Physics - electric field at the center of the square

Question

Using the symmetry of the arrangement, determine the direction of the electric field at the center of the square due to the four charges at the corners, given that qa = qb = -3.6 μC and qc = qd = 3.6 μC .  

Calculate the magnitude of the electric field, in newtons per coulomb, at the center of the square (the location of q), given that the square is 5.5 cm on a side. 

Answer 1

Assume that the four corners of the square are a, b, c, d going in a clockwise direction, starting from top left. Then according to the problem statement, the negative charges are placed at the top two corners of the square while the positive charges are on the bottom two corners.

By definition, Electric field at a point is the amount of force that one unit positive charge placed at that point would experience.

Electric Field at the center due to a positive charge Q (=3.6uC) at the bottom-left (corner d) is given by

E_d = kQ/r²

This field is directed away from the charge Q, toward the center. In this case k=1 (medium is air or vacuum). r is the distance from the corner to the center. This can easily be calculated using simple geometry, given the dimensions of the square. Remember to convert mm to m first, though.

Similarly the field at the center due to a positive charge Q (=3.6uC) at the bottom-right (corner c) is given by

E_c = kQ/r²

This field is also directed away from the charge Q towards the center. (draw the picture to visualize).

The field at the center due to negative charges located at the upper corners have the same magnitudes but are directed towards those negative charges.

E_a = E_b = kQ/r²

The electric field at the center, caused by the negative charges on the top corners is reinforced by the field due to positive charges on the bottom corners. So, there are two different fields acting at the center - one directed along the ca-diagonal, and the other directed along the db-diagonal. They are equal in strength.

The resultant field is a vector sum of the two. The problem is now reduced to one of geometry.

Hope this gives you enough direction & detail to be able to solve the problem on your own.