Answer to Question #248930 in Physics for tayyab

Question

A point charge is placed at each corner of a square with side length a. All charges have magnitude q. Two of the charges are positive and two are negative. What is the direction of the net electric field at the center of the square due to the four charges, and calculate its magnitude in terms of q and a?

Answer 1

Find the x-and y-components of the field:

The field from a single positive charge:

$$E_+=\frac{kq}{r^2}=\frac{kq}{(a\sqrt2/2)^2}=\frac{2kq}{a^2}.$$

The resultant field along x-axis is zero (the vectors cancel each other), along y-axis it is

$$E_p=2E_+\cos45°=\frac{4kq}{a^2}·\frac{\sqrt2}2=\frac{2kq\sqrt2}{a^2}.$$

The magnitude and direction of the resultant field from negative charges is absolutely the same:

$$E_n=2E_-\cos45°=\frac{4kq}{a^2}·\frac{\sqrt2}2=\frac{2kq\sqrt2}{a^2}.$$

The total field is

$$E=E_++E_-=\frac{4kq\sqrt2}{a^2}.$$