Obtain an expression for the electric potential at a point due to… - Vidyadip

Question

Obtain an expression for the electric potential at a point due to several point charges.

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Answer

Consider a point $P$ at distances $r_1, r_2, r_3, \ldots, r_N$ from point charges $q_1, q_2, q_3$,
$q _{ N }$, respectively.
The electric potentials of $P$ due to the individual charges are
$V _1=\frac{1}{4 \pi \varepsilon_0} \frac{q_1}{r_1},$
$V_2=\frac{1}{4 \pi \varepsilon_0} \frac{q_2}{r_2}$
$V _{ N }=\frac{1}{4 \pi \varepsilon_0} \frac{q_N}{r_N}$
Since potential is a scalar quantity, the potential of $P$ due to all the charges is the algebraic sum of the potentials due to the individual charges.
$\therefore V = V _1+ V _1+\ldots \ldots . . .+ V _{ N }$
$=\frac{1}{4 \pi \varepsilon_0}\left(\frac{q_1}{r_1}+\frac{q_2}{r_2}+\ldots+\frac{q_N}{r_N}\right)$
$=\frac{1}{4 \pi \varepsilon_0} \sum_{i=1}^N \frac{q_i}{r_i}$
$[$Note : Electric potential is a scalar quantity. To calculate the resultant potential due to two or more point charges, the potentials due to individual charges are added as simple scalars along with its sign, determined by the sign of the $q$ that produces $V.]$

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