Field and Potential from Square Array of Charges

## Question:

Small spherical charges of +2, -2, +3 & -6 (all x 10^-9) Coulombs are placed in order at the corners of a square of diagonal 0.20 metres.
(a)what is the electric potential at the centre?
(b)What is the size of electric field at the centre?
(c)Are there any points within the square at which electric potential is zero?
(d)Are there any points within the square at which electric field is zero?

Electric potential is a scalar quantity so for charges arranged in a square you can calculate the potential at any point from each sphere independently and just add them up. For the electric field which is a vector quantity calculate the vectors at any point independently for each charge and add them as vectors. To decide if there is a point of zero electric field or zero potential, write an expression for each based on summing the field or potential from each charge. Then setting those expressions equal to zero, solve for the x and y which satisfy those equations. That will tell you whether there is a point of zero field or zero potential in the square.

I suspect that the recipe I gave above for finding if the square contains any points of zero potential or field may contain some mathematical complications. As an alternative we might consider some symmetry arguments.

First with regard to the field, if all the charges had equal magnitude, then clearly the center of the square would be a point of zero field, where the contributions from the positive charges are equal and opposite and the contributions from the the negative charges are equal and opposite. Now if we allow the charge at a positive corner to increase, its contribution to the field will be larger than that from the opposite corner. Still somewhere on the line between those corners there will be a point where the field from those corner charges add to zero. When the other two corner charges are considered they will tend to shift the zero field point back toward the center of the square somewhat. Now if we let one of the negative charges increase in magnitude that will tend to move the point of zero field off the original diagonal but it will remain in the square as long as the increased charges are less than infinite.

Now considering the potential, if all the charges had equal magnitude, then the center of the square would be a point of zero potential, where the contributions from the negative charges and positive charges add to zero. Again, adjusting the amount of charge on two of the corners will just shift the location of this point around the square.

This information is brought to you by M. Casco Associates, a company dedicated to helping humankind reach the stars through understanding how the universe works. My name is James D. Jones. If I can be of more help, please let me know.

JDJ