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Lorentz force

In physics, the Lorentz force is the force on a point charge due to electromagnetic fields. It is given by the following equation in terms of the electric and magnetic fields:

mathbf{F} = q [mathbf{E} + (mathbf{v} times mathbf{B})],

where

F is the force (in newtons)
E is the electric field (in volts per meter)
B is the magnetic field (in teslas)
q is the electric charge of the particle (in coulombs)
v is the instantaneous velocity of the particle (in meters per second)
× is the vector cross product
and ∇ × are gradient and curl, respectively

or equivalently the following equation in terms of the vector potential and scalar potential:

mathbf{F} = q (- nabla phi - frac { partial mathbf{A} } { partial t } + mathbf{v} times (nabla times mathbf{A})),

where:

A and ɸ are the magnetic vector potential and electrostatic potential, respectively, which are related to E and B by
mathbf{E} = - nabla phi - frac { partial mathbf{A} } { partial t }
mathbf{B} = nabla times mathbf{A}.

Note that these are vector equations: All the quantities written in boldface are vectors (in particular, F, E, v, B, A).

The Lorentz force law has a close relationship with Faraday's law of induction.

A positively charged particle will be accelerated in the same linear orientation as the E field, but will curve perpendicularly to both the instantaneous velocity vector v and the B field according to the right-hand rule (in detail, if the thumb of the right hand points along v and the index finger along B, then the middle finger points along F).

The term qE is called the electric force, while the term qv × B is called the magnetic force. According to some definitions, the term "Lorentz force" refers specifically to the formula for the magnetic force:

mathbf{F}_{mag} = q(mathbf{v} times mathbf{B})
with the total electromagnetic force (including the electric force) given some other (nonstandard) name. This article will not follow this nomenclature: In what follows, the term "Lorentz force" will refer only to the expression for the total force.

The magnetic force component of the Lorentz force manifests itself as the force that acts on a current-carrying wire in a magnetic field. In that context, it is also called the Laplace force.

History

Hendrik Lorentz introduced this force in 1892. However, the discovery of the Lorentz force was before Lorentz's time. In particular, it can be seen at equation (77) in Maxwell's 1861 paper On Physical Lines of Force Later, Maxwell listed it as equation "D" of his 1864 paper, A Dynamical Theory of the Electromagnetic Field, as one of the eight original Maxwell's equations. In this paper the equation was written as follows:
mathbf{E} = mathbf{v} times (mu mathbf{H}) - frac{partialmathbf{A}}{partial t}-nabla phi
where
A is the magnetic vector potential,
,phi is the electrostatic potential,
H is the magnetic field H,
mu is magnetic permeability.

Although this equation is obviously a direct precursor of the modern Lorentz force equation, it actually differs in two respects:

  • It does not contain a factor of q, the charge. Maxwell didn't use the concept of charge. The definition of E used here by Maxwell is unclear. He uses the term electromotive force. He operated from Faraday's electro-tonic state A, which he considered to be a momentum in his vortex sea. The closest term that we can trace to electric charge in Maxwell's papers is the density of free electricity, which appears to refer to the density of the aethereal medium of his molecular vortices and that gives rise to the momentum A. Maxwell believed that A was a fundamental quantity from which electromotive force can be derived.
  • The equation here contains the information that what we nowadays call E, which today can be expressed in terms of scalar and vector potentials according to

mathbf{E} = - nabla phi - frac { partial mathbf{A} } { partial t }
The fact that E can be expressed this way is equivalent to one of the four modern Maxwell's equations, the Maxwell-Faraday equation.

Despite its historical origins in the original set of eight Maxwell's equations, the Lorentz force is no longer considered to be one of "Maxwell's equations" as the term is currently used (that is, as reformulated by Heaviside). It now sits adjacent to Maxwell's equations as a separate and essential law.

Significance of the Lorentz force

While the modern Maxwell's equations describe how electrically charged particles and objects give rise to electric and magnetic fields, the Lorentz force law completes that picture by describing the force acting on a moving point charge q in the presence of electromagnetic fields. The Lorentz force law describes the effect of E and B upon a point charge, but such electromagnetic forces are not the entire picture. Charged particles are possibly coupled to other forces, notably gravity and nuclear forces. Thus, Maxwell's equations do not stand separate from other physical laws, but are coupled to them via the charge and current densities. The response of a point charge to the Lorentz law is one aspect; the generation of E and B by currents and charges is another.

In real materials the Lorentz force is inadequate to describe the behavior of charged particles, both in principle and as a matter of computation. The charged particles in a material medium both respond to the E and B fields and 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).

Although one might suggest that these theories are only approximations intended to deal with large ensembles of "point particles", perhaps a deeper perspective is that the charge-bearing particles may respond to forces like gravity, or nuclear forces, or boundary conditions (see for example: boundary layer, boundary condition, Casimir effect, cross section (physics)) that are not electromagnetic interactions, or are approximated in a deus ex machina fashion for tractability.

Lorentz force law as the definition of E and B

In many textbook treatments of classical electromagnetism, the Lorentz Force Law is used as the definition of the electric and magnetic fields E and B. To be specific, the Lorentz Force is understood to be the following empirical statement:

The electromagnetic force on a test charge at a given point and time is a certain function of its charge and velocity, which can be parameterized by exactly two vectors E and B, in the functional form:

mathbf{F}=q[mathbf{E}+(mathbf{v}timesmathbf{B})].

If this empirical statement is valid (and, of course, countless experiments have shown that it is), then two vector fields E and B are thereby defined throughout space and time, and these are called the "electric field" and "magnetic field".

Note that the fields are defined everywhere in space and time, regardless of whether or not a charge is present to experience the force. In particular, the fields are defined with respect to what force a test charge would feel, if it were hypothetically placed there.

Note also that as a definition of E and B, the Lorentz force is only a definition in principle because a real particle (as opposed to the hypothetical "test charge" of infinitesimally-small mass and charge) would generate its own finite E and B fields, which would alter the electromagnetic force that it experiences. In addition, if the charge experiences acceleration, for example, if forced into a curved trajectory by some external agency, it emits radiation that causes braking of its motion. See, for example, Bremsstrahlung and synchrotron light. These effects occur through both a direct effect (called the radiation reaction force) and indirectly (by affecting the motion of nearby charges and currents).

Moreover, the electromagnetic force is not in general the same as the net force, due to gravity, electroweak and other forces, and any extra forces would have to be taken into account in a real measurement.

Lorentz force and Faraday's law of induction

Given a loop of wire in a magnetic field, Faraday's law of induction states:

mathcal{E} = -frac{dPhi_B}{dt}

where:

Phi_B is the magnetic flux through the loop,
mathcal{E} is the electromotive force (EMF) experienced,
t is time
The sign of the EMF is determined by Lenz's Law.

Using the Lorentz force law, the EMF around a closed path ∂Σ is given by:

mathcal{E} =oint_{part Sigma (t)} d boldsymbol{ell} cdot mathbf{F} / q = oint_{part Sigma (t)} d boldsymbol{ell} cdot left(mathbf {E} + mathbf{ v times B} right) ,

where d is an element of the curve ∂Σ(t), imagined to be moving in time. The flux ΦB in Faraday's law of induction can be expressed explicitly as:

frac {d Phi_B} {dt} = frac {d} {dt} iint_{Sigma (t)} d boldsymbol {A} cdot mathbf {B}(mathbf{r}, t) ,

where

Σ(t) is a surface bounded by the closed contour ∂Σ(t)
E is the electric field,
d is an infinitesimal vector element of the contour ∂Σ,
v is the velocity of the infinitesimal contour element d,
B is the magnetic field.
dA is an infinitesimal vector element of surface Σ , whose magnitude is the area of an infinitesimal patch of surface, and whose direction is orthogonal to that surface patch.
Both d and dA have a sign ambiguity; to get the correct sign, the right-hand rule is used, as explained in the article Kelvin-Stokes theorem.
The surface integral at the right-hand side of this equation is the explicit expression for the magnetic flux ΦB through Σ. Thus, incorporating the Lorentz law in Faraday's equation, we find:

oint_{part Sigma (t)} d boldsymbol{ell} cdot left(mathbf {E}(mathbf{r}, t) + mathbf{ v times B}(mathbf{r}, t) right) = -frac {d} {dt} iint_{Sigma (t)} d boldsymbol {A} cdot mathbf {B}(mathbf{r}, t) .

Notice that the ordinary time derivative appearing before the integral sign implies that time differentiation must include differentiation of the limits of integration, which vary with time whenever Σ(t) is a moving surface.

The above result can be compared with the version of Faraday's law of induction that appears in the modern Maxwell's equations, called here the Maxwell-Faraday equation:

nabla times mathbf{E} = -frac{partial mathbf{B}}{partial t} .

The Maxwell-Faraday equation also can be written in an integral form using the Kelvin-Stokes theorem:

oint_{partial Sigma (t)}d boldsymbol{ell} cdot mathbf{E}(mathbf{r}, t) = - iint_{Sigma (t)} d boldsymbol {A} cdot {{ partial mathbf {B}(mathbf{r}, t)} over partial t }

Comparison of the Faraday flux law with the integral form of the Maxwell-Faraday relation suggests:

frac {d} {dt} iint_{Sigma (t)} d boldsymbol {A} cdot mathbf {B}(mathbf{r}, t)= iint_{Sigma (t)} d boldsymbol {A} cdot {{ partial mathbf {B}(mathbf{r}, t)} over partial t } - oint_{part Sigma (t)} d boldsymbol{ell} cdot left(mathbf{ v times B}(mathbf{r}, t) right) .

which is a form of the Leibniz integral rule valid because div B = 0. The term in v × B accounts for motional EMF, that is the movement of the surface Σ, at least in the case of a rigidly translating body. In contrast, the integral form of the Maxwell-Faraday equation includes only the effect of the E-field generated by ∂B/∂t.

Often the integral form of the Maxwell-Faraday equation is used alone, and is written with the partial derivative outside the integral sign as:

oint_{partial Sigma}d boldsymbol{ell} cdot mathbf{E}(mathbf{r}, t) = - { partial over partial t } iint_{Sigma} d boldsymbol {A} cdot { mathbf {B}(mathbf{r}, t) } .

Notice that the limits ∂Σ and Σ have no time dependence. In the context of the Maxwell-Faraday equation, the usual interpretation of the partial time derivative is extended to imply a stationary boundary. On the other hand, 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 flux magnetic flux ΦB 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, ΦB 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 ΦB. 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 ΦB.

In a contrasting circumstance, when the loop is stationary and the B-field varies with time, the Maxwell-Faraday equation shows a nonconservative E-field is generated in the loop, which drives the carriers around the wire via the q E term in the Lorentz force. This situation also changes ΦB, producing an EMF predicted by Faraday's law of induction.

Naturally, in both cases, the precise value of current that flows in response to the Lorentz force depends on the conductivity of the loop.

Lorentz force in terms of potentials

If the scalar potential and vector potential replace E and B (see Helmholtz decomposition), the force becomes:
mathbf{F} = q(-nabla phi- frac{partial mathbf{A}}{partial mathbf{t}}+mathbf{v}times(nablatimesmathbf{A}))

or, equivalently (making use of the fact that v is a constant; see triple product),

mathbf{F} = q(-nabla phi- frac{partial mathbf{A}}{partial mathbf{t}}+ nabla(mathbf{v}cdotmathbf{A})-(mathbf{v}cdotnabla)mathbf{A} )
where
A is the magnetic vector potential
phi is the electrostatic potential
The symbols nabla,(nablatimes),(nablacdot) denote gradient, curl, and divergence, respectively.

The potentials are related to E and B by

mathbf{E} = - nabla phi - frac { partial mathbf{A} } { partial t }
mathbf{B} = nabla times mathbf{A}

Lorentz force in cgs units

The above-mentioned formulae use SI units which are the most common among experimentalists, technicians, and engineers. In cgs units, which are somewhat more common among theoretical physicists, one has instead
mathbf{F} = q_{cgs} cdot (mathbf{E}_{cgs} + frac{mathbf{v}}{c} times mathbf{B}_{cgs}).
where c is the speed of light. Although this equation looks slightly different, it is completely equivalent, since one has the following relations:

q_{cgs}=frac{q_{SI}}{sqrt{4pi epsilon_0}},   mathbf E_{cgs} =sqrt{4piepsilon_0},mathbf E_{SI}, and   mathbf B_{cgs} ={sqrt{4pi /mu_0}},{mathbf B_{SI}}

where ε0 and μ0 are the vacuum permittivity and vacuum permeability, respectively. In practice, unfortunately, the subscripts "cgs" and "SI" are always omitted, and the unit system has to be assessed from context.

Covariant form of the Lorentz force

Newton's law of motion can be written in covariant form in terms of the field strength tensor.

frac{d p^alpha}{d tau} = q u_beta F^{alpha beta}
where
tau is c times the proper time of the particle,
q is the charge,
u is the 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) ,

with γ = Lorentz factor defined above, and F is the field strength tensor (or electromagnetic tensor) and is written in terms of fields as:

F^{alpha beta} = begin{bmatrix}
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{bmatrix} .

The fields are transformed to a frame moving with constant relative velocity by:

acute{F}^{mu nu} = {Lambda^{mu}}_{alpha} {Lambda^{nu}}_{beta} F^{alpha beta}
,

where {Lambda^{mu}}_{alpha}

 is a Lorentz transformation.
Alternatively, using the four vector:

A^{alpha} = left(phi / c, A_x, A_y, A_z right) ,

related to the electric and magnetic fields by:

mathbf{E = -nabla} phi - partial_t mathbf{A}    mathbf{B = nabla times A } ,

the field tensor becomes:

F^{alpha beta} = frac {partial A^{beta}}{partial x_{alpha}} - frac {partial A^{alpha}}{partial x_{beta}} ,

where:

x_{alpha} = left(-ct, x, y, z right) .

Translation to vector notation

The mu =1 component (x-component) of the force is
gamma frac{d p^1}{d t} = frac{d p^1}{d tau} = q u_beta F^{1 beta} = qleft(-u^0 F^{10} + u^1 F^{11} + u^2 F^{12} + u^3 F^{13} right) .,

Here, tau is the proper time of the particle. Substituting the components of the electromagnetic tensor F yields

gamma frac{d p^1}{d t} = q left(-u^0 left(frac{-E_x}{c} right) + u^2 (B_z) + u^3 (-B_y) right) ,
Writing the four-velocity in terms of the ordinary velocity yields
gamma frac{d p^1}{d t} = q gamma left(c left(frac{E_x}{c} right) + v_y B_z - v_z B_y right) ,

gamma frac{d p^1}{d t} = q gamma left(E_x + left(mathbf{v} times mathbf{B} right)_x right) .,

The calculation of the mu = 2 or mu = 3 is similar yielding

gamma frac{d mathbf{p} }{d t} = frac{d mathbf{p} }{d tau} = q gamma left(mathbf{E} + (mathbf{v} times mathbf{B})right) ,

or, in terms of the vector and scalar potentials A and φ,

frac{d mathbf{p} }{d tau} = q gamma (- nabla phi - frac { partial mathbf{A} } { partial t } + mathbf{v} times (nabla times mathbf{A})) ,

which are the relativistic forms of Newton's law of motion when the Lorentz force is the only force present.

Force on a current-carrying wire

When a wire carrying an electrical current is placed in a magnetic field, each of the moving charges, which comprise the current, experiences the Lorentz force, and together they can create a macroscopic force on the wire (sometimes called the Laplace force). By combining the Lorentz force law above with the definition of electrical current, the following equation results, in the case of a straight, stationary wire:
mathbf{F} = I mathbf{L} times mathbf{B} ,

where

F = Force, measured in newtons
I = current in wire, measured in amperes
B = magnetic field vector, measured in teslas
times = vector cross product
L = a vector, whose magnitude is the length of wire (measured in metres), and whose direction is along the wire, aligned with the direction of conventional current flow.

Alternatively, some authors write

mathbf{F} = L mathbf{I} times mathbf{B}
where the vector direction is now associated with the current variable, instead of the length variable. The two forms are equivalent.

If the wire is not straight but curved, the force on it can be computed by applying this formula to each infinitesimal segment of wire dℓ, then adding up all these forces via integration. Formally, the net force on a stationary, rigid wire carrying a current I is

mathbf{F} = Ioint dboldsymbol{ell}times mathbf{B}(boldsymbol{ell} )
(This is the net force. In addition, there will usually be torque, plus other effects if the wire is not perfectly rigid.)

One application of this is Ampère's force law, which describes how two current-carrying wires can attract or repel each other, since each experiences a Lorentz force from the other's magnetic field. For more information, see the article: Ampère's force law.

EMF

The magnetic force (q v × B) component of the Lorentz force is responsible for motional electromotive force (or motional EMF), the phenomenon underlying many electrical generators. When a conductor is moved through a magnetic field, the magnetic force tries to push electrons through the wire, and this creates the EMF. The term "motional EMF" is applied to this phenomenon, since the EMF is due to the motion of the wire.

In other electrical generators, the magnets move, while the conductors do not. In this case, the EMF is due to the electric force (qE) term in the Lorentz Force equation. The electric field in question is created by the changing magnetic field, resulting in an induced EMF, as described by the Maxwell-Faraday equation (one of the four modern Maxwell's equations).

The two effects are not however symmetric. As one demonstration of this, a charge rotating around the magnetic axis of a stationary, cylindrically-symmetric bar magnet will experience a magnetic force, whereas if the charge is stationary and the magnet is rotating about its axis, there will be no force. This asymmetric effect is called Faraday's paradox.

Both of these EMF's, despite their different origins, can be described by the same equation, namely, the EMF is the rate of change of magnetic flux through the wire. (This is Faraday's law of induction, see above.) Einstein's theory of special 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 faces 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.

General references

The numbered references refer in part to the list immediately below.

  • : volume 2.

Numbered footnotes and references

Applications

The Lorentz force occurs in many devices, including:

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

See also

External links

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