## Potential Due to a Distribution of Electric Charges

The distribution of electric charges gives rise to a potential field. In the region of electric charges, every point in space has a potential
$V$
. This potential field has some a special property.
It is linear. If we have two charge distributions and bring them together the potential at each point is the sum of the two individual potentials.
The potential at a point A due to a point charge
$q$
at a point B is
$V=\frac{1}{4 \pi \epsilon_0} \frac{q}{r_{AB}}$
where
$r_{AB}$
is the distance from A to B and
$\epsilon_0 = 8.854 \times 10^{-12} F/m$
.
Suppose then that we have five charges as shown.
Each gives rise to a potential field, but the overall potential is
\begin{aligned} V &= V_1+V_2+V_3+V_4+V_5 \\ &= \frac{1}{4 \pi \epsilon_0} \frac{q_1}{d_1}+ \frac{1}{4 \pi \epsilon_0} \frac{q_2}{d_2}+ \frac{1}{4 \pi \epsilon_0} \frac{q_3}{d_3} + \frac{1}{4 \pi \epsilon_0} \frac{q_4}{d_4} + \frac{1}{4 \pi \epsilon_0} \frac{q_5}{d_5} \\ &= \frac{1}{4 \pi \epsilon_0} (\frac{q_1}{d_1} + \frac{q_2}{d_2} + \frac{q_3}{d_3} + \frac{q_4}{d_4} + \frac{q_5}{d_5}) \end{aligned}