Motion of Charged Particles in Magnetic Fields

The motion of charged particles in magnetic fields is a fundamental topic in physics, particularly in electromagnetism. Charged particles are influenced by magnetic fields due to the Lorentz force, which governs the dynamics of charged particles in both electric and magnetic fields.

Lorentz Force

The Lorentz force is the combination of electric and magnetic force on a point charge due to electromagnetic fields. The force ( \vec{F} ) on a charge ( q ) with velocity ( \vec{v} ) in an electric field ( \vec{E} ) and a magnetic field ( \vec{B} ) is given by:

[ \vec{F} = q(\vec{E} + \vec{v} \times \vec{B}) ]

For the case where only a magnetic field is present, the force simplifies to:

[ \vec{F} = q(\vec{v} \times \vec{B}) ]

This force is perpendicular to both the velocity of the particle and the magnetic field.

Motion in a Uniform Magnetic Field

When a charged particle enters a uniform magnetic field, it experiences a force that is perpendicular to its velocity and the magnetic field. This results in the particle following a circular or helical path, depending on whether there is a component of velocity along the direction of the magnetic field.

Circular Motion

For a particle with velocity perpendicular to the magnetic field, the motion is circular. The radius ( r ) of the circular path can be derived from the balance between the Lorentz force and the centripetal force required for circular motion:

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

Solving for ( r ), we get the radius of the circular path:

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

The frequency ( f ) of the circular motion, also known as the cyclotron frequency, is given by:

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

Helical Motion

If the particle has a velocity component parallel to the magnetic field, the motion becomes helical. The particle spirals along the direction of the magnetic field while also rotating in a circle in the plane perpendicular to the field.

Differences and Important Points

Here is a table summarizing the differences and important points regarding the motion of charged particles in magnetic fields:

Property	Circular Motion	Helical Motion
Magnetic Field	Perpendicular to ( \vec{v} )	Has a component along ( \vec{v} )
Path Shape	Circle	Helix
Lorentz Force	Centripetal	Centripetal + Parallel to ( \vec{B} )
Radius of Path	( r = \frac{mv}{qB} )	Same as circular motion
Frequency of Rotation	( f = \frac{qB}{2\pi m} )	Same as circular motion
Pitch of Helix	Not applicable	Distance along ( \vec{B} ) per turn

Examples

Example 1: Electron in a Magnetic Field

An electron with a velocity of ( 3 \times 10^6 ) m/s enters a magnetic field of ( 0.01 ) T perpendicular to its velocity. Calculate the radius of its path and the frequency of its motion.

Using the formulas for radius and frequency:

[ r = \frac{mv}{qB} = \frac{(9.11 \times 10^{-31} \text{ kg})(3 \times 10^6 \text{ m/s})}{(1.6 \times 10^{-19} \text{ C})(0.01 \text{ T})} \approx 0.0171 \text{ m} ]

[ f = \frac{qB}{2\pi m} = \frac{(1.6 \times 10^{-19} \text{ C})(0.01 \text{ T})}{2\pi(9.11 \times 10^{-31} \text{ kg})} \approx 8.79 \times 10^7 \text{ Hz} ]

Example 2: Proton in a Helical Path

A proton with a velocity of ( 5 \times 10^5 ) m/s enters a magnetic field of ( 0.02 ) T at an angle of ( 45^\circ ) to the field. Calculate the radius of the circular component of its path.

First, find the velocity component perpendicular to the magnetic field:

[ v_{\perp} = v \sin(45^\circ) = (5 \times 10^5 \text{ m/s}) \sin(45^\circ) \approx 3.54 \times 10^5 \text{ m/s} ]

Now calculate the radius:

[ r = \frac{mv_{\perp}}{qB} = \frac{(1.67 \times 10^{-27} \text{ kg})(3.54 \times 10^5 \text{ m/s})}{(1.6 \times 10^{-19} \text{ C})(0.02 \text{ T})} \approx 0.186 \text{ m} ]

In conclusion, the motion of charged particles in magnetic fields is characterized by the Lorentz force, which can result in circular or helical paths. The radius and frequency of these paths are determined by the charge, mass, velocity of the particle, and the strength of the magnetic field.