A charge of +q is located at the origin,?

Question

A charge of +q is located at the origin, while an identical charge is located on the x axis at x = 0.55 m. A third charge of +5 q is located on the x axis at such a place that the net electrostatic force on the charge at the origin quadruples, its direction remaining unchanged. Where should the third charge be located?

X = -------------------M

Answer 1

Force on origin charge

using F=kq^2/r^2

Dropping  common K

total force = 4 x original force: 4*q^2/(0.55)^2

total force = F2+F3 = 4*q^2/(0.55)^2

4*q^2/(0.55)^2 = q^2/(0.55)^2 + (5q*q)/(x)^2

4*q^2/(0.55)^2 - 1*q^2/(0.55)^2 = (5q*q)/(x)^2

or

3*q^2/(0.55)^2  = (5q*q)/(x)^2

x^2= (5q*q)/(3*q^2/(0.55)^2)

drop q^2

x=sqr(5 / (3 / (0.55^2))) = 0.71 meter

Where the third charge be located.

Answer 2

the force of the second charge acting on the first charge is given by:

F2=kq^2/(0.55)^2

the force of the third charge on the first charge is:

F3=5kq^2/x^2  and we want to solve for x

if the total force on the initial charge quadruples when F3 is added, we know that F3 must be 3 times F2 (so that F3+F2 is four times F2)

so, we know that F3=3F2

or:  5kq^2/x^2=3kq^2/(0.55)^2

dividing out common terms of k and q^2, we get:

x^2=(5/3)(0.55)^2 or

x=sqrt[5/3](0.55)=1.29x0.55=0.71m