I'm so close (I think). Electric Potential Question - Help Needed?

Question

A -3.2 { nC} charge is on the x-axis at x_1 = -9 {cm} and a 14.6 {nC} charge is on the x-axis at x_2 = 16 {cm}.

***At what point or points on the y-axis is the electric potential zero?

The way I've tried solving it is by setting (q1/r2) = (q2/r2)

So 3.2*sqrt((16 cm)^2 + y^2) = 14.6*sqrt((-9 cm)^2 + y^2)

Eventually getting -14644.52 = 202.92 y^2

And y = +/- 8.5

Unfortunately this answer isn't being marked as correct. Does anyone see something wrong with the way I'm trying to solve this?

Answer 1

No, you seem to be correct. (I think  you meant, and solved, q1/r1 = q2/r2, so q1r2 = q2r1.)

After squaring both sides and rearranging,

3.2^2*16*2 - 14.6^2*(-9)^2 indeed = -14644.52

And on the other side you have 14.6^2*y^2 - 3.2^2*y^2 =  202.92y^2

y = ±sqrt(14644.52/202.92) = ±8.495

But I think I know what the problem is. We've overlooked the fact that we're actually taking the square root of a negative number, thinking it's insignificant. But with the charges and x-axis distances given, there is no solution. Think: How can the ratio of distances ever be equal to the ratio of charges (roughly 4.6 to 1)? The ratio of x-distances is 16/9, about 1.8 to 1. So the ratio of the distances to a point on the y axis starts ~= 1.8 when y=0, and we know it approaches 1 as y increases. I believe this problem doesn't have a solution.

There will certainly be a point on the x axis with zero potential, somewhere to the left of the y axis. Along the y axis, the larger charge dominates.