Where on the X-Axis Does the Electric Field Equal Zero?

[confused1]

A charge of -3.4 × 10-9 C is at the origin and a charge of 6.9 × 10-9 C is on the x-axis at x = 3 m. At what location on the x-axis is the electric field zero?

HELP: Which region is the only possible one where the E-field could be zero (to the left of Q1, in between the charges, or to the right of Q2)? Now use Coulomb's Law.

HELP: The field cannot be zero in between the charges because the field from both charges points to the left there. To the left or right of the charges, the fields from the two charges point in opposite directions, so it appears that they can cancel on either side - but by looking at the magnitudes of the charges we can narrow the possibilities further.

The field cannot be zero to the right of Q2 because anywhere to the right of Q2 the magnitude of the field due to Q2 is greater than the magnitude of the field due to Q1 (Q2 is greater than Q1 and the distance from Q2 is always smaller than the distance from Q1).

The field must be zero to the left of Q1 at one point since somewhere to the left of Q1 the magnitude of the field due to Q2 is equal to the the magnitude of the field due to Q1 (Q2 is greater than Q1, but the distance from Q2 is always greater than the distance from Q1).

HELP: Now set up an expression for the magnitude of the field from each charge at some point to the left of Q1. For example: E1 = kQ1/r2 and E2 = kQ2/(r+3)2 where r is the magnitude of the distance from Q1. How how can you finish the problem?

Remember that at the point where the field is zero the field from one charge is equal and opposite the field from the other charge. Watch your signs, and be careful choosing which answer to use (your expression for r may be quadratic and give you two answers).

 

[lightgrav]

if you call x the distance from the small nearby charge,
and call D the distance from the Distant large charge,
then setting magnitudes of E-field contributions equal
gives D/x = sqrt(Q/q); use it to replace D in D - x = 3m