Two particles x and y having equal charge, after being accelerated through the same potential difference, enter a region of a uniform magnetic field. They describe circular paths of radii $R_1$ and $R_2$, respectively. The ratio of the mass of x to that of y is-

$(\frac{R_1}{R_2})^2$

The correct answer is Option (3) → $(\frac{R_1}{R_2})^2$

Charge on both particles: $q$

Energy gained on acceleration through potential $V$: $qV = \frac{1}{2}mv^2$

Thus, $v = \sqrt{\frac{2qV}{m}}$

In magnetic field $B$, radius of circular path: $R = \frac{mv}{qB}$

$R = \frac{m}{qB}\sqrt{\frac{2qV}{m}} = \frac{1}{B}\sqrt{\frac{2mV}{q}}$

So, $R \propto \sqrt{m}$

Therefore, $\frac{R_1}{R_2} = \sqrt{\frac{m_x}{m_y}}$

$\frac{m_x}{m_y} = \left(\frac{R_1}{R_2}\right)^2$

Answer: $\frac{m_x}{m_y} = \left(\frac{R_1}{R_2}\right)^2$