Potentials of Localised Charge Distributions

$$V=\underset{\text{Grainy Distribution}}{\underbrace{\frac{1}{4\pi ϵ\_0}\sum \_{i=1}^N\frac{q\_i}{r_i}}}=\underset{\text{Continuous Distribution}}{\underbrace{\frac{1}{4\pi ϵ\_0}\int \frac{1}{r},dq}}$$

With $r_i$ being the distance between $q\_i$ and the chosen point, it gives the Potential of charge distributions.

• This entire equation is based on the assumption that the reference point is at infinity $O=\infty$ and will only converge for problems with finite distributions

Remarks

• When calculating distributions, especially those of a point above lines with charges or a constant charge distribution, the potential does not require any projection terms $\cos \theta$ or $\sin \theta$ because $V$ itself is a scalar. Computing $E$, however requires the summation of only vertical components due to the horizontal ones being parallel to the distribution and cancel.