A singly charged ion of mass m is accelerated from rest by a potential differenc

Question

A singly charged ion of mass m is accelerated from rest by a potential difference ?V. It is then deflected by a uniform magnetic field (perpendicular to the ion's velocity) into a semicircle of radius R. Now a doubly charged ion of mass m' is accelerated through the same potential difference and deflected by the same magnetic field into a semicircle of radius R' = 2R. What is the ratio of the masses of the ions? m'/m =

Explanation / Answer

Note B is constant, and KE'/KE = Vq'/(Vq) = 3. "~" means "is proportional to". The cyclotron angular rate omega = QB/m (ref.), and r = v/omega ==> r = mv/(QB) ==> m ~ rQ/v with B constant. Since v ~ sqrt(KE/m), m ~ rQ*sqrt(m/KE) ==> m ~ (rQ)^2/KE. Then m'/m = (r'Q'/(rQ))^2 * KE/KE' = 48

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