two fixed point charges +4e and +e units are separated by a distance a .where should the third point charge be placed for it to be in equilibrium ?

Given,

q_{A} = +4e

q_{B }= +e

AB = x

Let, Q be the third point charge located at some point P.

AP = x

therefore, PB = (a-x)

Force on Q due to q_{A},

F_{1} = kq_{A}Q/x^{2}

Force on Q due to q_{B}

F_{2}= kQq_{B}/(a-x)^{2}

Since the charge is in equilibrium,

F_{1} = F_{2}

kq_{A}Q/x^{2} = k q_{B}Q/(a-x)^{2}

4e/x^{2} = e/(a-x)^{2}

4/x^{2 }= 1/(a-x)^{2 }(Taking square root on both sides)

2/x = 1/(a-x)

2(a-x) = x(1)

2a - 2x = x

2a = 3x

x = 2/3a

Therefore, the third point charge Q should be placed 2/3a from A.