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) → \( (R_1/R_2)^2 \)
The accelerating potential gives each particle a kinetic energy \( qV \), and in the magnetic field the magnetic force provides the centripetal force.
\( \displaystyle qV = \frac{1}{2}mv^{2} \;\Rightarrow\; mv = \sqrt{2mqV} \)
\( \displaystyle qvB = \frac{mv^{2}}{R} \;\Rightarrow\; R = \frac{mv}{qB} = \frac{\sqrt{2mqV}}{qB} \)
The charge \( q \), the potential \( V \) and the field \( B \) are the same for both particles, so
\( \displaystyle R \propto \sqrt{m} \;\Rightarrow\; m \propto R^{2} \)
\( \displaystyle \frac{m_1}{m_2} = \left(\frac{R_1}{R_2}\right)^{2} \)