Question

Two point electric charges of value `q` and `2q` are kept at a distance `d` apart from each other in air. A third charge `Q` is to be kept along the same line in such a way that the net force action on `q` and `2q` is zero. Calculate the position of charge `Q` in terms of `q` and `d`.
A. `(d)/(sqrt(2)-1)`
B. `(d)/(sqrt(2)+1)`
C. `(d)/(sqrt(3)+1)`
D. `(d)/(sqrt(3)-1)`

Answer 1

Correct Answer - b
Suppose that the charge `q, 2q` and `Q` are placed as shown in the figure

It follows that the net force on charge `q` and `2q` can be zero, only if the charge `Q` is of opposite sign to those of charges `q` and `2q`. Therefore, if charges `q` and `2q` are positive, then charge `Q` must be negative in nature. Let the distance of charge `Q` (negative) from `q` be equal to `x`.
For one charge `q` to be zero, `F_(AB)=F_(AC)`
Or `(1)/(4piepsilon_(0)).(q(2q))/(d^(2))= (1)/(4pi epsilon_(0)).(qQ)/(x^(2))` ....(ii)
or `(Q)/(q)=(2x^(2))/(d^(2))`
For force on charge `2q` to be zero `F_(BA)=F_(BC)`
or `(1)/(4 piepsilon_(0)).(q(2q))/(d^(2))= (1)/(4piepsilon_(0)).((2q)Q)/((d-x)^(2))`
or `(Q)/(q)=((d-x)^(2))/(d^(2))`....(ii)
For the equations (i) and (ii), we get
`x=(d)/(sqrt(2)+1)` or `(d)/(sqrt(2)-1)`