Cyclotron Frequency: Understanding the Expression

The question asks for the correct expression for cyclotron frequency. The cyclotron is a particle accelerator that uses a high-frequency alternating voltage to accelerate charged particles like protons or ions, making them move in a spiral path under the influence of a constant magnetic field.

When a charged particle moves in a uniform magnetic field, the magnetic force acts perpendicular to the velocity of the particle, causing it to move in a circular path. This magnetic force provides the necessary centripetal force for the circular motion.

Deriving the Cyclotron Frequency Formula

Let's consider a charged particle with charge \(q\) and mass \(m\) moving with velocity \(v\) in a uniform magnetic field \(B\). The magnetic force (\(F_B\)) acting on the particle is given by:

\[F_B = qvB \sin\theta\]

For the particle to move in a circular path, the velocity vector must be perpendicular to the magnetic field vector, so \(\theta = 90^\circ\), and \(\sin 90^\circ = 1\). Therefore, the magnetic force is:

\[F_B = qvB\]

This magnetic force acts as the centripetal force (\(F_C\)) required for circular motion, which is given by:

\[F_C = \frac{mv^2}{r}\]

Where \(r\) is the radius of the circular path. Equating the magnetic force to the centripetal force, we get:

\[qvB = \frac{mv^2}{r}\]

From this equation, we can find the radius \(r\) of the circular path:

\[r = \frac{mv}{qB}\]

The angular frequency (\(\omega\)) of the circular motion is related to velocity and radius by:

\[\omega = \frac{v}{r}\]

Substitute the expression for \(r\) into the equation for angular frequency:

\[\omega = \frac{v}{\left(\frac{mv}{qB}\right)}\]

\[\omega = \frac{qvB}{m}\]

The linear frequency (\(V_C\), often denoted as \(f\)) is related to the angular frequency by:

\[V_C = \frac{\omega}{2\pi}\]

Substitute the expression for \(\omega\) into the equation for linear frequency:

\[V_C = \frac{\left(\frac{qB}{m}\right)}{2\pi}\]

Thus, the cyclotron frequency expression is:

\[V_C = \frac{qB}{2\pi m}\]

Analyzing the Options for Cyclotron Frequency

Let's examine the given options and compare them with our derived expression for cyclotron frequency:

• Option 1: \(\mathrm{V}_{\mathrm{C}}=\frac{q B}{2 \pi m}\)
  This expression matches our derived formula for cyclotron frequency.
• Option 2: \(\mathrm{V}_{\mathrm{C}}=\frac{\mu I}{2 R}\)
  This expression represents the magnetic field at the center of a circular coil, not cyclotron frequency.
• Option 3: \(\mathrm{V}_{\mathrm{C}}=\frac{\mu N I}{2 \pi R}\)
  This expression is related to the magnetic field generated by a current loop or solenoid, not cyclotron frequency.
• Option 4: \(\mathrm{V}_{\mathrm{C}}=\frac{2 \pi m}{q B}\)
  This expression is the reciprocal of the cyclotron frequency, representing the time period of revolution.

Based on the derivation, the correct expression for cyclotron frequency is \(\mathrm{V}_{\mathrm{C}}=\frac{q B}{2 \pi m}\).