Two particles $$X$$ and $$Y$$ having equal charges are being accelerated through the same potential difference. Thereafter, they enter normally in a region of uniform magnetic field and describes circular paths of radii $$R_1$$ and $$R_2$$ respectively. The mass ratio of $$X$$ and $$Y$$ is:

We need to find the mass ratio $$\frac{m_X}{m_Y}$$ of two particles with equal charges accelerated through the same potential difference, making circular paths of radii $$R_1$$ and $$R_2$$.

When a particle of mass m and charge q is accelerated through a potential $$V$$, its kinetic energy satisfies $$\frac{1}{2}mv^2 = qV \Rightarrow v = \sqrt{\frac{2qV}{m}}$$. In a magnetic field, the circular radius is given by $$R = \frac{mv}{qB} = \frac{m}{qB}\sqrt{\frac{2qV}{m}} = \frac{\sqrt{2mV/q}}{B} = \frac{1}{B}\sqrt{\frac{2mV}{q}}$$, so that $$R \propto \sqrt{m}$$ when q, V, and B are constant.

Hence, $$\frac{R_1}{R_2} = \sqrt{\frac{m_X}{m_Y}}$$ and thus $$\frac{m_X}{m_Y} = \left(\frac{R_1}{R_2}\right)^2$$.

Therefore, the answer is Option B.