Potential

Potential and Capacitance
Potential at a point is defined as the work done per unit charge when a test charge is brought to that point from infinity.

This potential at a point is not absolute potential rather a potential difference between potential at that point and potential at infinity. The potential at infinity is assumed to be zero.

If a charge q is placed at origin, then the potential at a point with position vector \(\overrightarrow{r}\) due to this charge is given by
\(V\left( \overrightarrow{r}\right)\) =\(\frac{1}{4\pi \epsilon_{0}}\frac{Q}{r}\)

If a point dipole of dipole moment \(\overrightarrow{p}\) is placed at the origin, then the potential at a point with position vector \(\overrightarrow{r} due to this dipole is given by
\((V\left( \overrightarrow{r}\right) =\dfrac{1}{4\pi \varepsilon {0}}\dfrac{\overrightarrow{p}\cdot \widehat{r}}{r^{2}}\)

If a dipole has charges -q and q separated by a distance, d, such that r >> d, then also

\((V\left( \overrightarrow{r}\right) =\frac{1}{4\pi \varepsilon {0}}\frac{\overrightarrow{p}\cdot \widehat{r}}{r^{2}})\)

For a configuration of charges \((q_1 , q_2, q_3 … q_n )\) with position vectors \((\overrightarrow{r_1}), (\overrightarrow{r_2}), (\overrightarrow{r_3})… (\overrightarrow{r_n})\), the potential at a point P is given by the superposition principle

\(V=\frac{1}{4\pi \varepsilon {0}}\left( \frac{q{1}}{r_{1P}}+\frac{q_{2}}{r_{2P}}+\ldots +\frac{q_{n}}{r_{nP}}\right)\) where r_{1P} \)is the distance between r_{1} and {P}