Lorentz Force

Lorentz Force

Mathematical Definition

The Lorentz force (F) experienced by a particle with charge (q) moving with velocity (v) in an electric field (E) and a magnetic field (B) is given by the vector equation:

Where:

• F is the force vector in Newtons (N).
• q is the charge in Coulombs (C).
• E is the electric field vector in Volts per meter (V/m) or Newtons per Coulomb (N/C).
• v is the velocity vector in meters per second (m/s).
• B is the magnetic field vector in Teslas (T).
• × denotes the vector cross product.

Key Concepts

• Electric Force: The component qE acts in the direction of the electric field for a positive charge (and opposite for a negative charge). It accelerates the particle regardless of its velocity.
• Magnetic Force: The component q(v × B) is always perpendicular to both the velocity vector and the magnetic field vector. This means the magnetic force can change the direction of the particle's velocity, but never its speed (it does no work).
• Cyclotron Motion: If a charged particle enters a uniform magnetic field perpendicularly and with no electric field, it will move in a circular path. The radius of this circle is known as the gyroradius or Larmor radius.
• Crossed Fields: When electric and magnetic fields are perpendicular to each other, they can act as a velocity selector. Only particles with a specific velocity (v = E/B) will pass through undeflected, as the electric and magnetic forces perfectly cancel out.

Historical Context

The expression for the force was derived by the Dutch physicist Hendrik Lorentz in 1895. However, earlier forms of the magnetic force on a current-carrying wire were investigated by André-Marie Ampère, and the force on a moving charge was studied by J.J. Thomson and Oliver Heaviside prior to Lorentz's comprehensive formulation.

Real-world Applications

• Mass Spectrometry: Used to separate ions according to their mass-to-charge ratio based on their trajectories in electric and magnetic fields.
• Particle Accelerators: Cyclotrons and synchrotrons use powerful magnetic fields to steer charged particles and electric fields to accelerate them.
• Cathode Ray Tubes: Older televisions and oscilloscopes used the Lorentz force to steer a beam of electrons to specific points on a fluorescent screen.
• Hall Effect Sensors: Magnetic field sensors based on the deflection of charge carriers in a semiconductor.

Related Concepts

• Magnetic Field Visualization — visualizing the fields that cause the magnetic component of the force.
• Electric Field Lines — visualizing the fields responsible for the electric component.
• Moment of Inertia — relevant for understanding rotation in physical systems.