In the figure below determine the point other than infinity

In the figure below, determine the point (other than infinity) at which the electric field is zero. (Let q_1 = -2.40 mu C and q_2 = 6.60 mu C.)

Solution

Let q1 be at the origin (0, 0) and q2 at x2 = 1.0 m

The electric field of a finite number of point charge is given by

E = kq/r^2

A positive test charge placed between the two charges would be pulled to the left by q1 and pushed to the left by q2. A positive test charge placed to the right of q2 would be pushed to the right by q2 more strongly (because q2 > q1 and r2 < r1) than it would be pulled to the left by q1. So the only place to look for equilibrium is to the left of q1.(where x<0 and r2 = r1 + x2)

E = - kq1/r1^2 - kq2/(r1+x2)^2 = 0

q1/r1^2 = -q2/(r1+x2)^2

(r1 + x2)/r1 = (-q2/q1)^0.5

r1 = 1.0/((6.6/2.4)^0.5 - 1) = 1.519 m

So E = 0 at r1 = 1.519 m

or x = -1.519 m