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?
## Answer:

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.

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JDJ