A proton, deutron and an $\alpha$-particle enter a magnetic field perpendicular to field with same velocity. What is the ratio of the radii of circular paths?

$\text{Force on a charged particle due circular motion: }F=\frac{m{v}^2}{r}$

$\text{Force on a charged particle due to magnetic field: }{F}_B=qvB$

$F=F_B\Rightarrow \frac{m{v}^2}{r}=qvB\Rightarrow r=\frac{mv}{qB}$

$\text{Here}v\text{and}\text{are constant.}$

$\Rightarrow r\propto \frac{m}{q}$

$\text{For proton:}{r}_p=1\times k$

$\text{For deutron:}{r}_d=2k$

$\text{For an-}\alpha \text{particle:}{r}_{\alpha }=2k$

$\text{Therefore, }{r}_p:r_d:r_{\alpha }=1:2:2$