Magnetic Forces on Moving Charges

Expert reviewed • 23 November 2024 • 6 minute read

When charged particles move through magnetic fields, they experience forces that create fascinating patterns of motion. Understanding these interactions is crucial for many applications, from particle accelerators to aurora borealis.

The Magnetic Force

When a charged particle moves through a magnetic field, it experiences a force given by:

Fm=qvBsin⁡θF_m = qvB\sin\theta

where:

  • FmF_m is the magnetic force (in Newtons, N)
  • qq is the particle's charge (in Coulombs, C)
  • vv is the particle's velocity (in meters per second, m/s)
  • BB is the magnetic field strength (in Tesla, T)
  • θ\theta is the angle between the velocity and magnetic field vectors

For particles moving perpendicular to the magnetic field (θ=90°\theta = 90°), this simplifies to:

Fm=qv⊥BF_m = qv_⊥B

Direction of Force

The direction of the magnetic force follows the right-hand palm rule:

  • Point your fingers in the direction of the magnetic field (B)
  • Point your thumb in the direction of the particle's velocity (v)
  • For a positive charge, the force is directed out of your palm
  • For a negative charge, the force is in the opposite direction

Circular Motion in Magnetic Fields

Since the magnetic force always acts perpendicular to the particle's velocity, it creates circular motion. The magnetic force provides the centripetal force needed for this circular motion:

Fm=FcF_m = F_c qvB=mv2rqvB = \frac{mv^2}{r}

Solving for the radius gives:

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

This equation shows that:

  • Heavier particles orbit in larger circles
  • Faster particles orbit in larger circles
  • Stronger magnetic fields create tighter orbits
  • Particles with larger charges orbit in smaller circles

Practical Applications

The Mass Spectrometer

The relationship between particle mass and orbital radius is used in mass spectrometry to separate ions of different masses. By measuring the radius of curvature, scientists can determine the mass-to-charge ratio of particles.

Particle Deflection Control

When both electric and magnetic fields are present, their forces can be balanced to allow particles to pass through undeflected. This principle is used in velocity selectors and particle accelerators. The required electric field strength (E) for undeflected motion is given by:

E=vBE = vB

where the electric field must be oriented perpendicular to both the magnetic field and the particle's velocity.

Practice Question 1

An electron (charge = -1.60 × 10⁻¹⁹ C, mass = 9.11 × 10⁻³¹ kg) enters a magnetic field of 0.05 T with a velocity of 300 m/s perpendicular to the field. Determine the radius of the ciruclar path the electron will follow. Also find the magnetic force that it experiences. 1. The electron will follow a circular path with radius:

r=mvqB=(9.11×10−31)(300)(1.60×10−19)(0.05)=3.41×10−5 mr = \frac{mv}{qB} = \frac{(9.11 × 10⁻³¹)(300)}{(1.60 × 10⁻¹⁹)(0.05)} = 3.41 × 10⁻⁵ \text{ m}

  • The magnetic force experienced is: Fm=qvB=(1.60×10−19)(300)(0.05)=2.40×10−15 NF_m = qvB = (1.60 × 10⁻¹⁹)(300)(0.05) = 2.40 × 10⁻¹⁵ \text{ N}

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Understanding Charged Particle Behavior in Fields