Two point charges, one with charge +q and the other with charge +9q, are placed a distance "d" apart. It is possible to place a third charge so that the net electrostatic force on all three charges is zero. What are the sign, magnitude, and position of this third charge (in terms of q and d)?
The Attempt at a Solution
Since both of these charges are positive, I believe that the third charge needs to be negative, to "hold" the first two in place. If it were positive than at least one would be pushed away and not held in place.
In regards to magnitude, the Coulomb equation would yield (9kq^2)/(d^2). Since this is obviously depended on distance as an inverse square law, I think that this charge will be placed in the middle of the two positive charges, closer to the smaller one. The problem is I don't know how to determine the exact position of the charge. I'm thinking that if it is placed at say (d/4), then the magnitude would be (9kq^2)/(16d^2).
My problem with this question is twofold. First, I don't know how to find the actual distance, which prevents me from finding the magnitude. Secondly, I have thought about this problem at length and mostly every answer I've gotten so far is from thinking. I don't really have any math to back it up. Do you set the two Coulombs equations equal to each other? How do you go about solving this? Many thanks!