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Last updated date: 09th Dec 2023
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Determine the electrical potential energy for a system of three point charges.

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Hint: Electrical potential energy, charge at a point is defined as the amount of work done, bringing the charge from infinity to that point. It is denoted by $U$. The difference in potential energy between two points in an electric field is called electrical potential energy.

First we calculate the difference between two point charges, later we can discuss about three point charges. Electric potential energy of the system for two point charges. Suppose assume that two charges ${q_1}$ and ${q_2}$ are situated at a distance of $r$.The electrical potential energy is:
$U = \dfrac{1}{{4\pi {e_ \circ }}} \times \dfrac{{{q_1}{q_2}}}{r}$
When charge ${q_1}$is bought from infinity to certain position, no work is done. There is no other charge to repel each other. Now, the position of charge is ${q_2}$
${V_1} = \dfrac{1}{{4\pi {e_ \circ }}} \times \dfrac{{{q_1}}}{r}$ this equation is from electric field,
Therefore the work done bringing the charge ${q_2}$ to infinity to its own position
$W = U = {V_1}{q_2}$
In the above equation we are substituting the ${V_1}$ then we get,
$U = \dfrac{1}{{4\pi {e_ \circ }}} \times \dfrac{{{q_1}{q_2}}}{r} \to \left( 1 \right)$
Both the charges are in the same nature, the potential energy is positive for unlike charges it will be negative.

Electrical potential energy of a system of three charges: Consider three charges ${q_1}, {q_2}, {q_3}$, the charges ${q_2}$ and ${q_3}$ initially at finite distance from the charge ${q_1}$ work done bringing charge ${q_2}$ from infinity to point,
${W_{12}} = \dfrac{1}{{4\pi {e_ \circ }}} \times \dfrac{{{q_1}{q_2}}}{{{r_{12}}}} \to \left( 2 \right)$
Work done bringing charges ${q_3}$ then we get,
${W_{123}} = {V_1}{q_3} + {V_2}{q_3} \\ \Rightarrow {W_{123}} = \dfrac{1}{{4\pi {e_ \circ }}} \times \dfrac{{{q_1}{q_3}}}{{{r_{31}}}} + \dfrac{1}{{4\pi {e_ \circ }}} \times \dfrac{{{q_2}{q_3}}}{{{r_{23}}}} \to \left( 3 \right) \\$
Therefore the total work done is, work done by two point charges and work done by three point charges, here we get the total work done at some infinite point
$W = {W_{12}} + {W_{123}} \\ \Rightarrow W = \dfrac{1}{{4\pi {e_ \circ }}} \times \dfrac{{{q_1}{q_2}}}{{{r_{12}}}} + \dfrac{1}{{4\pi {e_ \circ }}} \times \dfrac{{{q_1}{q_3}}}{{{r_{31}}}} + \dfrac{1}{{4\pi {e_ \circ }}} \times \dfrac{{{q_2}{q_3}}}{{{r_{23}}}} \\ \Rightarrow W = \dfrac{1}{{4\pi {e_ \circ }}}\left[ {\dfrac{{{q_1}{q_2}}}{{{r_{12}}}} + \dfrac{{{q_1}{q_3}}}{{{r_{31}}}} + \dfrac{{{q_2}{q_3}}}{{{r_{23}}}}} \right] \\$
Total work done in electrical potential energy is stored in the form of potential energy,
The electrical potential energy for a system of three point charges is,
$\therefore U = \dfrac{1}{2}\left[ {\dfrac{1}{{4\pi {e_ \circ }}}\sum\limits_{allpairs} {\dfrac{{{q_i}{q_j}}}{{{r_{ij}}}}} } \right] \to \left( 4 \right)$
In the above equation (4) we have multiplied $\dfrac{1}{2}$ because each pair comes two times.

Note:The electric potential is charged at a point to bring charge from infinity point to certain point; according to the above data we have calculated the electrical potential charge for two points and for three points. Same method is used to calculate the electrical potential at three points.