Question

A particle of mass $M$ and charge $q$ moving with velocity $v$ enters $Region-2$ normal to the boundry as shown in the figure. $Region-2$ has uniform magnetic field $B$ perpendicular to the plane of the paper. The length of the $Region-2$ is $l$. Choose the correct choice.
(A) The particle enters $Region-3$ only, if its velocity $v>qlB/m$.
(B) The particle enters $Region-3$ only, if its velocity $v<qlB/m$
(C) Path length of the particle in $Region-2$ is maximum when velocity $v=qlB/m$.
(D) Time spend in $Region-2$ is same for any velocity $v$ as long as the particle return to $Region-1$.

• A. $A$ only true
• B. $A,C$ true
• C. $A,C,D$ true
• D. All are true

Answer 1

The correct option is C $A,C,D$ true

When the particles enter in magnetic field, a force tends to move the particle in circular path.

The radius of the circular path is given by:

$R=\frac{mv}{qB}$

Now, for the particle to enter region $III$, $r>l$ (path shown by dotted line)

$\Rightarrow \frac{mv}{qB}>l\Rightarrow v>\frac{qlB}{m}$

To cover maximum pat in region $II$, $r=l$

$\therefore 1=\frac{\left(mv\right)}{\left(q\left(B\right)\right)}rArrv=\frac{\left(qlB\right)}{\left(m\right)}$

The time taken by the particle at any velocity in the region $II$ will be same if the particle returns back to the region $I$ because the radius of the particle will adjust accordingly.

Hence, the correct option is $Option\left(c\right)$ - $\left(A\right),\left(C\right)and\left(D\right)true$