Four point charges each having charge q are located at the corners of a square having sides of length a. (a) find an expression for the total electric potential at the center of the square due to the four charges. (use any variable or symbol stated above along with the following as necessary: ke.)

Explanation:

Electric Potential of Point Charges

The electric potential from a point charge Q at a distance r from the charge is

Where k is the Coulomb's constant. The total electric potential for a system of point charges is equal to the scalar sum of their individual potentials. The potential is not a vector, so there is no direction or vectors to deal with.

We are required to compute the total electric potential in the center of the square. We need to know the distance from each corner to the center. The diagonal of the square is

where a is the length of the side.

The distance from any corner to the center is half that diagonal, thus

The total potential in the center is  

Please note all the potentials must be calculated including the sign of the charges. Since all the charges are equal to Q, and the distances are the same, the total potential is 4 times the individual potential of each charge.

Operating

Thus:

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