Lorentz force

Lorentz force acting on fast-moving charged particles in a bubble chamber. Positive and negative charge trajectories curve in opposite directions.

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 (sometimes called Laplace force), 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

Lorentz force F on a charged particle (of charge q) in motion (instantaneous velocity v). The E field and B field vary in space and time.

The Lorentz force F acting on a point particle with electric charge q, moving with velocity v, due to an external electric field E and magnetic field B, is given by (SI definition of quantities):

$\mathbf {F} =q\left(\mathbf {E} +\mathbf {v} \times \mathbf {B} \right)$

Here, × is the vector cross product, and all quantities in bold are vectors. In component form, the force is written as: $${\begin{aligned}F_{x}&=q\left(E_{x}+v_{y}B_{z}-v_{z}B_{y}\right),\\[0.5ex]F_{y}&=q\left(E_{y}+v_{z}B_{x}-v_{x}B_{z}\right),\\[0.5ex]F_{z}&=q\left(E_{z}+v_{x}B_{y}-v_{y}B_{x}\right).\end{aligned}}$$

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: $$\mathbf {F} \left(\mathbf {r} (t),{\dot {\mathbf {r} }}(t),t,q\right)=q\left[\mathbf {E} (\mathbf {r} ,t)+{\dot {\mathbf {r} }}(t)\times \mathbf {B} (\mathbf {r} ,t)\right]$$ in which r is the position vector of the charged particle, t is time, and the overdot is a time derivative.

The total electromagnetic force consists of two parts: the electric force qE, which acts in the direction of the electric field and accelerates the particle linearly, and the magnetic force q(v × B), 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 (for a positive charge). 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: $$\mathbf {v} \cdot \mathbf {F} =q\mathbf {v} \cdot (\mathbf {E} +\mathbf {v} \times \mathbf {B} )=q\,\mathbf {v} \cdot \mathbf {E} .$$Here, the magnetic term vanishes because a vector is always perpendicular to its cross product with another vector; the scalar triple product $\mathbf {v} \cdot (\mathbf {v} \times \mathbf {B} )$ is zero. Thus, only the electric field can transfer energy to or from a particle and change its kinetic energy.

Continuous charge distribution

Lorentz force (per unit 3-volume) f on a continuous charge distribution (charge density ρ) in motion. The 3-current density J corresponds to the motion of the charge element dq in volume element dV and varies throughout the continuum.

The Lorentz force law also 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 $\mathrm {d} q$, the infinitesimal force is given by: $$\mathrm {d} \mathbf {F} =\mathrm {d} q\left(\mathbf {E} +\mathbf {v} \times \mathbf {B} \right)$$ Dividing both sides by the volume $\mathrm {d} V$ of the charge element gives the force density $$\mathbf {f} =\rho \left(\mathbf {E} +\mathbf {v} \times \mathbf {B} \right),$$ where $\rho$ is the charge density and $\mathbf {f}$ is the force per unit volume. Introducing the current density ${\textstyle \mathbf {J} =\rho \mathbf {v} }$, this can be rewritten as:

$\mathbf {f} =\rho \mathbf {E} +\mathbf {J} \times \mathbf {B}$

The total force is the volume integral over the charge distribution: $$\mathbf {F} =\int \left(\rho \mathbf {E} +\mathbf {J} \times \mathbf {B} \right)\mathrm {d} V.$$

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:$$\mathbf {f} =\nabla \cdot {\boldsymbol {\sigma }}-{\dfrac {1}{c^{2}}}{\dfrac {\partial \mathbf {S} }{\partial t}}$$ where ${\boldsymbol {\sigma }}$ is the Maxwell stress tensor, $\nabla \cdot$ denotes the tensor divergence, $c$ is the speed of light, and $\mathbf {S}$ 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. (See Covariant formulation of classical electromagnetism for more details.)

The power density corresponding to the Lorentz force, the rate of energy transfer to the material, is given by:$$\mathbf {J} \cdot \mathbf {E} .$$

Inside a material, the total charge and current densities can be separated into free and bound parts. In terms of free charge density $\rho _{\rm {f}}$, free current density $\mathbf {J} _{\rm {f}}$, polarization $\mathbf {P}$, and magnetization $\mathbf {M}$, the force density becomes $$\mathbf {f} =\left(\rho _{\rm {f}}-\nabla \cdot \mathbf {P} \right)\mathbf {E} +\left(\mathbf {J} _{\rm {f}}+\nabla \times \mathbf {M} +{\frac {\partial \mathbf {P} }{\partial t}}\right)\times \mathbf {B} .$$This form accounts for the torque applied to a permanent magnet by the magnetic field. The associated power density is $$\left(\mathbf {J} _{f}+\nabla \times \mathbf {M} +{\frac {\partial \mathbf {P} }{\partial t}}\right)\cdot \mathbf {E} .$$

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 (i.e. forces on e.g. an electron) 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 instead $$\mathbf {F} =q_{\mathrm {G} }\left(\mathbf {E} _{\mathrm {G} }+{\frac {\mathbf {v} }{c}}\times \mathbf {B} _{\mathrm {G} }\right),$$ where c is the speed of light. Although this equation looks slightly different, it is equivalent, since one has the following relations: $$q_{\mathrm {G} }={\frac {q_{\mathrm {SI} }}{\sqrt {4\pi \varepsilon _{0}}}},\quad \mathbf {E} _{\mathrm {G} }={\sqrt {4\pi \varepsilon _{0}}}\,\mathbf {E} _{\mathrm {SI} },\quad \mathbf {B} _{\mathrm {G} }={\sqrt {4\pi /\mu _{0}}}\,{\mathbf {B} _{\mathrm {SI} }},\quad c={\frac {1}{\sqrt {\varepsilon _{0}\mu _{0}}}}.$$ where ε₀ is the vacuum permittivity and μ₀ the vacuum permeability. In practice, the subscripts "G" and "SI" are omitted, and the used convention (and unit) must be determined from context.

Force on a current-carrying wire

Right-hand rule for the force on a current-carrying wire in a magnetic field B

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: $$\mathbf {F} =I{\boldsymbol {\ell }}\times \mathbf {B} ,$$ where I is the current and ℓ 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 $\mathrm {d} {\boldsymbol {\ell }}$, then adding up all these forces by integration. In this case, the net force on a stationary wire carrying a steady current is $$\mathbf {F} =I\int (\mathrm {d} {\boldsymbol {\ell }}\times \mathbf {B} ).$$

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 $\mathbf {F}$ acting on the rotor creates a torque, making the motor spin. Hence, though the Lorentz force law does not apply when the magnetic field $\mathbf {B}$ is generated by the current $I$, it does apply when the current $I$ is induced by the movement of magnetic field $\mathbf {B}$.

Electromagnetic induction

A conducting rod moving through a uniform magnetic field. The magnetic component of the Lorentz force pushes electrons to one end, resulting in charge separation.

The magnetic component (qv × B) of the Lorentz force is responsible for motional electromotive force (motional EMF)—the fundamental mechanism underlying induction motors and generators. When a conducting rod moves through a magnetic field that is perpendicular to both the rod and its direction of motion, the mobile electrons inside experience a magnetic force that drives them along the rod. This causes a separation of charge between the two ends of the rod. As charge accumulates, an electric field develops that eventually balances the magnetic force. The resulting electric potential difference reflects the induced motional EMF.

In other electrical machines such as synchronous generators, the magnetic field moves, while the conductors do not. In this case, the EMF is due to the electric force component (qE) of the Lorentz force. The electric field in question is created by the changing magnetic field, resulting in an induced EMF called the transformer EMF, as described by the Maxwell–Faraday equation.

Both of these EMFs, despite their apparently distinct origins, adhere to Faraday's law of induction.

Motional EMF, induced by moving a conductor through a magnetic field.

Transformer EMF, induced by a changing magnetic field.

Einstein's special theory of relativity was partially motivated by the desire to better understand this link between the two effects. In fact, the electric and magnetic fields are different facets of the same electromagnetic field, and in moving from one inertial frame to another, the solenoidal vector field portion of the E-field can change in whole or in part to a B-field or vice versa.

Equivalence of Lorentz force law and Faraday's law of induction

The Lorentz force, together with Maxwell's Equations, can be used to derive Faraday's law of induction and vice versa.

Faraday's law states that the electromotive force (EMF) is given by the rate of change of magnetic flux through a wire loop: $${\mathcal {E}}=-{\frac {\mathrm {d} \Phi _{B}}{\mathrm {d} t}},$$ where the magnetic flux is given by the surface integral: $$\Phi _{B}=\int _{\Sigma (t)}\mathbf {B} (\mathbf {r} ,t)\cdot \mathrm {d} \mathbf {A} ,$$ with dA a vector area element of the surface Σ(t); bounded by the closed contour ∂Σ(t), at time t. The sign of the EMF is determined by Lenz's law, which states that the magnetic field created by the induced current opposes changes in the initial magnetic field. This is valid for not only a stationary wire – but also for a moving wire.

Let ∂Σ(t) represent a closed linear circuit. Then the EMF acting in the circuit given by: $${\mathcal {E}}=\oint _{\partial \Sigma (t)}{\frac {\mathbf {F} }{q}}\cdot \mathrm {d} {\boldsymbol {\ell }}$$ with dℓ a vector element of the contour ∂Σ(t). Equating both integrals leads to the field theory form of Faraday's law, given by: $${\mathcal {E}}=\oint _{\partial \Sigma (t)}\mathbf {E} '(\mathbf {r} ,t)\cdot \mathrm {d} {\boldsymbol {\ell }}=-{\frac {\mathrm {d} }{\mathrm {d} t}}\int _{\Sigma (t)}\mathbf {B} (\mathbf {r} ,t)\cdot \mathrm {d} \mathbf {A} ,$$ where $\mathbf {E} '(\mathbf {r} ,t)=\mathbf {F} /q(\mathbf {r} ,t)$ denotes the "effective" electric field. This result can be compared with the version of Faraday's law that appears in Maxwell's equations, called the (integral form of) Maxwell–Faraday equation: $$\oint _{\partial \Sigma (t)}\mathbf {E} (\mathbf {r} ,t)\cdot \mathrm {d} {\boldsymbol {\ell }}=-\int _{\Sigma (t)}{\frac {\partial \mathbf {B} (\mathbf {r} ,t)}{\partial t}}\cdot \mathrm {d} \mathbf {A} .$$

The two equations are equivalent if the wire is not moving, i.e., $\mathbf {E} '(\mathbf {r} ,t)=\mathbf {E} (\mathbf {r} ,t)$ when $\mathbf {v} =0$.

In case the circuit is moving with a velocity $\mathbf {v}$ in some direction, then, using the Leibniz integral rule and that div B = 0, gives $$\oint _{\partial \Sigma (t)}\mathbf {E} '(\mathbf {r} ,t)\cdot \mathrm {d} {\boldsymbol {\ell }}=-\int _{\Sigma (t)}{\frac {\partial \mathbf {B} (\mathbf {r} ,t)}{\partial t}}\cdot \mathrm {d} \mathbf {A} +\oint _{\partial \Sigma (t)}\left(\mathbf {v} \times \mathbf {B} (\mathbf {r} ,t)\right)\cdot \mathrm {d} {\boldsymbol {\ell }}.$$ Substituting the Maxwell-Faraday equation then gives $$\oint _{\partial \Sigma (t)}\mathbf {E} '(\mathbf {r} ,t)\cdot \mathrm {d} {\boldsymbol {\ell }}=\oint _{\partial \Sigma (t)}\mathbf {E} (\mathbf {r} ,t)\cdot \mathrm {d} {\boldsymbol {\ell }}+\oint _{\partial \Sigma (t)}\left(\mathbf {v} \times \mathbf {B} (\mathbf {r} ,t)\right)\mathrm {d} {\boldsymbol {\ell }}$$ since this is valid for any wire position it implies that $$\mathbf {F} =q\,\mathbf {E} (\mathbf {r} ,\,t)+q\,\mathbf {v} \times \mathbf {B} (\mathbf {r} ,\,t).$$

Faraday's law of induction holds whether the loop of wire is rigid and stationary, or in motion or in process of deformation, and it holds whether the magnetic field is constant in time or changing. However, there are cases where Faraday's law is either inadequate or difficult to use, and application of the underlying Lorentz force law is necessary. See inapplicability of Faraday's law.

If the magnetic field is fixed in time and the conducting loop moves through the field, the magnetic flux Φ_B linking the loop can change in several ways. For example, if the B-field varies with position, and the loop moves to a location with different B-field, Φ_B will change. Alternatively, if the loop changes orientation with respect to the B-field, the B ⋅ dA differential element will change because of the different angle between B and dA, also changing Φ_B. As a third example, if a portion of the circuit is swept through a uniform, time-independent B-field, and another portion of the circuit is held stationary, the flux linking the entire closed circuit can change due to the shift in relative position of the circuit's component parts with time (surface ∂Σ(t) time-dependent). In all three cases, Faraday's law of induction then predicts the EMF generated by the change in Φ_B.

Note that the Maxwell-Faraday's equation implies that the electric field E is non-conservative when the magnetic field B varies in time.

Role in the electromagnetic theory

Maxwell's equations describe how electric charges and currents generate electric and magnetic fields. The Lorentz force law completes the picture by describing how those fields act on a moving point charge q. Together, they form a self-consistent framework: the fields influence the motion of charges, and charges and currents generate the fields. In a broader physical context, charged particles may also be subject to other forces, such as gravity or nuclear interactions, which lie outside the scope of classical electrodynamics.

Trajectory of a particle with a positive or negative charge q under the influence of a magnetic field B, which is directed perpendicularly out of the screen

Beam of electrons moving in a circle, due to the presence of a magnetic field. Purple light revealing the electron's path in this Teltron tube is created by the electrons colliding with gas molecules.

Charged particles experiencing the Lorentz force

In many textbook treatments of classical electromagnetism, the Lorentz force law is also used to define the electric and magnetic fields E and B. The force F acting on a test particle of charge q and velocity v is taken to follow a form that uniquely determines the fields:

The electromagnetic force F on a test charge at a given point and time is a certain function of its charge q and velocity v, which can be parameterized by exactly two vectors E and B, in the functional form: $$\mathbf {F} =q(\mathbf {E} +\mathbf {v} \times \mathbf {B} )$$

This definition remains valid even for particles approaching the speed of light (that is, magnitude of v, |v| ≈ c). The fields E and B are thereby defined throughout space and time by the force a test charge would experience, regardless of whether any charge is actually present.

In real materials the Lorentz force is inadequate to describe the collective behavior of charged particles, both in principle and as a matter of computation. The charged particles in a material medium not only respond to the E and B fields but also generate these fields. Complex transport equations must be solved to determine the time and spatial response of charges, for example, the Boltzmann equation or the Fokker–Planck equation or the Navier–Stokes equations. For example, see magnetohydrodynamics, fluid dynamics, electrohydrodynamics, superconductivity, stellar evolution. An entire physical apparatus for dealing with these matters has developed. See for example, Green–Kubo relations and Green's function (many-body theory).

History

Lorentz's theory of electrons. Formulas for the Lorentz force (I, ponderomotive force) and the Maxwell equations for the divergence of the electrical field E (II) and the magnetic field B (III), La théorie electromagnétique de Maxwell et son application aux corps mouvants, 1892, p. 451. V is the velocity of light.

Early attempts to quantitatively describe the electromagnetic force were made in the mid-18th century. It was proposed that the force on magnetic poles, by Johann Tobias Mayer and others in 1760, and electrically charged objects, by Henry Cavendish in 1762, obeyed an inverse-square law. However, in both cases the experimental proof was neither complete nor conclusive. It was not until 1784 when Charles-Augustin de Coulomb, using a torsion balance, was able to definitively show through experiment that this was true. Soon after the discovery in 1820 by Hans Christian Ørsted that a magnetic needle is acted on by a voltaic current, André-Marie Ampère that same year was able to devise through experimentation the formula for the angular dependence of the force between two current elements. In all these descriptions, the force was always described in terms of the properties of the matter involved and the distances between two masses or charges rather than in terms of electric and magnetic fields.

The modern concept of electric and magnetic fields first arose in the theories of Michael Faraday, particularly his idea of lines of force, later to be given full mathematical description by Lord Kelvin and James Clerk Maxwell. From a modern perspective it is possible to identify in Maxwell's 1865 formulation of his field equations a form of the Lorentz force equation in relation to electric currents, although in the time of Maxwell it was not evident how his equations related to the forces on moving charged objects. J. J. Thomson was the first to attempt to derive from Maxwell's field equations the electromagnetic forces on a moving charged object in terms of the object's properties and external fields. Interested in determining the electromagnetic behavior of the charged particles in cathode rays, Thomson published a paper in 1881 wherein he gave the force on the particles due to an external magnetic field as $$\mathbf {F} ={\frac {q}{2}}\mathbf {v} \times \mathbf {B} .$$ Thomson derived the correct basic form of the formula, but, because of some miscalculations and an incomplete description of the displacement current, included an incorrect scale-factor of a half in front of the formula. Oliver Heaviside invented the modern vector notation and applied it to Maxwell's field equations; he also (in 1885 and 1889) had fixed the mistakes of Thomson's derivation and arrived at the correct form of the magnetic force on a moving charged object. Finally, in 1895, Hendrik Lorentz derived the modern form of the formula for the electromagnetic force which includes the contributions to the total force from both the electric and the magnetic fields. Lorentz began by abandoning the Maxwellian descriptions of the ether and conduction. Instead, Lorentz made a distinction between matter and the luminiferous aether and sought to apply the Maxwell equations at a microscopic scale. Using Heaviside's version of the Maxwell equations for a stationary ether and applying Lagrangian mechanics (see below), Lorentz arrived at the correct and complete form of the force law that now bears his name.

Lorentz force in terms of potentials

See also: Mathematical descriptions of the electromagnetic field, Maxwell's equations, and Helmholtz decomposition

The E and B fields can be replaced by the magnetic vector potential A and (scalar) electrostatic potential ϕ by $${\begin{aligned}\mathbf {E} &=-\nabla \phi -{\frac {\partial \mathbf {A} }{\partial t}}\\[1ex]\mathbf {B} &=\nabla \times \mathbf {A} \end{aligned}}$$ where ∇ is the gradient, ∇⋅ is the divergence, and ∇× is the curl.

The force becomes $$\mathbf {F} =q\left[-\nabla \phi -{\frac {\partial \mathbf {A} }{\partial t}}+\mathbf {v} \times (\nabla \times \mathbf {A} )\right].$$

Using an identity for the triple product this can be rewritten as $$\mathbf {F} =q\left[-\nabla \phi -{\frac {\partial \mathbf {A} }{\partial t}}+\nabla \left(\mathbf {v} \cdot \mathbf {A} \right)-\left(\mathbf {v} \cdot \nabla \right)\mathbf {A} \right].$$

(Notice that the coordinates and the velocity components should be treated as independent variables, so the del operator acts only on $\mathbf {A}$, not on $\mathbf {v}$; thus, there is no need of using Feynman's subscript notation in the equation above.) Using the chain rule, the convective derivative of $\mathbf {A}$ is: $${\frac {\mathrm {d} \mathbf {A} }{\mathrm {d} t}}={\frac {\partial \mathbf {A} }{\partial t}}+(\mathbf {v} \cdot \nabla )\mathbf {A}$$ so that the above expression becomes: $$\mathbf {F} =q\left[-\nabla (\phi -\mathbf {v} \cdot \mathbf {A} )-{\frac {\mathrm {d} \mathbf {A} }{\mathrm {d} t}}\right].$$

With v = ẋ and $${\frac {\mathrm {d} }{\mathrm {d} t}}\left[{\frac {\partial }{\partial {\dot {\mathbf {x} }}}}\left(\phi -{\dot {\mathbf {x} }}\cdot \mathbf {A} \right)\right]=-{\frac {\mathrm {d} \mathbf {A} }{\mathrm {d} t}},$$ we can put the equation into the convenient Euler–Lagrange form

$\mathbf {F} =q\left[-\nabla _{\mathbf {x} }(\phi -{\dot {\mathbf {x} }}\cdot \mathbf {A} )+{\frac {\mathrm {d} }{\mathrm {d} t}}\nabla _{\dot {\mathbf {x} }}(\phi -{\dot {\mathbf {x} }}\cdot \mathbf {A} )\right]$

where $$\nabla _{\mathbf {x} }={\hat {x}}{\dfrac {\partial }{\partial x}}+{\hat {y}}{\dfrac {\partial }{\partial y}}+{\hat {z}}{\dfrac {\partial }{\partial z}}$$ and $$\nabla _{\dot {\mathbf {x} }}={\hat {x}}{\dfrac {\partial }{\partial {\dot {x}}}}+{\hat {y}}{\dfrac {\partial }{\partial {\dot {y}}}}+{\hat {z}}{\dfrac {\partial }{\partial {\dot {z}}}}.$$

Lorentz force and analytical mechanics

See also: Magnetic vector potential § Interpretation as Potential Momentum

The Lagrangian for a charged particle of mass m and charge q in an electromagnetic field equivalently describes the dynamics of the particle in terms of its energy, rather than the force exerted on it. The classical expression is given by: $$L={\frac {m}{2}}\mathbf {\dot {r}} \cdot \mathbf {\dot {r}} +q\mathbf {A} \cdot \mathbf {\dot {r}} -q\phi$$ where A and ϕ are the potential fields as above. The quantity $V=q(\phi -\mathbf {A} \cdot \mathbf {\dot {r}} )$ can be identified as a generalized, velocity-dependent potential energy and, accordingly, $\mathbf {F}$ as a non-conservative force. Using the Lagrangian, the equation for the Lorentz force given above can be obtained again.

Derivation of Lorentz force from classical Lagrangian (SI units)

For an A field, a particle moving with velocity v = ṙ has potential momentum $q\mathbf {A} (\mathbf {r} ,t)$, so its potential energy is $q\mathbf {A} (\mathbf {r} ,t)\cdot \mathbf {\dot {r}}$. For a ϕ field, the particle's potential energy is $q\phi (\mathbf {r} ,t)$.

The total potential energy is then: $$V=q\phi -q\mathbf {A} \cdot \mathbf {\dot {r}}$$ and the kinetic energy is: $$T={\frac {m}{2}}\mathbf {\dot {r}} \cdot \mathbf {\dot {r}}$$ hence the Lagrangian: $${\begin{aligned}L&=T-V\\[1ex]&={\frac {m}{2}}\mathbf {\dot {r}} \cdot \mathbf {\dot {r}} +q\mathbf {A} \cdot \mathbf {\dot {r}} -q\phi \\[1ex]&={\frac {m}{2}}\left({\dot {x}}^{2}+{\dot {y}}^{2}+{\dot {z}}^{2}\right)+q\left({\dot {x}}A_{x}+{\dot {y}}A_{y}+{\dot {z}}A_{z}\right)-q\phi \end{aligned}}$$

Lagrange's equations are $${\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial {\dot {x}}}}={\frac {\partial L}{\partial x}}$$ (same for y and z). So calculating the partial derivatives: $${\begin{aligned}{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial {\dot {x}}}}&=m{\ddot {x}}+q{\frac {\mathrm {d} A_{x}}{\mathrm {d} t}}\\&=m{\ddot {x}}+q\left[{\frac {\partial A_{x}}{\partial t}}+{\frac {\partial A_{x}}{\partial x}}{\frac {dx}{dt}}+{\frac {\partial A_{x}}{\partial y}}{\frac {dy}{dt}}+{\frac {\partial A_{x}}{\partial z}}{\frac {dz}{dt}}\right]\\[1ex]&=m{\ddot {x}}+q\left[{\frac {\partial A_{x}}{\partial t}}+{\frac {\partial A_{x}}{\partial x}}{\dot {x}}+{\frac {\partial A_{x}}{\partial y}}{\dot {y}}+{\frac {\partial A_{x}}{\partial z}}{\dot {z}}\right]\\\end{aligned}}$$ $${\frac {\partial L}{\partial x}}=-q{\frac {\partial \phi }{\partial x}}+q\left({\frac {\partial A_{x}}{\partial x}}{\dot {x}}+{\frac {\partial A_{y}}{\partial x}}{\dot {y}}+{\frac {\partial A_{z}}{\partial x}}{\dot {z}}\right)$$ equating and simplifying: $$m{\ddot {x}}+q\left({\frac {\partial A_{x}}{\partial t}}+{\frac {\partial A_{x}}{\partial x}}{\dot {x}}+{\frac {\partial A_{x}}{\partial y}}{\dot {y}}+{\frac {\partial A_{x}}{\partial z}}{\dot {z}}\right)=-q{\frac {\partial \phi }{\partial x}}+q\left({\frac {\partial A_{x}}{\partial x}}{\dot {x}}+{\frac {\partial A_{y}}{\partial x}}{\dot {y}}+{\frac {\partial A_{z}}{\partial x}}{\dot {z}}\right)$$ $${\begin{aligned}F_{x}&=-q\left({\frac {\partial \phi }{\partial x}}+{\frac {\partial A_{x}}{\partial t}}\right)+q\left[{\dot {y}}\left({\frac {\partial A_{y}}{\partial x}}-{\frac {\partial A_{x}}{\partial y}}\right)+{\dot {z}}\left({\frac {\partial A_{z}}{\partial x}}-{\frac {\partial A_{x}}{\partial z}}\right)\right]\\[1ex]&=qE_{x}+q[{\dot {y}}(\nabla \times \mathbf {A} )_{z}-{\dot {z}}(\nabla \times \mathbf {A} )_{y}]\\[1ex]&=qE_{x}+q[\mathbf {\dot {r}} \times (\nabla \times \mathbf {A} )]_{x}\\[1ex]&=qE_{x}+q(\mathbf {\dot {r}} \times \mathbf {B} )_{x}\end{aligned}}$$ and similarly for the y and z directions. Hence the force equation is: $$\mathbf {F} =q(\mathbf {E} +\mathbf {\dot {r}} \times \mathbf {B} )$$

The relativistic Lagrangian is $$L=-mc^{2}{\sqrt {1-\left({\frac {\dot {\mathbf {r} }}{c}}\right)^{2}}}+q\mathbf {A} (\mathbf {r} )\cdot {\dot {\mathbf {r} }}-q\phi (\mathbf {r} )$$

The action is the relativistic arclength of the path of the particle in spacetime, minus the potential energy contribution, plus an extra contribution which quantum mechanically is an extra phase a charged particle gets when it is moving along a vector potential.

Derivation of Lorentz force from relativistic Lagrangian (SI units)

The equations of motion derived by extremizing the action (see matrix calculus for the notation): $${\frac {\mathrm {d} \mathbf {P} }{\mathrm {d} t}}={\frac {\partial L}{\partial \mathbf {r} }}=q{\partial \mathbf {A} \over \partial \mathbf {r} }\cdot {\dot {\mathbf {r} }}-q{\partial \phi \over \partial \mathbf {r} }$$ $$\mathbf {P} -q\mathbf {A} ={\frac {m{\dot {\mathbf {r} }}}{\sqrt {1-\left({\frac {\dot {\mathbf {r} }}{c}}\right)^{2}}}}$$ are the same as Hamilton's equations of motion: $${\frac {\mathrm {d} \mathbf {r} }{\mathrm {d} t}}={\frac {\partial }{\partial \mathbf {p} }}\left({\sqrt {(\mathbf {P} -q\mathbf {A} )^{2}+(mc^{2})^{2}}}+q\phi \right)$$ $${\frac {\mathrm {d} \mathbf {p} }{\mathrm {d} t}}=-{\frac {\partial }{\partial \mathbf {r} }}\left({\sqrt {(\mathbf {P} -q\mathbf {A} )^{2}+(mc^{2})^{2}}}+q\phi \right)$$ both are equivalent to the noncanonical form: $${\frac {\mathrm {d} }{\mathrm {d} t}}{m{\dot {\mathbf {r} }} \over {\sqrt {1-\left({\frac {\dot {\mathbf {r} }}{c}}\right)^{2}}}}=q\left(\mathbf {E} +{\dot {\mathbf {r} }}\times \mathbf {B} \right).$$ This formula is the Lorentz force, representing the rate at which the EM field adds relativistic momentum to the particle.

Relativistic form of the Lorentz force

Covariant form of the Lorentz force

Field tensor

Main articles: Covariant formulation of classical electromagnetism and Mathematical descriptions of the electromagnetic field

Using the metric signature (1, −1, −1, −1), the Lorentz force for a charge q can be written in covariant form:

${\frac {\mathrm {d} p^{\alpha }}{\mathrm {d} \tau }}=qF^{\alpha \beta }U_{\beta }$

where p^α is the four-momentum, defined as $$p^{\alpha }=\left(p_{0},p_{1},p_{2},p_{3}\right)=\left(\gamma mc,p_{x},p_{y},p_{z}\right),$$ τ the proper time of the particle, F^αβ the contravariant electromagnetic tensor $$F^{\alpha \beta }={\begin{pmatrix}0&-E_{x}/c&-E_{y}/c&-E_{z}/c\\E_{x}/c&0&-B_{z}&B_{y}\\E_{y}/c&B_{z}&0&-B_{x}\\E_{z}/c&-B_{y}&B_{x}&0\end{pmatrix}}$$ and U is the covariant 4-velocity of the particle, defined as: $$U_{\beta }=\left(U_{0},U_{1},U_{2},U_{3}\right)=\gamma \left(c,-v_{x},-v_{y},-v_{z}\right),$$ in which $$\gamma (v)={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}={\frac {1}{\sqrt {1-{\frac {v_{x}^{2}+v_{y}^{2}+v_{z}^{2}}{c^{2}}}}}}$$ is the Lorentz factor.

The fields are transformed to a frame moving with constant relative velocity by: $$F'^{\mu \nu }={\Lambda ^{\mu }}_{\alpha }{\Lambda ^{\nu }}_{\beta }F^{\alpha \beta }\,,$$ where Λ^μ_α is the Lorentz transformation tensor.

Translation to vector notation

The α = 1 component (x-component) of the force is $${\frac {\mathrm {d} p^{1}}{\mathrm {d} \tau }}=qU_{\beta }F^{1\beta }=q\left(U_{0}F^{10}+U_{1}F^{11}+U_{2}F^{12}+U_{3}F^{13}\right).$$

Substituting the components of the covariant electromagnetic tensor F yields $${\frac {\mathrm {d} p^{1}}{\mathrm {d} \tau }}=q\left[U_{0}\left({\frac {E_{x}}{c}}\right)+U_{2}(-B_{z})+U_{3}(B_{y})\right].$$

Using the components of covariant four-velocity yields $${\frac {\mathrm {d} p^{1}}{\mathrm {d} \tau }}=q\gamma \left[c\left({\frac {E_{x}}{c}}\right)+(-v_{y})(-B_{z})+(-v_{z})(B_{y})\right]=q\gamma \left(E_{x}+v_{y}B_{z}-v_{z}B_{y}\right)=q\gamma \left[E_{x}+\left(\mathbf {v} \times \mathbf {B} \right)_{x}\right]\,.$$

The calculation for α = 2, 3 (force components in the y and z directions) yields similar results, so collecting the three equations into one: $${\frac {\mathrm {d} \mathbf {p} }{\mathrm {d} \tau }}=q\gamma \left(\mathbf {E} +\mathbf {v} \times \mathbf {B} \right),$$ and since differentials in coordinate time dt and proper time dτ are related by the Lorentz factor, $$dt=\gamma (v)\,d\tau ,$$ so we arrive at $${\frac {\mathrm {d} \mathbf {p} }{\mathrm {d} t}}=q\left(\mathbf {E} +\mathbf {v} \times \mathbf {B} \right).$$

This is precisely the Lorentz force law, however, it is important to note that p is the relativistic expression, $$\mathbf {p} =\gamma (v)m_{0}\mathbf {v} \,.$$

Lorentz force in spacetime algebra (STA)

The electric and magnetic fields are dependent on the velocity of an observer, so the relativistic form of the Lorentz force law can best be exhibited starting from a coordinate-independent expression for the electromagnetic and magnetic fields ${\mathcal {F}}$, and an arbitrary time-direction, $\gamma _{0}$. This can be settled through spacetime algebra (or the geometric algebra of spacetime), a type of Clifford algebra defined on a pseudo-Euclidean space, as $$\mathbf {E} =\left({\mathcal {F}}\cdot \gamma _{0}\right)\gamma _{0}$$ and $$i\mathbf {B} =\left({\mathcal {F}}\wedge \gamma _{0}\right)\gamma _{0}$$ ${\mathcal {F}}$ is a spacetime bivector (an oriented plane segment, just like a vector is an oriented line segment), which has six degrees of freedom corresponding to boosts (rotations in spacetime planes) and rotations (rotations in space-space planes). The dot product with the vector $\gamma _{0}$ pulls a vector (in the space algebra) from the translational part, while the wedge-product creates a trivector (in the space algebra) who is dual to a vector which is the usual magnetic field vector. The relativistic velocity is given by the (time-like) changes in a time-position vector $v={\dot {x}}$, where $$v^{2}=1,$$ (which shows our choice for the metric) and the velocity is $$\mathbf {v} =cv\wedge \gamma _{0}/(v\cdot \gamma _{0}).$$

The proper form of the Lorentz force law ('invariant' is an inadequate term because no transformation has been defined) is simply

$F=q{\mathcal {F}}\cdot v$

Note that the order is important because between a bivector and a vector the dot product is anti-symmetric. Upon a spacetime split like one can obtain the velocity, and fields as above yielding the usual expression.

Lorentz force in general relativity

In the general theory of relativity the equation of motion for a particle with mass $m$ and charge $e$, moving in a space with metric tensor $g_{ab}$ and electromagnetic field $F_{ab}$, is given as

$$m{\frac {du_{c}}{ds}}-m{\frac {1}{2}}g_{ab,c}u^{a}u^{b}=eF_{cb}u^{b},$$

where $u^{a}=dx^{a}/ds$ ($dx^{a}$ is taken along the trajectory), $g_{ab,c}=\partial g_{ab}/\partial x^{c}$, and $ds^{2}=g_{ab}dx^{a}dx^{b}$.

The equation can also be written as

$$m{\frac {du_{c}}{ds}}-m\Gamma _{abc}u^{a}u^{b}=eF_{cb}u^{b},$$

where $\Gamma _{abc}$ is the Christoffel symbol (of the torsion-free metric connection in general relativity), or as

$$m{\frac {Du_{c}}{ds}}=eF_{cb}u^{b},$$

where $D$ is the covariant differential in general relativity.

Applications

The Lorentz force occurs in many devices, including:

• Cyclotrons and other circular path particle accelerators
• Mass spectrometers
• Velocity filters
• Magnetrons
• Lorentz force velocimetry

In its manifestation as the Laplace force on an electric current in a conductor, this force occurs in many devices, including:

• Electric motors
• Railguns
• Linear motors
• Loudspeakers
• Magnetoplasmadynamic thrusters
• Electrical generators
• Homopolar generators
• Linear alternators

See also

• Hall effect
• Gravitomagnetism
• Formulation of Maxwell's equations in special relativity
• Moving magnet and conductor problem
• Abraham–Lorentz force
• Larmor formula
• Cyclotron radiation
• Magnetoresistance
• Scalar potential
• Helmholtz decomposition
• Field line
• Coulomb's law
• Electromagnetic buoyancy

References

• Dahl, Per F (1997). Flash of the Cathode Rays: A History of J J Thomson's Electron. CRC Press. doi:10.1201/9781482268461. ISBN 978-0-429-18192-4.
• Darrigol, Olivier (2000). Electrodynamics from Ampère to Einstein. Oxford ; New York: Clarendon Press. ISBN 0-19-850594-9.
• Feynman, Richard Phillips; Leighton, Robert B.; Sands, Matthew L. (2006). The Feynman lectures on physics. Vol. 2. Pearson / Addison-Wesley. ISBN 0-8053-9047-2.
• Griffiths, David J. (2023). Introduction to Electrodynamics. Cambridge University Press. doi:10.1017/9781009397735. ISBN 978-1-009-39773-5.
• Huray, Paul G. (2009). Maxwell's Equations. Hoboken, N.J: John Wiley & Sons. ISBN 978-0-470-54276-7.
• Jackson, John David (1998). Classical Electrodynamics. New York: John Wiley & Sons. ISBN 978-0-471-30932-1.
• Nahin, Paul (2002). Oliver Heaviside. Johns Hopkins University Press. doi:10.56021/9780801869099. ISBN 978-0-8018-6909-9.
• Purcell, Edward M.; Morin, David J. (2013). Electricity and Magnetism:. Cambridge University Press. doi:10.1017/cbo9781139012973. ISBN 978-1-139-01297-3.
• Sadiku, Matthew N. O. (2018). Elements of electromagnetics (7th ed.). New York/Oxford: Oxford University Press. ISBN 978-0-19-069861-4.
• Serway, Raymond A.; Jewett, John W. Jr. (2004). Physics for scientists and engineers, with modern physics. Belmont, California: Thomson Brooks/Cole. ISBN 0-534-40846-X.
• Srednicki, Mark A. (2007). Quantum field theory. Cambridge, England; New York City: Cambridge University Press. ISBN 978-0-521-86449-7.

External links

• Lorentz force (demonstration)
• Interactive Java applet on the magnetic deflection of a particle beam in a homogeneous magnetic field Archived 2011-08-13 at the Wayback Machine by Wolfgang Bauer