A proton and a deuteron with the same initial | vedclass

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(B) For a charged particle moving in a perpendicular magnetic field,the magnetic force provides the centripetal force: $\frac{mv^2}{r} = Bqv$. This si

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$1: \sqrt{2}$

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(B) For a charged particle moving in a perpendicular magnetic field,the magnetic force provides the centripetal force: $\frac{mv^2}{r} = Bqv$. This simplifies to the radius formula: $r = \frac{mv}{Bq} = \frac{p}{Bq}$,where $p$ is the momentum. Since the kinetic energy $E$ is the same for both,we use the relation $p = \sqrt{2mE}$. Substituting this into the radius formula: $r = \frac{\sqrt{2mE}}{Bq}$. For a proton $(p)$ and a deuteron $(d)$,the charges are equal $(q_p = q_d = e)$ and the kinetic energies are equal $(E_p = E_d = E)$. The ratio of the radii is $\frac{r_p}{r_d} = \frac{\sqrt{2m_p E} / (Be)}{\sqrt{2m_d E} / (Be)} = \sqrt{\frac{m_p}{m_d}}$. Given that the mass of a deuteron is twice the mass of a proton $(m_d = 2m_p)$,we have $\frac{r_p}{r_d} = \sqrt{\frac{m_p}{2m_p}} = \sqrt{\frac{1}{2}} = \frac{1}{\sqrt{2}}$.

$A$ proton and a deuteron with the same initial kinetic energy enter a magnetic field in a direction perpendicular to the direction of the field. The ratio of the radii of the circular trajectories described by them is

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