D.3.2 Charged particle motion in magnetic fields

Circular Motion of Charged Particles in a Magnetic Field

When a charged particle moves through a uniform magnetic field at an angle, it experiences a magnetic force that is always perpendicular to its velocity.

This force acts as a centripetal force, causing the particle to move in a circular path.

Deriving the Radius of the Circular Path

The magnetic force acting on a charged particle is given by:

$$F=qvB$$

where:

• $q$ is the charge of the particle.
• $v$ is the velocity of the particle.
• $B$ is the magnetic field strength.

This force provides the centripetal force required to keep the particle in a circular path:

$$F=\frac{m{v}^2}{r}$$

• Equating the two expressions for force:

$$qvB=\frac{m{v}^2}{r}$$

• Solving for the radius $r$:

$$r=\frac{mv}{qB}$$

Time for One Revolution

• The time $T$ for the particle to complete one full revolution is:

$$T=\frac{2\pi r}{v}$$

• Substituting the expression for $r$:

$$T=\frac{2\pi }{v}\cdot \frac{mv}{qB}=\frac{2\pi m}{qB}$$

Helical Motion: Combination of Circular and Linear Motion

If a charged particle enters a magnetic field with a velocity that has both perpendicular and parallel components to the field, its motion becomes helical.

Components of Motion

• Perpendicular Component ( $v\perp$):
  • Causes the particle to move in a circular path.
  • The radius of the circle is given by:

$$r=\frac{m{v}_{\perp }}{qB}$$

• Parallel Component ($v\Vert$):
  • Causes the particle to move in a straight line along the direction of the magnetic field.

This results in a helical path.

Pitch of the Helix

The pitch of the helix is the distance the particle travels along the field in one complete revolution:

$$p=v_{\parallel }T=v_{\parallel }\cdot \frac{2\pi m}{qB}$$

Determining the Charge-to-Mass Ratio

The charge-to-mass ratio ($q/m$) of a particle can be determined using its motion in a magnetic field.

Experimental Setup

Accelerate the Particle:

• The particle is accelerated through a potential difference $V$, gaining kinetic energy:

$$\frac{1}{2}m{v}^2=qV$$

• Solving for $v$:

$$v=\sqrt{\frac{2qV}{m}}$$

Deflect the Particle:

• The particle enters a magnetic field, moving in a circular path of radius $r$.
• Using the formula for the radius:

$$r=\frac{mv}{qB}$$

• Substituting the expression for $v$:

$$r=\frac{m}{qB}\cdot \sqrt{\frac{2qV}{m}}$$

• Rearranging to find $q/m$:

$$\frac{q}{m}=\frac{2V}{B^2{r}^2}$$

Applications: Mass Spectrometry and Cyclotrons

Mass Spectrometry

Mass spectrometers use magnetic fields to separate ions based on their mass-to-charge ratio ($m/q$).

Ionization:

• Atoms or molecules are ionized to create charged particles.

Acceleration:

• Ions are accelerated through a potential difference, gaining kinetic energy.

Deflection:

• Ions enter a magnetic field and follow circular paths.
• The radius of the path depends on the mass-to-charge ratio:

$$r=\frac{mv}{qB}$$

Detection:

• Ions with different $m/q$ ratios are detected at different positions, allowing for identification and analysis.

Cyclotrons

Cyclotrons are devices that accelerate charged particles to high speeds using a combination of electric and magnetic fields.

Magnetic Field:

• The magnetic field forces particles to move in a circular path.

Electric Field:

• An alternating electric field accelerates the particles each time they cross the gap between the two halves of the cyclotron.

Increasing Radius:

• As the particles gain energy, their speed increases, causing them to spiral outward in larger circles.

Reflection and Broader Connections