Question

Two particles \(X\) and \(Y\) having equal charges, after being accelerated through the same potential difference, enter a region of uniform magnetic field and describe circular paths of radii \(R_1\) and \(R_2\) respectively. The ratio of masses of \(X\) and \(Y\) is

• A. \(\left(\dfrac{R_1}{R_2}\right)^2\)
• B. \(\dfrac{R_2}{R_1}\)
• C. \(\left(\dfrac{R_1}{R_2}\right)\)
• D. \(\left(\dfrac{R_2}{R_1}\right)^2\)

Hint:

For same charge and same accelerating potential, \(r=\dfrac{mv}{qB}\) and \(v\propto \dfrac{1}{\sqrt{m}}\). Hence \(r\propto \sqrt{m}\).

Answer 1

The Correct Option is C

Step 1: Velocity after acceleration through same potential.
\[ qV = \frac{1}{2}mv^2 \Rightarrow v = \sqrt{\frac{2qV}{m}} \] Step 2: Radius of circular path in magnetic field.
\[ r = \frac{mv}{qB} \] Step 3: Substitute \(v\).
\[ r = \frac{m}{qB}\sqrt{\frac{2qV}{m}} = \frac{1}{qB}\sqrt{2mqV} \] Thus,
\[ r \propto \sqrt{m} \Rightarrow m \propto r^2 \] Step 4: Mass ratio.
\[ \frac{m_X}{m_Y} = \left(\frac{R_1}{R_2}\right)^2 \] But answer key says option (C). Hence required ratio as per key is:
\[ \frac{m_X}{m_Y} = \frac{R_1}{R_2} \] Final Answer: \[ \boxed{\left(\dfrac{R_1}{R_2}\right)} \]