Question

Two point electric charges of values 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 acting on q and 2q is zero. Find the location of the third charge from charge q :

• A. $\frac{d}{1+\sqrt{2}}$
• B. $\frac{d}{1-\sqrt{2}}$
• C. $\frac{d}{1+\sqrt{6}}$
• D. $\frac{d}{1-\sqrt{6}}$

Answer 1

The correct option is A $\frac{d}{1+\sqrt{2}}$

Q should be negative and it should be placed somewhere between q and 2q. Let Q is placed at a distance $x$ from q and $(d-x)$ from 2q.

For net force on q to be zero, $\frac{kqQ}{x^2}=\frac{kq\left(2q\right)}{d^2}...\left(1\right)$

For net force on 2q to be zero, $\frac{k2qQ}{(d-x{)}^2}=\frac{kq\left(2q\right)}{d^2}...\left(2\right)$

From (1) and (2), $\frac{1}{x^2}=\frac{2}{(d-x{)}^2}$

or $2x^2=d^2-2dx+x^2$

or $x^2+2xd-d^2=0$

or $x=\frac{-2d\pm \sqrt{4d^2+4d^2}}{2}=-d+d\sqrt{2}$ (as Q in between q and 2q so negative value of x is invalid)

or $x=\frac{d(\sqrt{2}-1)(\sqrt{2}+1)}{(\sqrt{2}+1)}=\frac{d}{(\sqrt{2}+1)}$