A proton and a deuteron,both having the same | vedclass

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Question

(A) The magnetic force provides the necessary centripetal force for circular motion: $\frac{mv^2}{R} = qvB$,which simplifies to $R = \frac{mv}{qB}$.Si

Vedclass · Accepted Answer

${R_d} = \sqrt{2} \,{R_p}$

Vedclass · Answer

(A) The magnetic force provides the necessary centripetal force for circular motion: $\frac{mv^2}{R} = qvB$,which simplifies to $R = \frac{mv}{qB}$.Since the kinetic energy $E = \frac{1}{2}mv^2$,we have $mv = \sqrt{2mE}$.Substituting this into the radius formula: $R = \frac{\sqrt{2mE}}{qB}$.For a proton,${R_p} = \frac{\sqrt{2{m_p}E}}{{q_p}B}$. For a deuteron,${R_d} = \frac{\sqrt{2{m_d}E}}{{q_d}B}$.Since a deuteron has the same charge as a proton $(q_d = q_p)$ and its mass is twice that of a proton $(m_d = 2{m_p})$,we get:$\frac{{R_d}}{{R_p}} = \sqrt{\frac{m_d}{m_p}} = \sqrt{\frac{2{m_p}}{{m_p}}} = \sqrt{2}$.Therefore,${R_d} = \sqrt{2} {R_p}$.

$A$ proton and a deuteron,both having the same kinetic energy,enter perpendicularly into a uniform magnetic field $B$. For the motion of the proton and deuteron on circular paths of radii ${R_p}$ and ${R_d}$ respectively,the correct statement is:

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