THere are a number of ways to do this, some more difficult than others.
If you note the E due to q2 is upward, and the E due to q1 is horizontal, then the E due to q3 must add to zero. That is, the E due to q3 in the horizontal direction must be equal and opposite to q1, and The E due to q3 in the vertical must be equal and opposite to Q2
So dealing with q3 first.
letting o be the distance of the horizontal (o^2=d^2+4d^2=5d^2; or o=dsqrt5)
Ehorizontal=k q/o^2*2d/o=2kqd/o^3
Evertical= kq/o^2*d/o=kqd/o^3
So now setting the vertical of q3 equal to the vertical of q2
k*q2/d^2=kqd/o^3
q2=q (d/o)^3=q(d/dsqrt5)^3=q/5sqrt5
now, setting the horizontal of q3 equal to the horizontal of q1
kq1/4d^2=2kqd/o^3
q1=8q (d/o)^3 and you can finish the algebra.
Check all this math, I did it in my head keying it in.
A rectangle has a length of 2d and a height of d. Each of the following three charges is located at a corner of the rectangle: +q1 (upper left corner), +q2 (lower right corner), and -q (lower left corner). The net electric field at the (empty) upper right corner is zero. Find the magnitudes of q1 and q2. Express your answers in terms of q.
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