Force is a vector, and it's critical to add forces as vectors whenever there is more than one force being exerted. Consider some two-dimensional situations, all involving the net force the positive charge at the center of a square experiences because of equal-magnitude charges placed at each corner of the square.

Configuration 1. The net force on the positive center charge is toward the top-right.
What are the signs of the equal-magnitude charges occupying each corner?
How many possible configurations can you come up with that will produce the desired force?

1. 1
2. 2
3. 3
4. 4
5. either 0 or more than 4

There are two configurations that will result in a force toward the top right corner.

If each side of the square has a length of 2L, and each charge has a magnitude Q, what is the magnitude of the net force on the center charge?

Two of the forces cancel, and the other two are the same size and direction. The distance from the center charge to any other charge is:

r = L, so r² = 2L²

Therefore F5

=

2kQ2 r2

=

2kQ2 2L2

=

kQ2 L2

Configuration 2. The net force on the positive center charge is straight down.
What are the signs of the equal-magnitude charges occupying each corner?
How many possible configurations can you come up with that will produce the desired force?

1. 1
2. 2
3. 3
4. 4
5. either 0 or more than 4

There is only one way to achieve this result.

If each side of the square has a length of 2L, and each charge has a magnitude Q, what is the magnitude of the net force on the center charge?

All the horizontal components cancel and all the vertical components add. Once again:
r = L, so r² = 2L²

Therefore F5

=

4kQ2 r2

x

1

=

4kQ2 2L2

x

1

=

kQ2 L2

Configuration 3. The net force on the positive center charge is zero.
What are the signs of the equal-magnitude charges occupying each corner?
How many possible configurations can you come up with that will produce the desired force?

1. 1
2. 2
3. 3
4. 4
5. either 0 or more than 4

There are four ways to get the forces to cancel.