Lorentz force
In electromagnetism, the Lorentz force is the force exerted on a charged particle by electric and magnetic fields. It determines how charged particles move in electromagnetic environments and underlies many physical phenomena, from the operation of electric motors and particle accelerators to the behavior of plasmas.
The Lorentz force has two components. The electric force acts in the direction of the electric field for positive charges and opposite to it for negative charges, tending to accelerate the particle in a straight line. The magnetic force is perpendicular to both the particle's velocity and the magnetic field, and it causes the particle to move along a curved trajectory, often circular or helical in form, depending on the directions of the fields.
Variations on the force law describe the magnetic force on a current-carrying wire, and the electromotive force in a wire loop moving through a magnetic field, as described by Faraday's law of induction.
Together with Maxwell's equations, which describe how electric and magnetic fields are generated by charges and currents, the Lorentz force law forms the foundation of classical electrodynamics. While the law remains valid in special relativity, it breaks down at small scales where quantum effects become important. In particular, the intrinsic spin of particles gives rise to additional interactions with electromagnetic fields that are not accounted for by the Lorentz force.
Historians suggest that the law is implicit in a paper by James Clerk Maxwell, published in 1865. Hendrik Lorentz arrived at a complete derivation in 1895, identifying the contribution of the electric force a few years after Oliver Heaviside correctly identified the contribution of the magnetic force.
Definition and properties
Point particle
The Lorentz force acting on a point particle with electric charge, moving with velocity, due to an external electric field and magnetic field, is given by :Here, is the vector cross product, and all quantities in bold are vectors. In component form, the force is written as:
In general, the electric and magnetic fields depend on both position and time. As a charged particle moves through space, the force acting on it at any given moment depends on its current location, velocity, and the instantaneous values of the fields at that location. Therefore, explicitly, the Lorentz force can be written as:
in which is the position vector of the charged particle, is time, and the overdot is a time derivative.
The total electromagnetic force consists of two parts: the electric force, which acts in the direction of the electric field and accelerates the particle linearly, and the magnetic force, which acts perpendicularly to both the velocity and the magnetic field. Some sources refer to the Lorentz force as the sum of both components, while others use the term to refer to the magnetic part alone.
The direction of the magnetic force is often determined using the right-hand rule: if the index finger points in the direction of the velocity, and the middle finger points in the direction of the magnetic field, then the thumb points in the direction of the force. In a uniform magnetic field, this results in circular or helical trajectories, known as cyclotron motion.
In many practical situations, such as the motion of electrons or ions in a plasma, the effect of a magnetic field can be approximated as a superposition of two components: a relatively fast circular motion around a point called the guiding center, and a relatively slow drift of this point. The drift speeds may differ for various species depending on their charge states, masses, or temperatures. These differences may lead to electric currents or chemical separation.
While the magnetic force affects the direction of a particle's motion, it does no mechanical work on the particle. The rate at which the energy is transferred from the electromagnetic field to the particle is given by the dot product of the particle's velocity and the force:
Here, the magnetic term vanishes because a vector is always perpendicular to its cross product with another vector; the scalar triple product is zero. Thus, only the electric field can transfer energy to or from a particle and change its kinetic energy.
Some textbooks use the Lorentz force law as the fundamental definition of the electric and magnetic fields. That is, the fields and are uniquely defined at each point in space and time by the hypothetical force a test particle of charge and velocity would experience there, even if no charge is present. This definition remains valid even for particles approaching the speed of light. However, some argue that using the Lorentz force law as the definition of the electric and magnetic fields is not necessarily the most fundamental approach possible.
Continuous charge distribution
The Lorentz force law can also be given in terms of continuous charge distributions, such as those found in conductors or plasmas. For a small element of a moving charge distribution with charge , the infinitesimal force is given by:Dividing both sides by the volume of the charge element gives the force density
where is the charge density and is the force per unit volume. Introducing the current density, this can be rewritten as:
The total force is the volume integral over the charge distribution:
Using Maxwell's equations and vector calculus identities, the force density can be reformulated to eliminate explicit reference to the charge and current densities. The force density can then be written in terms of the electromagnetic fields and their derivatives:
where is the Maxwell stress tensor, denotes the tensor divergence, is the speed of light, and is the Poynting vector. This form of the force law relates the energy flux in the fields to the force exerted on a charge distribution.
The power density corresponding to the Lorentz force, the rate of energy transfer to the material, is given by:
Inside a material, the total charge and current densities can be separated into free and bound parts. In terms of free charge density, free current density , polarization , and magnetization, the force density becomes
This form accounts for the torque applied to a permanent magnet by the magnetic field. The associated power density is
Formulation in the Gaussian system
The above-mentioned formulae use the conventions for the definition of the electric and magnetic field used with the SI, which is the most common. However, other conventions with the same physics are possible and used. In the conventions used with the older CGS-Gaussian units, which are somewhat more common among some theoretical physicists as well as condensed matter experimentalists, one has insteadwhere is the speed of light. Although this equation looks slightly different, it is equivalent, since one has the following relations:
where is the vacuum permittivity and the vacuum permeability. In practice, the subscripts "G" and "SI" are omitted, and the used convention must be determined from context.
Force on a current-carrying wire
When a wire carrying a steady electric current is placed in an external magnetic field, each of the moving charges in the wire experience the Lorentz force. Together, these forces produce a net macroscopic force on the wire. For a straight, stationary wire in a uniform magnetic field, this force is given by:where is the current and is a vector whose magnitude is the length of the wire, and whose direction is along the wire, aligned with the direction of the current.
If the wire is not straight or the magnetic field is non-uniform, the total force can be computed by applying the formula to each infinitesimal segment of wire, then adding up all these forces by integration. In this case, the net force on a stationary wire carrying a steady current is
One application of this is Ampère's force law, which describes the attraction or repulsion between two current-carrying wires. Each wire generates a magnetic field, described by the Biot–Savart law, which exerts a Lorentz force on the other wire. If the currents flow in the same direction, the wires attract; if the currents flow in opposite directions, they repel. This interaction provided the basis of the former definition of the ampere, as the constant current that produces a force of 2 × 10-7 newtons per metre between two straight, parallel wires one metre apart.
Another application is an induction motor. The stator winding AC current generates a moving magnetic field which induces a current in the rotor. The subsequent Lorentz force acting on the rotor creates a torque, making the motor spin. Hence, though the Lorentz force law does not apply when the magnetic field is generated by the current, it does apply when the current is induced by the movement of magnetic field.
Electromagnetic induction
The Lorentz force acting on electric charges in a conducting loop can produce a current by pushing charges around the circuit. This effect is the fundamental mechanism underlying induction motors and generators. It is described in terms of electromotive force, a quantity which plays a central role in the theory of electromagnetic induction. In a simple circuit with resistance, an emf gives rise to a current according to the Ohm's law.Both components of the Lorentz force—the electric and the magnetic—can contribute to the emf in a circuit, but through different mechanisms. In both cases, the induced emf is described by Faraday's flux rule, which states that the emf around a closed loop is equal to the negative rate of change of the magnetic flux through the loop:
The magnetic flux is defined as the surface integral of the magnetic field over a surface bounded by the loop:
The flux can change either because the loop moves or deforms over time, or because the field itself varies in time. These two possibilities correspond to the two mechanisms described by the flux rule:
- Motional emf: The circuit moves through a static but non-uniform magnetic field.
- Transformer emf: The circuit remains stationary while the magnetic field changes over time
The flux rule can be derived from the Maxwell–Faraday equation and the Lorentz force law. In some cases, especially in extended systems, the flux rule may be difficult to apply directly or may not provide a complete description, and the full Lorentz force law must be used.