In cyclotron, an ion is made to travel successively along semi circles of increasing radius under the action of magnetic field. Which of the following factor on which angular velocity of the ion is not dependent?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → radius of the circle and speed of the ion
In a cyclotron the magnetic force \( qvB \) on the ion provides the centripetal force for its circular path.
\( \displaystyle q v B = \frac{m v^{2}}{r} \;\Rightarrow\; r = \frac{m v}{q B} \)
\( \displaystyle \omega = \frac{v}{r} = \frac{q B}{m} \)
The angular velocity depends on the charge, the mass and the magnetic field, but not on the radius or the speed. As the ion speeds up, its radius grows in the same proportion, so the time for each semicircle stays the same. This is why a fixed-frequency oscillator can keep accelerating the ion (the cyclotron frequency \( \nu = qB/2\pi m \)).
NCERT (2026-27):
Motion in a Magnetic Field – “… If \( \omega \) is the angular frequency, then \( v = \omega r \). So,
\( \displaystyle \omega = 2\pi\nu = qB/m \qquad [4.6(a)] \)
which is independent of the velocity or energy. Here \( \nu \) is the frequency of rotation. The independence of \( \nu \) from energy has important application in the design of a cyclotron.”