hat{u}$ to be the unit vector perpendicular to the first plane, $P_1$, then we have two ‘scenarios’:

• Electric field between plates, where ${\hat{u}}_1=-{\hat{u}}_2$
• Electric fields outside plate, where ${\hat{u}}_1={\hat{u}}_2$

$$\sum \vec{E}=\frac{{\sigma }_1}{2{ϵ}_0}{\hat{u}}_1+\frac{{\sigma }_2}{2{ϵ}_0}{\hat{u}}_2$$

Between plates:

$$\sum \vec{E}=\frac{{\sigma }_1}{2{ϵ}_0}{\hat{u}}_1-\frac{{\sigma }_2}{2{ϵ}_0}{\hat{u}}_1=\frac{{\sigma }_1-{\sigma }_2}{2{ϵ}_0}\hat{u}$$

Outside plates

\sum \vec{E} = \dfrac{\sigma_{1}+\sigma_{2}}{2\epsilon_{0}}\hat{u}$$