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(A) The radius $R$ of a circular path of a charged particle in a uniform magnetic field $B$ is given by the formula $R = \frac{mv}{qB}$.Given that the

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$4: 3$

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(A) The radius $R$ of a circular path of a charged particle in a uniform magnetic field $B$ is given by the formula $R = \frac{mv}{qB}$.Given that the masses $m$ are the same and the magnetic field $B$ is uniform,the radius is proportional to the ratio of speed to charge: $R \propto \frac{v}{q}$.Therefore,the ratio of the radii $R_1$ and $R_2$ is given by:$\frac{R_1}{R_2} = \frac{v_1}{q_1} 	imes \frac{q_2}{v_2} = \left(\frac{v_1}{v_2}ight) 	imes \left(\frac{q_2}{q_1}ight)$.Given the ratios $\frac{v_1}{v_2} = \frac{2}{3}$ and $\frac{q_1}{q_2} = \frac{1}{2}$ (which implies $\frac{q_2}{q_1} = \frac{2}{1}$).Substituting these values:$\frac{R_1}{R_2} = \left(\frac{2}{3}ight) 	imes \left(\frac{2}{1}ight) = \frac{4}{3}$.Thus,the ratio of the radii is $4:3$.

Two ions having same mass have charges in the ratio $1: 2$. They are projected normally in a uniform magnetic field with their speeds in the ratio $2: 3$. The ratio of the radii of their circular trajectories is -

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