A particle of mass $m$ and charge $q$ is in an electric and magnetic field given by

$\vec E = 2\hat i + 3\hat j ;\, B = 4\hat j + 6\hat k$

The charged particle is shifted from the origin to the point $P(x = 1 ;\, y = 1)$ along a straight path. The magnitude of the total work done is

At $t$ = $0$, a positively charged particle of mass $m$ is projected from the origin with velocity $u_0$ at an angle $37^o $ from the $x-$axis as shown in the figure. A constant magnetic field ${\vec B_0} = {B_0}\hat j$  is present in space. After a time interval $t_0$ velocity of particle may be:-

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 $I$

A charged particle projected in a limited magnetic field according to figure. The charged  particle does not strike to the opposite plate provided

A proton of mass $1.67\times10^{-27}\, kg$ and charge $1.6\times10^{-19}\, C$ is projected with a speed of $2\times10^6\, m/s$ at an angle of $60^o$ to the $X-$ axis. If a uniform magnetic field of $0.104\, tesla$ is applied along the $Y-$ axis, the path of the proton is