Under the influence of a uniform magnetic field, a charged particle moves with constant speed v in a circle of

Question

Under the influence of a uniform magnetic field, a charged particle moves with constant speed v in a circle of radius R. The time period of rotation of the particle -

Accepted Answer

1. **Force Balance:** When a charged particle (mass $m$, charge $q$) moves perpendicular to a magnetic field ($B$), the magnetic Lorentz force provides the necessary centripetal force for circular motion.
   $$qvB = \frac{mv^2}{R}$$ .
2. **Angular Velocity:** From the force equation, we can find the angular velocity ($\omega = v/R$):
   $$\frac{v}{R} = \frac{qB}{m} \Rightarrow \omega = \frac{qB}{m}$$
3. **Time Period:** The time period ($T$) of rotation is derived from the angular velocity:
   $$T = \frac{2\pi}{\omega} = \frac{2\pi m}{qB}$$ .
4. **Conclusion:** The expression for the time period depends only on the mass ($m$), charge ($q$), and magnetic field ($B$). It does not contain velocity ($v$) or radius ($R$). This property is fundamental to the operation of the cyclotron.

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