Three charges in a line

Three charged balls are placed in a line.
Ball 1 has an unknown charge and sign, but we know that it is a distance 2r to the left of ball 2.
Ball 2 is positive, with a charge of +Q.
Ball 3 has an unknown non-zero charge and sign, but we know that it is a distance r to the right of ball 2.

Ball 3 is in equilibrium - it feels no net electrostatic force due to the other two balls.

What is the sign of the charge on ball 1?

  1. Positive
  2. Negative
  3. We can't tell unless we know the sign of the charge on ball 3.











Even though we don't know much about ball 3, we have enough information to determine that ball 1 has a negative charge. Ball 3 is in equilibrium because it experiences equal-and-opposite forces from the other two balls. Flipping the sign of the charge on ball 3 reverses both these forces, so they still cancel.

What is the charge (sign and magnitude) of ball 1?

Is it possible to answer this question, or do we need to know the sign and/or magnitude of the charge of ball 3?










Again, focus on the fact that the two forces acting on ball 3 are equal-and-opposite. Changing the magnitude of the charge on ball 3 changes both forces by the same amount, so they still cancel.

All we need to know is that ball 1 is three times further from ball 3 than ball 2 is. Because the distance is squared in the force equation the factor of 3 becomes a factor of 9, so we need a similar factor in the numerator, where the charge is. Thus, Q1 = -9Q.

The neat thing here is that we don't need to know anything about ball 3. It can be anything - we can put whatever charge we like at the location of ball 3 and it will feel no net force because of ball 1 and ball 2.

Thus, ball 3 isn't special - it's the location that's special. So, let's get rid of ball 3 from the picture and think about how the two charged balls influence the point where ball 3 was.

Essentially we think about it like this.

Ball 2's effect on ball 3 is given by Coulomb's Law:

F =
k Q2 Q3
r2

Ball 2's effect on the point where ball 3 was is given by:

Electric Field :   E =
k Q2
r2

We can say that the electric field from ball 1 and the electric field from ball 2 cancel out at the location where ball 3 was.