The following particles given below are accelerated through the same potential difference \( V \) and are injected into a uniform magnetic field \( B \) perpendicular to its direction. Arrange these particles in decreasing order of the radii of their circular paths
(A) electron
(B) proton
(C) alpha particle
(D) nucleus of tritium
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (D), (C), (B), (A)
The particle gains kinetic energy \( qV \) from the potential difference. It then moves on a circle whose radius is fixed by its momentum, \( r = p/(qB) \).
\( \displaystyle qV = \frac{p^{2}}{2m} \ \Rightarrow\ p = \sqrt{2mqV} \)
\( \displaystyle r = \frac{p}{qB} = \frac{1}{B}\sqrt{\frac{2mV}{q}} \)
With the same \( V \) and \( B \), the radius depends only on the mass-to-charge ratio:
\( \displaystyle r \propto \sqrt{\frac{m}{q}} \)
Values of \( m/q \) in units of (proton mass)/\( e \):
\( \text{(A) electron: } (1/1836)/1 \approx 5.4\times10^{-4} \)
\( \text{(B) proton: } 1/1 = 1 \)
\( \text{(C) alpha particle: } 4/2 = 2 \)
\( \text{(D) nucleus of tritium: } 3/1 = 3 \)
Decreasing order of radius:
\( \text{(D)} \gt \text{(C)} \gt \text{(B)} \gt \text{(A)} \)