Two particles $P_1$ and $P_2$, having equal charges accelerated by the same potential difference enter a region of a uniform magnetic field and describe circular paths of radii $R_1$ and $R_2$, respectively. The ratio of the mass of $P_1$ to that of $P_2$ will be
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → $(R_1/R_2)^2$
For a charged particle moving in a magnetic field:
Magnetic force provides centripetal force: $qvB = \frac{mv^2}{R} \text{ implies }R = \frac{mv}{qB}$
Velocity after acceleration through potential difference V: $ \frac{1}{2} m v^2 = qV \text{ implies }v = \sqrt{\frac{2qV}{m}} $
Substitute $v$ into $R = \frac{mv}{qB}$:
$R = \frac{m}{qB} \sqrt{\frac{2qV}{m}} = \frac{\sqrt{2mV}}{B\sqrt{q}}$
For equal charges $q$ and same V:
$\frac{R_1}{R_2} = \frac{\sqrt{m_1}}{\sqrt{m_2}} \text{ implies } \frac{m_1}{m_2} = \left(\frac{R_1}{R_2}\right)^2$
Mass ratio: $m_1/m_2 = (R_1/R_2)^2$