91Ó°ÊÓ

The electric potential energy per unit charge at any point in space is termed as electric potential at that point.

The formula for the electric potential at a distance r from a point charge Q is as follows:

\(V = \frac{{kQ}}{r}\)

Here, k is the Coulomb’s constant.

The length of the side of a square is l.

The charges at the three corners are \( + 3Q\), \( + Q\) and \( - 2Q\).

The schematic diagram for the problem can be drawn as:

The expression for the potential at point A due to point charge\( + Q\)can be written as:

\({V_1} = \frac{{kQ}}{{\sqrt 2 l}}\)

The expression for the potential at point A due to point charge\( - 2Q\)can be written as:

\(\begin{aligned}{V_2} &= \frac{{k\left( { - 2Q} \right)}}{l}\\{V_2} &= \frac{{ - 2kQ}}{l}\end{aligned}\)

The expression for the potential at point A due to point charge\( + 3Q\)can be written as:

\(\begin{aligned}{V_3} &= \frac{{k\left( { + 3Q} \right)}}{l}\\{V_3} &= \frac{{3kQ}}{l}\end{aligned}\)

The total potential at point A can be calculated as:

\(\begin{aligned}{V_{\rm{A}}} &= {V_1} + {V_2} + {V_3}\\{V_{\rm{A}}} &= \left( {\frac{{kQ}}{{\sqrt 2 l}}} \right) - \left( {\frac{{2kQ}}{l}} \right) + \left( {\frac{{3kQ}}{l}} \right)\\{V_{\rm{A}}} &= \frac{{kQ}}{l}\left( {\frac{1}{{\sqrt 2 }} - 2 + 3} \right)\\{V_{\rm{A}}} &= \frac{{kQ}}{l}\left( {1 + \frac{1}{{\sqrt 2 }}} \right)\end{aligned}\)

Solve further as,

\(\begin{aligned}{V_{\rm{A}}} &= \frac{{kQ}}{l}\left( {\frac{{\sqrt 2 + 1}}{{\sqrt 2 }}} \right)\\{V_{\rm{A}}} &= \frac{{kQ}}{{\sqrt 2 l}}\left( {\sqrt 2 + 1} \right)\\{V_{\rm{A}}} &= \frac{{\sqrt 2 kQ}}{{2L}}\left( {\sqrt 2 + 1} \right)\end{aligned}\)

Thus, the total potential at point A is \(\frac{{\sqrt 2 kQ}}{{2L}}\left( {\sqrt 2 + 1} \right)\).