Question

The surface of a planet is found to be uniformly charged. When a particle of mass $m$ and no charge is thrown at an angle from the surface of the planet, it has a parabolic trajectory as in projectile motion with horizontal range $L$. A particle of mass $m$ and charge $q$ with the same initial conditions has a range $\frac{L}{2}$. The range of particle of mass $m$ and charge $2q$ with the same initial conditions is :

• A. $L$
• B. $\frac{L}{2}$
• C. $\frac{L}{3}$
• D. $\frac{L}{4}$

Answer 1

The correct option is C $\frac{L}{3}$

Let particle is thrown at angle $\alpha$ with initial velocity $u$
then when particle is neutral in charge then
$R=\frac{u^2\sin 2\alpha }{g}$...(1)
$R=L$ which is given
Now when particle has a charge q then $R=L/2$
and $g\prime =g+\left(qE/m\right)$
therefore
$R\prime =\frac{u^2\sin 2\alpha }{g\prime }$
$\frac{L}{2}=\frac{u^2\sin 2\alpha }{(g+(qE/m\left)\right)}$ ----(2)
Dividing eq. (1) by eq.(2)
we get
$2=\frac{(g+(qE/m\left)\right)}{g}$
which gives
$\frac{qE}{m}=g$
Now when the particle has charge 2q
$R\prime \prime =\frac{u^2\sin 2\alpha }{g\prime \prime }$
$R\prime \prime =\frac{u^2\sin 2\alpha }{(g+(2qE/m\left)\right)}$
$R\prime \prime =\frac{u^2\sin 2\alpha }{(g+2g)}$
$R\prime \prime =\frac{u^2\sin 2\alpha }{3g}$
$R\prime \prime =\left(L/3\right)$