A particle of mass $m$ and charge $\mathrm{q}$, moving with velocity $\mathrm{V}$ enters Region $II$ normal to the boundary as shown in the figure. Region $II$ has a uniform magnetic field B perpendicular to the plane of the paper. The length of the Region $II$ is $\ell$. Choose the correct choice$(s)$.

Figure: $Image$

$(A)$ The particle enters Region $III$ only if its velocity $V>\frac{q / B}{m}$

$(B)$ The particle enters Region $III$ only if its velocity $\mathrm{V}<\frac{\mathrm{q} / \mathrm{B}}{\mathrm{m}}$

$(C)$ Path length of the particle in Region $II$ is maximum when velocity $V=\frac{q / B}{m}$

$(D)$ Time spent in Region $II$ is same for any velocity $V$ as long as the particle returns to Region $I$

Answer the following questions:

$(a)$ A magnetic field that varies in magnitude from point to point but has a constant direction (east to west) is set up in a chamber. A charged particle enters the chamber and travels undeflected along a straight path with constant speed. What can you say about the initial velocity of the particle?

$(b)$ A charged particle enters an environment of a strong and non-uniform magnetic field varying from point to point both in magnitude and direction, and comes out of it following a complicated trajectory. Would its final speed equal the initial speed if it suffered no collisions with the environment?

$(c)$ An electron travelling west to east enters a chamber having a uniform electrostatic field in north to south direction. Specify the direction in which a uniform magnetic field should be set up to prevent the electron from deflecting from its straight line path.

A particle of charge $q$ and velocity $v$ passes undeflected through a space with non-zero electric field $E$ and magnetic field $B$. The undeflecting conditions will hold if.

A proton with a kinetic energy of $2.0\,eV$ moves into a region of uniform magnetic field of magnitude $\frac{\pi}{2} \times 10^{-3}\,T$. The angle between the direction of magnetic field and velocity of proton is $60^{\circ}$. The pitch of the helical path taken by the proton is $……….cm$ (Take, mass of proton $=1.6 \times 10^{-27}\,kg$ and Charge on proton $=1.6 \times 10^{-19}\,kg)$

Consider a thin metallic sheet perpendicular to the plane of the paper moving with speed $'v'$ in a uniform magnetic field $B$ going into the plane of the paper (See figure). If charge densities ${\sigma _1}$ and ${\sigma _2}$ are induced on the left and right surfaces, respectively, of the sheet then (ignore fringe effects)