Understanding Charged Particle Motion in Magnetic Fields

When a charged particle enters a homogeneous magnetic field, it experiences a magnetic force known as the Lorentz force. If the velocity of the particle is perpendicular to the magnetic field, this force acts perpendicular to both the velocity and the magnetic field direction.

In the case where the charged particle's velocity is exactly perpendicular to the uniform magnetic field, the Lorentz force provides the necessary centripetal force, causing the particle to move in a circular path. The radius of this circular path is called the radius of curvature.

Forces Acting on the Charged Particle

The magnitude of the Lorentz force (\(F_B\)) on a charged particle with charge \(q\) and velocity \(v\) moving perpendicular to a magnetic field of strength \(B\) is given by:

\(F_B = |q|vB\)

For the particle to move in a circle of radius \(r\), a centripetal force (\(F_c\)) is required, which is given by:

\(F_c = \frac{mv^2}{r}\)

where \(m\) is the mass of the particle.

Deriving the Radius of Curvature Formula

Since the magnetic force provides the centripetal force for circular motion (when velocity is perpendicular to the magnetic field), we can equate the two forces:

\(F_B = F_c\)

\(|q|vB = \frac{mv^2}{r}\)

We can rearrange this equation to solve for the radius of curvature, \(r\):

\(r = \frac{mv^2}{|q|vB}\)

Simplifying the equation by cancelling one \(v\) term from the numerator and denominator:

\(r = \frac{mv}{|q|B}\)

Relating Radius to Momentum

The momentum (\(p\)) of a particle is defined as the product of its mass (\(m\)) and velocity (\(v\)):

\(p = mv\)

Substituting the momentum \(p\) into the equation for the radius of curvature, we get:

\(r = \frac{p}{|q|B}\)

Analyzing Proportionality

From the formula \(r = \frac{p}{|q|B}\), we can see how the radius of curvature (\(r\)) is related to different quantities:

• The radius \(r\) is directly proportional to the momentum (\(p\)) of the particle, assuming charge \(|q|\) and magnetic field \(B\) are constant: \(r \propto p\).
• The radius \(r\) is inversely proportional to the magnitude of the charge (\(|q|\)) of the particle, assuming momentum \(p\) and magnetic field \(B\) are constant: \(r \propto \frac{1}{|q|}\).
• The radius \(r\) is inversely proportional to the strength of the magnetic field (\(B\)), assuming momentum \(p\) and charge \(|q|\) are constant: \(r \propto \frac{1}{B}\).

The question asks what the radius of curvature is proportional to among the given options. The derivation clearly shows that the radius is directly proportional to the momentum of the particle.

Evaluating the Options

Let's look at the provided options based on the formula \(r = \frac{mv}{|q|B}\) or \(r = \frac{p}{|q|B}\):

1. energy of particle:

   The kinetic energy is \(KE = \frac{1}{2}mv^2 = \frac{p^2}{2m}\). While \(r\) depends on \(v\) (and thus related to energy), the relationship is not a simple proportionality like \(r \propto KE\). Specifically, \(v = \sqrt{\frac{2KE}{m}}\), so \(r = \frac{m\sqrt{\frac{2KE}{m}}}{|q|B} = \frac{\sqrt{2mKE}}{|q|B}\). The radius is proportional to the square root of energy (\(r \propto \sqrt{KE}\)).

2. charge of particle:

   As shown above, \(r \propto \frac{1}{|q|}\), which means the radius is inversely proportional to the charge, not directly proportional.

3. momentum of particle:

   As shown above, \(r \propto p\), which means the radius is directly proportional to the momentum of the particle. This matches our derivation.

4. strength of the field:

   As shown above, \(r \propto \frac{1}{B}\), which means the radius is inversely proportional to the strength of the field, not directly proportional.

Based on the analysis, the radius of curvature of a charged particle's passage in a homogeneous magnetic field (when velocity is perpendicular to the field) is directly proportional to the momentum of the particle.

Revision Table: Factors Affecting Radius of Curvature

Additional Information on Charged Particle Motion

It is important to note that the derivation \(r = \frac{mv}{|q|B}\) or \(r = \frac{p}{|q|B}\) is valid specifically when the charged particle's velocity is perpendicular to the uniform magnetic field. If the velocity has a component parallel to the magnetic field, that component remains unaffected by the magnetic force. The component of velocity perpendicular to the field causes circular motion, while the parallel component causes the particle to move along the field lines. The combination of linear motion along the field and circular motion perpendicular to the field results in a helical path.

The frequency of the circular motion, known as the cyclotron frequency (\(\omega_c\)), is given by \(\omega_c = \frac{|q|B}{m}\). Interestingly, this frequency does not depend on the velocity or energy of the particle (as long as the motion is non-relativistic and perpendicular to the field). The period of motion is \(T_c = \frac{2\pi m}{|q|B}\).

Devices like mass spectrometers and cyclotrons utilize the principles of charged particle motion in magnetic fields to separate particles by mass-to-charge ratio or accelerate them to high energies, respectively.