A charge + q is located at the origin, while an identical charge is located on the x axis at x = + 0.91 m. A third charge of + 3q is located on the x axis at such a place that the net electrostatic force on the charge at the origin doubles, its direction remaining unchanged. Where should the third charge be located?

Charge should be located at x = 0.7071m

Explanation:

We are given;

r = 0.5m

From coulombs law, we know that;

F = k•q1•q2/r²

k is electrostatic constant with a value of 9 x 10^ (9) N. m²/C²

Now force between q and q is;

F1 = 9 x 10^ (9) • (q²) / 0.5² = 3.6 x 10^ (10) q²

Now force between q and 2q is;

F2 = 9 x 10^ (9) • (q x 2q) / r²

F2 = 1.8 x 10^ (10) q²/r

Net force = F1 + F2

F_net = 3.6 x 10^ (10) q² + 1.8 x 10^ (10) q²/r²

To get the required location,

F_net = 2F1

Thus,

3.6 x 10^ (10) q² + 1.8 x 10^ (10) q²/r² = 2 (3.6 x 10^ (10) q²)

Divide through by q² to get;

3.6 x 10^ (10) + (1.8 x 10^ (10)) / r = 2 (3.6 x 10^ (10))

(1.8 x 10^ (10)) / r² = (7.2 x 10^ (10)) - (3.6 x 10^ (10))

(1.8 x 10^ (10)) / r² = 3.6 x 10^ (10)

r² = (1.8 x 10^ (10)) / 3.6 x 10^ (10)

r² = 0.5

r = √0.5

r = 0.7071m