2.4. ELECTRIC POTENTIAL DUE TO A GROUP OR SYSTEM OF POINT CHARGES (PRINCIPLE OF SUPERPOSITION OF POTENTIALS)
State principle of superposition of electric potentials and use it to find the expression for the electric potential at a point due to a group of point charges.
Principle of superposition of potentials states that the net potential at any point due to $n$ discrete charges is given by the algebraic sum of their individual potentials at that point.
Consider $n$ discrete positive point charges $q_1, q_2, q_3, \dots, q_n$ at distances $r_1, r_2, r_3, \dots, r_n$ respectively from a point P.
Potential at P due to charge $q_1$, $V_1 = \frac{1}{4\pi\epsilon_0} \frac{q_1}{r_1}$
Potential at P due to charge $q_2$, $V_2 = \frac{1}{4\pi\epsilon_0} \frac{q_2}{r_2}$
Potential at P due to charge $q_n$, $V_n = \frac{1}{4\pi\epsilon_0} \frac{q_n}{r_n}$
Applying principle of superposition of potentials for a group of charges, we get net electric potential V at P due to n charges
$ V = V_1 + V_2 + \dots + V_n$
or
$V = \frac{1}{4\pi\epsilon_0} \frac{q_1}{r_1} + \frac{1}{4\pi\epsilon_0} \frac{q_2}{r_2} + \dots + \frac{1}{4\pi\epsilon_0} \frac{q_n}{r_n}$
or
$V = \frac{1}{4\pi\epsilon_0} \sum_{i=1}^{n} \frac{q_i}{r_i}$
Electric potential due to a group of charges in terms of position vectors
Let $\vec{r}_1, \vec{r}_2, \dots, \vec{r}_n$ be the position vectors of charges $q_1, q_2, \dots, q_n$ respectively. Let P be the point having position vector $\vec{r}_0$, where electric potential due to the group of charges is to be calculated.
Electric potential at P due to all these charges is given by
$ V = \frac{1}{4\pi\epsilon_0} \frac{q_1}{|\vec{r}_0 - \vec{r}_1|} + \frac{1}{4\pi\epsilon_0} \frac{q_2}{|\vec{r}_0 - \vec{r}_2|}+$
$+ \dots + \frac{1}{4\pi\epsilon_0} \frac{q_n}{|\vec{r}_0 - \vec{r}_n|}$
or
$V = \frac{1}{4\pi\epsilon_0} \sum_{i=1}^{n} \frac{q_i}{|\vec{r}_0 - \vec{r}_i|}$