Question

Figure shows a charge $+Q$ clamped at a point in free space. From a large distance another charge particle of charge $-q$ and mass of $m$ is thrown towards $+Q$ with an impact parameter $d$ as shown with speed $v$. How many positions for minimum separation would be attained for the particle.

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

The correct option is B $2$

Since the electrostatic force acting on $-q$ charge always passes through the clamped charge as shown in figure.


So the net torque on $-q$ charge about clamped charge is zero i.e., ${\tau }_{net}=0$. Thus, the angular momentum is conserved in this case. so,

$L_i=L_f......\left(1\right)$
where,
$L_i=mvd=$Initial angular momentum of the system

$L_f=m{v}_0{r}_{min}=$Final angular momentum of the system

here, $r_{min}$ is the minimum seperation which should be perpendicular to the trajectory where the instantaneous velocity is $\left(v_0\right)$.

From $\left(1\right)$, we can say that

$mvd=m{v}_0{r}_{min}$

$\Rightarrow v_0=\frac{vd}{r_{min}}.....\left(2\right)$

Applying conservation of energy principle, we get

${P.E}_i+{K.E}_i={P.E}_f+{K.E}_f...\left(3\right)$

Here,
Since initially the particle was at very large distance, we can say that , initial potential energy, $P.E_i=0$

initial kinetic energy, $K.E_i=\frac{1}{2}m{v}^2$

final potential energy when they are at $r_{min}$
$P.E_f=\frac{-KqQ}{r_{min}}$ $[\because K=\frac{1}{4\pi {\epsilon }_0}]$

final kinetic energy, $K.E_f=\frac{1}{2}m{v}_0^2$

Substituting the values in $\left(3\right)$, we get

$\Rightarrow 0+\frac{1}{2}m{v}^2=\frac{-KQq}{r_{min}}+\frac{1}{2}m{v_0}^2$

$\Rightarrow \frac{1}{2}m{v}^2=\frac{-KQq}{r_{min}}+\frac{1}{2}m{v_0}^2....\left(4\right)$

Using $\left(2\right)$ and $\left(4\right)$,

$\frac{1}{2}m{v}^2=-\frac{KQq}{r_{min}}+\frac{1}{2}m\frac{v^2{d}^2}{r_{min}^2}$

Since, the above equation is the quadratic equation of $r_{min}$, on solving we will get two values of $r_{min}$. One will be positive and other will be negative. So, there will be one possible value of $r_{min}$.

Hence, option (a) is the correct answer.