A particle of mass $\mathrm{m}$ and charge $\mathrm{q}$ moving with velocity $\mathrm{v}$ enters region-b form region-a along the normal to the boundary as shown is the figure. Region-b has a uniform magnetic field $B$ perpendicular to the plane of the paper. Also, region-b has length L. Choose the correct statements :

• Correct answer
  A

  The particle enters region-c only if $\mathrm{v}>\frac{\mathrm{qLB}}{\mathrm{m}}$
• B

  The particle enters region-c only if $\mathrm{v} < \frac{\mathrm{qLB}}{\mathrm{m}}$
• Correct answer
  C

  Path of the particle is a circle in region-b
• Correct answer
  D

  Time spent in region -b is independent of velocity v

Solution

Particle enter region 'c' if Radius is greater than ' $L$ ' $\Rightarrow \frac{m V}{q B}>L \Rightarrow V>\frac{q B L}{m}$ $\because$ B is uniform and $\overrightarrow{\mathrm{V}} \perp \overrightarrow{\mathrm{B}}$ so. Path is circular in region b $$ \text { Time spent }=\frac{\text { Angle cov ered }}{\omega} \quad \omega=\frac{\mathrm{qB}}{\mathrm{m}} $$ Now angle covered depens on whether particle goes to region c or comes back to region 'a'

Hence, the correct answer is options: 'A', 'C', 'D'.