Question

As shown in the figure the uniform magnetic field between the two identical plates is $B$. There is a hole in the plate, If through this hole a particle of charge $q$, mass $m$ and kinetic energy $E$ enters the magnetic field, then the particle will not collide with the upper plate provided :

• A. $B>\frac{2mE}{qd}$
• B. $B>\frac{\sqrt{2mE}}{qd}$
• C. $B<\frac{2mE}{qd}$
• D. $B<\frac{\sqrt{2mE}}{qd}$

Answer 1

The correct option is B $B>\frac{\sqrt{2mE}}{qd}$

The particle will bend in the magnetic field.
The particle will not collide with the upper plate if radius of the semicircular trajectory is less than the distance d between the plates.
Equating Lorentz force and centripetal force for circular motion,
$qvB=\frac{m{v}^2}{r}$
$\therefore r=\frac{mv}{qB}$
Since $r<d$, $\frac{mv}{qB}<d$
$\therefore B>\frac{mv}{qd}=\frac{m\sqrt{\frac{2E}{m}}}{qd}=\frac{\sqrt{2mE}}{qd}$