What is the use of a mass spectrometer?

An $\alpha$-particle,a proton,and a deuteron enter a uniform transverse magnetic field $B$ with the same accelerating potential $V$. Find the ratio of the radii of the paths followed by these particles.

If $\alpha$ and $\beta$ particles are moving with equal velocity perpendicular to the magnetic flux density $B$,then the radii of their paths will be

An electron moves with a speed of $2 \times 10^5\, m/s$ along the $+x$ direction in a magnetic field $\vec{B} = (\hat{i} - 4\hat{j} - 3\hat{k})\,T$. The magnitude of the force (in newton) experienced by the electron is (charge on electron $= 1.6 \times 10^{-19}\,C$).

The region between $y = 0$ and $y = d$ contains a magnetic field $\vec B = B\hat k$. $A$ particle of mass $m$ and charge $q$ enters the region with a velocity $\vec v = v\hat i$. If $d = \frac{mv}{2qB}$,the acceleration of the charged particle at the point of its emergence at the other side is

Two charged particles of mass $m$ and charge $q$ each are projected from the origin simultaneously with the same speed $V$ in a transverse magnetic field $B$. If $\vec{r}_1$ and $\vec{r}_2$ are the position vectors of the particles (with respect to the origin) at $t = \frac{\pi m}{qB}$, then the value of $\vec{r}_1 \cdot \vec{r}_2$ at that time is: