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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 $L / 2$. The range of particle of mass $m$ and charge $2 q$, with the same initial conditions is
$L$
$\frac{L}{2}$
$\frac{L}{3}$
$\frac{L}{4}$
Solution
(c)
For uncharged particle in projectile motion, range is
$L=\frac{u^2 \sin 2 \theta}{g} \quad \dots(i)$
For a charged particle, acceleration is $\left(g+\frac{q E}{m}\right)$
So, its range will be $\frac{L}{2}=\frac{u^2 \sin 2 \theta}{\left(g+\frac{q E}{m}\right)}$ Equating for $L$ from above equations, we get
$\frac{1}{g} =\frac{2}{g+\frac{q E}{m}}$
$\Rightarrow \quad g=q E / m \quad \dots(ii)$
Now, for a particle of mass $m$ and charge $2 q$, range will be
$R=\frac{u^2 \sin 2 \theta}{g+\frac{2 q E}{m}}$
Here, $\quad q \frac{E}{m}=g$ [From Eq. $(ii)$]
$=\frac{u^2 \sin 2 \theta}{3 g}$
$\text { Again, } \frac{u^2 \sin 2 \theta}{g}=L$ [From Eq. $(i)$]
$=\frac{L}{3}$
