Question

A line of charge is bent into a square with a side of 2 cm and a charge density of 3 microC/cm. Using Coulomb's law, determine the electric field at all points along the axis of the square (the line through the center of the square and perpendicular to it).

Answer 1

The drawing for this problem looks like:

The electric field due to the line charge distributed in the square is equal to the contribution of each side of the square.

The contribution of the side "A" has the following direction for the net electric field:

let's call the vector direction for the electric field due to the side A.

and the can apply the same procedure for the other sides, and we get:

and now we can write the net electric field due to all the sides of the square as:

The electric field due to each side has the same module, because the distance r:

is the same for all of them.

and now let's find the direction of the net electric field:

Now, let's find the electric field E due to one side:

and as: we can write the differential electric field as:

and now we can write the components in X and Y directions:

Let's find the X component:

let's compute now the constribution for the Y component of the electric field:

and now that we do have both components, we can write the magnitude of the electric field due to one of the sides of the square with linear charge :

In the case of the square, the radius r is equal to:

and plugginf the givens, we get: