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-
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( \left(\displaystyle\frac{R_1}{R_2}\right)^{2} \)
A charge moving perpendicular to a uniform magnetic field moves in a circle of radius \( r = mv/(qB) \). The speed comes from the accelerating potential difference by energy conservation.
\( \displaystyle qV = \frac{1}{2}mv^{2} \;\Rightarrow\; mv = \sqrt{2mqV} \)
\( \displaystyle r = \frac{\sqrt{2mqV}}{qB} \)
The charge \( q \), the potential difference \( V \) and the field \( B \) are the same for both particles, so
\( \displaystyle r \propto \sqrt{m} \;\Rightarrow\; m \propto r^{2} \)
\( \displaystyle \frac{m_x}{m_y} = \left(\frac{R_1}{R_2}\right)^{2} \)