${H^ + },\,H{e^ + }$ and ${O^{ + + }}$ ions having same kinetic energy pass through a region of space filled with uniform magnetic field $B$ directed perpendicular to the velocity of ions. The masses of the ions ${H^ + },\,H{e^ + }$and ${O^{ + + }}$ are respectively in the ratio $1:4:16$. As a result

A charged particle of charge $q$ and mass $m$, gets deflected through an angle $\theta$ upon passing through a square region of side $a$, which contains a uniform magnetic field $B$ normal to its plane. Assuming that the particle entered the square at right angles to one side, what is the speed of the particle?

A particle of mass $m$ and charge $q$ moves with a constant velocity $v$ along the positive $x$ direction. It enters a region containing a uniform magnetic field $B$ directed along the negative $z$ direction, extending from $x = a$ to $x = b$. The minimum value of $v$ required so that the particle can just enter the region $x > b$ is

A particle of mass $m = 1.67 \times 10^{-27}\, kg$ and charge $q = 1.6 \times 10^{-19} \, C$ enters a region of uniform magnetic field of strength $1$ $tesla$ along the direction shown in the figure. the radius of the circular portion of the path is :-

A charge particle is moving in a uniform magnetic field $(2 \hat{i}+3 \hat{j}) T$. If it has an acceleration of $(\alpha \hat{i}-4 \hat{j}) m / s ^{2}$, then the value of $\alpha$ will be.

An electron emitted by a heated cathode and accelerated through a potential difference of $ 2.0 \;kV$, enters a region with uniform magnetic field of $0.15\; T$. Determine the trajectory of the electron if the field

$(a)$ is transverse to its initial velocity,

$(b)$ makes an angle of $30^o$ with the initial velocity