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