Question

A uniform electric field $\left(\overrightarrow{E}\right)$ and magnetic field $\left({\overrightarrow{B}}_0\right)$ exist in a region as shown in figure. Both the fields are mutually perpendicular.

A positively charged particle having charge $q$ and mass $m$ is projected vertically upward, and it's crossed the line $\text{AB}$ after time $t_0$. Find the speed of projection, if the particle moves with constant velocity after time $t_0$.
$($Given, $g$ is acceleration due to gravity, and $qE=mg)$ $($ Here, the electric field and gravity exist everywhere but the magnetic field starts only after crossing the line $AB)$

• A. $g{t}_0$
• B. $2g{t}_0$
• C. $\frac{g{t}_0}{4}$
• D. $\frac{g{t}_0}{2}$

Answer 1

The correct option is B $2g{t}_0$

Let the velocity of the particle be $v$, when it crosses the line $\text{AB}$.

The value of magnetic force is,

$F_m=qv{B}_0\sin {90}^{\circ }=qv{B}_0$


Now resolving the magnetic force in vertical & horizontal direction and applying the equilibrium condition $(\because \overrightarrow{v}=\text{constant, or}\overrightarrow{a}=0)$ for particle at shown position.

$F_m\cos \theta =mg........\left(i\right)$

And, $F_m\sin \theta =qE=F_E$

$F_m\sin \theta =qE.........\left(ii\right)$

From equations $\left(i\right)\&\left(ii\right)$;

$\tan \theta =\frac{qE}{mg}$

$\tan \theta =1(\text{given,}qE=mg)$

$\therefore \theta ={45}^{\circ }$

After time $t_0$, the velocity of the particle along $x-$ axis is given by

$v_x=v_x+a_{x}t$

$v_x=0+\left(\frac{qE}{m}\right)t_0$

$v\cos \theta =\left(\frac{qE}{m}\right)t_0=g{t}_0.....\left(iii\right)$

Similarly, $v_y=u_y+a_{y}t$

$v\sin \theta =u-g{t}_0.....\left(iv\right)$

Dividing equations $\left(iv\right)\text{to}\left(iii\right)$, we get

$\frac{u-g{t}_0}{g{t}_0}=\tan \theta =\tan {45}^{\circ }=1$

$\therefore u=2g{t}_0$

Hence, option $\left(b\right)$ is correct.

Why this question ? Tip: The net force on a particle must be zero when it achieves constant velocity after time $t_0$. Concept : Draw FBD of particle and balance the acting external forces $(mg,F_E,F_m)$ to achieve equilibrium.