Question

In the diagram shown, the charge $+Q$ is fixed. Another charge $+2q$ and mass $M$ is projected from a distance $R$ from the fixed charge. Minimum separation between the two charges if the velocity becomes $\frac{1}{\sqrt{3}}$ times of the projected velocity, at this moment is (Assume gravity to be absent):

• A. $\frac{\sqrt{3}}{2}R$
• B. $\frac{1}{\sqrt{3}}R$
• C. $\frac{1}{2}$
• D. $Noneofthese$

Answer 1

The correct option is A $\frac{\sqrt{3}}{2}R$

The only force on the charges is electrostatic force, which is an internal force on this two charge system(+Q applies force on +2q and vice-versa). So conserving linear momentum is out of option, but angular momentum can be conserved along an axis passing through the +Q charge(As torque along this axis is zero because force is passing through the axis itself all the time). Also, at minimum distance, velocity of approach must be zero, i.e. velocity must be perpendicular to the line joining the charges.Hence:

$\underset{L_i}{\to }=\underset{L_f}{\to }$

$\Rightarrow MVRSin\left({30}^{\circ }\right)=M\left(V/\sqrt{3}\right)r$

$\Rightarrow r=\sqrt{3}/2\times R.$