Lorentz Force on Moving Charge

When a charged particle moves through a magnetic field, it experiences a force — not because of any collision or direct contact, but because of the fundamental interaction between moving charges and magnetic fields. This force is called the magnetic Lorentz force, named after the Dutch physicist Hendrik Lorentz. Unlike gravity or electric forces, the magnetic force never acts along the direction of motion; it always pushes the charge sideways. This seemingly strange behaviour is what makes magnetic forces so useful in particle accelerators, electric motors, and even the aurora borealis. The key insight is that a stationary charge in a magnetic field feels nothing — absolutely no force at all. The moment that charge starts moving, however, the field exerts a push perpendicular to both the velocity and the field direction. Think of it like rowing a boat across a river current: the current pushes you sideways, not forwards or backwards. The faster you row (the greater the charge's speed), the stronger the sideways push from the magnetic field. Similarly, a larger charge or a stronger field both increase the force. Because the magnetic force is always perpendicular to the velocity, it does no work on the charge — it cannot speed it up or slow it down. Instead, it continuously changes the direction of motion, causing charged particles to travel in circular or helical paths. This is the principle behind cyclotrons and the way Earth's magnetic field traps energetic particles in the Van Allen radiation belts.

Key takeaways
  • The role of charge and velocity
  • The cross product and direction
  • The angle between v and B
  • Circular motion as a consequence

The role of charge and velocity

The magnetic force exists only when a charge is moving. A charge q moving with velocity v through a magnetic field B experiences a force. If either the charge is zero or the particle is stationary (v = 0), the force is zero. The magnitude of the force increases with the size of the charge, the speed of the particle, and the strength of the magnetic field.

The cross product and direction

The direction of the force is given by the cross product F = qv × B. This means the force is always perpendicular to both the velocity v and the field B. Use the right-hand rule: point your fingers in the direction of v, curl them towards B, and your thumb points in the direction of v × B. For a negative charge (like an electron), the force is in the opposite direction — flip your thumb.

The angle between v and B

The magnitude of the force depends on the sine of the angle θ between the velocity and the magnetic field: F = qvB sin θ. If the particle moves parallel to the field (θ = 0° or 180°), sin θ = 0 and the force is zero. Maximum force occurs when the particle moves perpendicular to the field (θ = 90°), giving F = qvB. A particle moving at an angle between 0° and 90° will follow a helical (corkscrew) path.

Circular motion as a consequence

When a charged particle moves perpendicular to a uniform magnetic field, the constant sideways force causes it to move in a circle. The magnetic force provides the centripetal force: qvB = mv²/r, which gives the radius of circular motion as r = mv/(qB). This radius is called the cyclotron radius or Larmor radius. A heavier particle or faster particle curves less (larger r), while a stronger field or larger charge curves it more (smaller r).

The Key Equation

F⃗=qv⃗×B⃗\vec{F} = q\vec{v} \times \vec{B}
  • \vec{F} — Magnetic force (Newton (N)): The force experienced by the moving charge; always perpendicular to both v and B.
  • q — Electric charge (Coulomb (C)): The charge of the particle; can be positive or negative, which reverses the force direction.
  • \vec{v} — Velocity (Metres per second (m/s)): The velocity vector of the charged particle at the instant the force is calculated.
  • \vec{B} — Magnetic field (magnetic flux density) (Tesla (T)): The magnetic field vector at the location of the charge; determines both magnitude and direction of the force.

Worked Example

Problem. A proton (charge q = +1.6 × 10⁻¹⁹ C) moves east with a speed of 3.0 × 10⁶ m/s through a uniform magnetic field of magnitude 0.50 T directed vertically upward. Calculate the magnitude of the magnetic force on the proton and state its direction.

F=qvBsin⁡θ=(1.6×10−19)(3.0×106)(0.50)(1)=2.4×10−13 NF = qvB\sin\theta = (1.6\times10^{-19})(3.0\times10^{6})(0.50)(1) = 2.4\times10^{-13}\text{ N}

Answer. The magnetic force on the proton has magnitude 2.4 × 10⁻¹³ N, directed southward (perpendicular to both the velocity and the magnetic field).

Figure
Diagram
Lorentz Force on a Moving Charge××××××××××××××××××××××××××××××B intopage (×)+qvF = qv×BRight-hand rule: fingers → v, curl → B, thumb → FCharge moves right; B points into page; Force is upward
Origins · 1 minHendrik Antoon Lorentz and how this was discovered
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Hendrik Antoon Lorentz
1853–1928 · Dutch

Lorentz formulated the complete expression for the electromagnetic force on a moving charge — combining both electric and magnetic components — in 1895. His work provided the theoretical foundation for understanding how charged particles interact with electromagnetic fields, and his equations later became central to Einstein's special relativity. He shared the 1902 Nobel Prize in Physics with Pieter Zeeman for their work on the influence of magnetism on radiation. Einstein described Lorentz as 'the greatest and most powerful thinker he had ever met,' and Lorentz had essentially derived key results of special relativity before Einstein — but he interpreted them differently, believing in an absolute reference frame called the 'aether.'

  • Magnetic Force on a Current-Carrying Conductor — A current is simply many moving charges; the force on a conductor is the collective Lorentz force on all charge carriers, making this a direct extension of the same principle.
  • Magnetic Fields and Field Lines — Understanding the direction and strength of magnetic fields is essential for applying the Lorentz force law correctly, since B appears directly in the force equation.
  • Sources of Magnetic Fields — Knowing how magnetic fields are produced by moving charges and currents helps explain why the Lorentz force is fundamental to all magnetic phenomena.
  • Circular Motion of Charged Particles in Magnetic Fields — The Lorentz force being perpendicular to velocity is exactly why charged particles undergo circular or helical motion — this concept builds directly on the force law.