Question

Two free point charges +q and +4q are a distance L apart. A third charge is 

placed so that the entire system is in equilibrium. Find the location, 

magnitude and sign of the third charge.

Answer 1

$k\frac{q\cdot 4q}{L^2}=k\frac{q\cdot q_0}{x^2}$

and

$k\frac{q\cdot 4q}{L^2}=k\frac{4q\cdot q_0}{(L-x)^2}$

We have

$k\frac{q\cdot q_0}{x^2}=k\frac{4q\cdot q_0}{(L-x)^2} \to 3x^2+2Lx-L^2=0\to x=L/3$ to the right of the charge $q$ . Answer

$k\frac{q\cdot 4q}{L^2}=k\frac{q\cdot q_0}{(L/3)^2}\to q_0=-\frac{4}{9}q$ . Answer