$A$ particle of charge per unit mass $\alpha$ is released from the origin with a velocity $\vec{v} = v_0 \hat{i}$ in a uniform magnetic field $\vec{B} = -B_0 \hat{k}$. If the particle passes through $(0, y, 0)$,then $y$ is equal to

Electrons moving with different speeds enter a uniform magnetic field in a direction perpendicular to the field. They will move along circular paths.

$A$ charge $q_0$ moving with velocity $\overrightarrow{v}$ in a magnetic field of induction $\overrightarrow{B}$ experiences a force $\overrightarrow{F}$. The angle between $\overrightarrow{v}$ and $\overrightarrow{B}$ is $\theta$. The speed of $q_0$ after one second will be

An electron beam is moving between two parallel plates having an electric field of $1.125 \times 10^{-6} \, N/C$. $A$ magnetic field of $3 \times 10^{-10} \, T$ is also applied so that the beam of electrons does not deflect. The velocity of the electron is ............ $m/s$.

Consider the motion of a positive point charge in a region where there are simultaneous uniform electric and magnetic fields $\vec{E}=E_0 \hat{j}$ and $\vec{B}=B_0 \hat{j}$. At time $t=0$,this charge has velocity $\vec{v}$ in the $x-y$ plane,making an angle $\theta$ with the $x$-axis. Which of the following option$(s)$ is(are) correct for time $t>0$?
$(A)$ If $\theta=0^{\circ}$,the charge moves in a circular path in the $x-z$ plane.
$(B)$ If $\theta=0^{\circ}$,the charge undergoes helical motion with constant pitch along the $y$-axis.
$(C)$ If $\theta=10^{\circ}$,the charge undergoes helical motion with its pitch increasing with time,along the $y$-axis.
$(D)$ If $\theta=90^{\circ}$,the charge undergoes linear but accelerated motion along the $y$-axis.

$A$ proton beam enters a magnetic field of $10^{-4} \ T$ normally. Given the specific charge $\frac{q}{m} = 10^{11} \ C/kg$ and velocity $v = 10^7 \ m/s$,what is the radius of the circular path described by the beam in meters?