6)

In the figure above, four charges are arranged. If the magnitudes of all the charges q are all the same and the distance r between them is as shown above, what is the magnitude of the net force on the bottom right charge in terms of q, r, and k (where )?

Although you do not care about the direction of our net vector in the end, you do have to take orientation into account to find our magnitude. Number the charges from the top left and going clockwise as 1, 2, 3, and 4. Drawing out the different forces on the bottom right charge (charge 3), you get

Now solve each of these force vectors

These two are the simpler force vectors to solve. In order to solve the charge 1’s force on charge 3, the distance first needs to be solved. Because they are ordered in a square using a diagonal creates a 45° - 45° - 90° triangle. The distance in this case is . So,

First, combine the with to get an equivalent vector that will be in the same direction as ,

They form a 45° - 45° - 90° triangle when laid tail-to-end so,

This vector points in the same direction as , so to get the magnitude of the net force, simply add on 3 with ,