Circular Motion Of A Charged Particle

A charged particle moving perpendicular to a uniform field follows a circle whose radius comes from equating the magnetic force to the centripetal requirement: $qvB=\frac{m{v}^2}{r}$, giving $r=\frac{mv}{qB}$. Expressing momentum through kinetic energy, $mv=\sqrt{2mK}$, so $r=\frac{\sqrt{2mK}}{qB}$. For equal kinetic energy $K$ and equal field $B$, and since the magnitudes of electron and proton charge are the same, the radius depends only on $\sqrt{m}$. The proton is roughly $1836$ times more massive than the electron, so it traces a substantially larger circle, with $r\propto \sqrt{m}$. The option claiming the electron traces the larger circle inverts the mass dependence. The option stating equal radii ignores that radius grows with momentum, not energy alone. The option claiming mass independence contradicts the derived $\sqrt{m}$ factor. This analysis underlies mass spectrometry, an NCERT application. A consistency check: heavier particles carry more momentum at the same energy, so a larger radius is physically expected. This square-root dependence on mass is precisely what allows a mass spectrometer to separate ions: feeding particles of equal energy into a uniform field spreads them onto arcs of differing radii according to their masses, so heavier isotopes land farther out on the detector. The same reasoning explains why, in fusion and particle-physics contexts, controlling the field strength tunes how tightly charged particles of a given energy are confined.