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Electricity · Magnetism
Electrodynamics
 · Free space · Lorentz force law · EMF · Electromagnetic induction · Faraday’s law · Displacement current · Maxwell’s equations · EM field · Electromagnetic radiation · Liénard-Wiechert Potentials · Maxwell tensor · Eddy current ·
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Trajectory of a particle with charge q, under the influence of a magnetic field B (directed perpendicularly out of the screen), for different values of q.
Trajectory of a particle with charge q, under the influence of a magnetic field B (directed perpendicularly out of the screen), for different values of q. Electromagnetism is the Physics of the Electromagnetic field: a field which exerts a Force on particles that possess the property of In Physics, magnetism is one of the Phenomena by which Materials exert attractive or repulsive Forces on other Materials. Classical electromagnetism (or classical electrodynamics) is a theory of Electromagnetism that was developed over the course of the 19th century most prominently In Classical physics, free space is a concept of Electromagnetic theory, corresponding to a theoretically "perfect" Vacuum, and sometimes Electromotive force ( emf, \mathcal{E} is a term used to characterize electrical devices such as Voltaic cells thermoelectric devices electrical Faraday's law of induction describes an important basic law of electromagnetism which is involved in the working of Transformers Inductors and many forms of Faraday's law of induction describes an important basic law of electromagnetism which is involved in the working of Transformers Inductors and many forms of Displacement current is a quantity that arises in a changing electric field In Classical electromagnetism, Maxwell's equations are a set of four Partial differential equations that describe the properties of the electric The electromagnetic field is a physical field produced by electrically charged objects. Electromagnetic radiation takes the form of self-propagating Waves in a Vacuum or in Matter. The Liénard-Wiechert potential describes the electromagnetic effect of a moving Electric charge. The Maxwell Stress Tensor (also known as Maxwell's Stress Tensor is used to calculate the stresses on objects in magnetic or electrical fields An eddy current (also known as Foucault current) is an electrical phenomenon discovered by French physicist Léon Foucault in
This article is about the equation governing the electromagnetic force. For a qualitative overview of the electromagnetic force, see Electromagnetic force. In Physics, the electromagnetic force is the force that the Electromagnetic field exerts on electrically charged particles

In physics, the Lorentz force is the force on a point charge due to electromagnetic fields. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Hendrik Antoon Lorentz ( July 18, 1853 &ndash February 4, 1928) was a Dutch Physicist who shared the 1902 Nobel In Physics, a force is whatever can cause an object with Mass to Accelerate. A point charge is an idealized model of a particle which has an Electric charge. The electromagnetic field is a physical field produced by electrically charged objects. It is given by the following equation in terms of the electric and magnetic fields:[1]

\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[2]
\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). In Physics, the space surrounding an Electric charge or in the presence of a time-varying Magnetic field has a property called an electric field (that can In Physics, a magnetic field is a Vector field that permeates space and which can exert a magnetic force on moving Electric charges In Physics, a force is whatever can cause an object with Mass to Accelerate. The newton (symbol N) is the SI derived unit of Force, named after Isaac Newton in recognition of his work on Classical In Physics, the space surrounding an Electric charge or in the presence of a time-varying Magnetic field has a property called an electric field (that can The volt (symbol V) is the SI derived unit of electric Potential difference or Electromotive force. The metre or meter is a unit of Length. It is the basic unit of Length in the Metric system and in the International In Physics, a magnetic field is a Vector field that permeates space and which can exert a magnetic force on moving Electric charges The tesla (symbol T) is the SI derived unit of Magnetic field B (which is also known as "magnetic flux density" and "magnetic Electric charge is a fundamental conserved property of some Subatomic particles which determines their Electromagnetic interaction. The coulomb (symbol C) is the SI unit of Electric charge. It is named after Charles-Augustin de Coulomb. In Physics, velocity is defined as the rate of change of Position. The metre or meter is a unit of Length. It is the basic unit of Length in the Metric system and in the International The second ( SI symbol s) sometimes abbreviated sec, is the name of a unit of Time, and is the International System of Units In Mathematics, the cross product is a Binary operation on two vectors in a three-dimensional Euclidean space that results in another vector which In Vector calculus, the gradient of a Scalar field is a Vector field which points in the direction of the greatest rate of increase of the scalar cURL is a Command line tool for transferring files with URL syntax. The magnetic potential provides a mathematical way to define a Magnetic field in Classical electromagnetism. At a point in space the electric potential is the Potential energy per unit of charge that is associated with a static (time-invariant Electric field The magnetic potential provides a mathematical way to define a Magnetic field in Classical electromagnetism. At a point in space the electric potential is the Potential energy per unit of charge that is associated with a static (time-invariant Electric field In Typography, emphasis is the exaggeration of words in a text with a font in a different style from the rest of the text&mdashto emphasise them

The interesting feature of the second form of the Lorentz force law is its clean separation of the portion of the force due to the irrotational or grad φ portion of the force, which is due to electrical charges, and the solenoidal portion of the force or A-field portion, which corresponds to the part that appears as magnetic or as electric force depending upon the relative velocity of the frame of reference. In Vector calculus a conservative vector field is a Vector field which is the Gradient of a Scalar potential. In Vector calculus a solenoidal vector field (also known as an incompressible vector field) is a Vector field v with Divergence zero

The Lorentz force law has a close relationship with Faraday's law of induction. Faraday's law of induction describes an important basic law of electromagnetism which is involved in the working of Transformers Inductors and many forms of

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). For the related yet different principle relating to electromagnetic coils see Right hand grip rule.

The term qE is called the electric force, while the term qv × B is called the magnetic force. [3] According to some definitions, the term "Lorentz force" refers specifically to the formula for the magnetic force:[4]

\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. Electric current is the flow (movement of Electric charge. The SI unit of electric current is the Ampere. A wire is a single usually cylindrical, elongated string of drawn Metal. In Physics, a magnetic field is a Vector field that permeates space and which can exert a magnetic force on moving Electric charges In that context, it is also called the Laplace force.

Contents

History

Lorentz introduced this force in 1892. [5] 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. A Dynamical Theory of the Electromagnetic Field which was written in the year 1864 is the third of James Clerk Maxwell 's papers concerned with Electromagnetism In this paper the equation was written as follows:

\mathbf{E} = \mathbf{v} \times (\mu \mathbf{H}) - \frac{\partial\mathbf{A}}{\partial t}-\nabla \phi

where

A is the magnetic vector potential,
\,\phi is the electrostatic potential,
H is the magnetic field H,
μ is magnetic permeability. The magnetic potential provides a mathematical way to define a Magnetic field in Classical electromagnetism. At a point in space the electric potential is the Potential energy per unit of charge that is associated with a static (time-invariant Electric field In Physics, a magnetic field is a Vector field that permeates space and which can exert a magnetic force on moving Electric charges In Electromagnetism, permeability is the degree of Magnetization of a material that responds linearly to an applied Magnetic field.

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

\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. Faraday's law of induction describes an important basic law of electromagnetism which is involved in the working of Transformers Inductors and many forms of [8]

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). In Classical electromagnetism, Maxwell's equations are a set of four Partial differential equations that describe the properties of the electric It now sits adjacent to Maxwell's equations as a separate and essential law. [1]

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. [1][9] 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. The Boltzmann equation, also often known as the Boltzmann transport equation, devised by Ludwig Boltzmann, describes the statistical distribution of The Fokker–Planck equation describes the Time evolution of the Probability density function of the position of a particle and can be generalized to other observables The Navier–Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of viscous Fluid substances such For example, see magnetohydrodynamics, fluid dynamics, electrohydrodynamics, superconductivity, stellar evolution. Magnetohydrodynamics (MHD ( magnetofluiddynamics or hydromagnetics) is the Academic discipline which studies the dynamics of electrically Fluid dynamics is the sub-discipline of Fluid mechanics dealing with fluid flow: Fluids ( Liquids and Gases in motion Electrohydrodynamics (EHD, also known as electro-fluid-dynamics (EFD or electrokinetics, is the study of the dynamics of electrically conducting Superconductivity is a phenomenon occurring in certain Materials generally at very low Temperatures characterized by exactly zero electrical resistance Stellar evolution is the process by which a Star undergoes a sequence of radical changes during its lifetime An entire physical apparatus for dealing with these matters has developed. See for example, Green–Kubo relations and Green's function (many-body theory). Green–Kubo relations give exact mathematical expression for transport coefficients in terms of integrals of time correlation functions In Many-body theory, the term Green's function (or Green function) is sometimes used interchangeably with Correlation function, but refers specifically

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. In Physics and Fluid mechanics, a boundary layer is that layer of Fluid in the immediate vicinity of a bounding surface In Mathematics, in the field of Differential equations a boundary value problem is a Differential equation together with a set of additional restraints In Physics, the Casimir effect and the Casimir-Polder force are physical forces arising from a quantized field. In nuclear and Particle physics, the concept of a cross section is used to express the likelihood of interaction between particles A deus ex machina ( lat. ˈdeːus eks ˈmaːkʰina literally "god from a/the machine" is an improbable [10]

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. [11] 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}\times\mathbf{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". In physical theories, a test particle is an idealized model of an object whose physical properties (usually Mass, Charge, or size) are assumed In Mathematics a vector field is a construction in Vector calculus which associates a vector to every point in a (locally Euclidean space.

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. Bremsstrahlung ( pronounced, from German de ''bremsen'' "to brake" and de ''Strahlung'' "radiation" i This article is mostly concerned with applications of Synchrotron radiation. These effects occur through both a direct effect (called the radiation reaction force) and indirectly (by affecting the motion of nearby charges and currents). In the Physics of Electromagnetism, the Abraham-Lorentz force is the Recoil Force on an accelerating Charged particle caused

Moreover, the electromagnetic force is not in general the same as the net force, due to gravity, electroweak and and other forces, and any extra forces would have to be taken into account in a real measurement. Gravitation is a natural Phenomenon by which objects with Mass attract one another In Particle physics, the electroweak interaction is the unified description of two of the four Fundamental interactions of nature Electromagnetism and the

Lorentz force and Faraday's law of induction

Main article: Faraday's law of induction

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

\mathcal{E} = -\frac{d\Phi_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. Faraday's law of induction describes an important basic law of electromagnetism which is involved in the working of Transformers Inductors and many forms of In Physics, a magnetic field is a Vector field that permeates space and which can exert a magnetic force on moving Electric charges Magnetic flux, represented by the Greek letter Φ ( Phi) is a measure of quantity of Magnetism, taking into account the strength and the extent of a Magnetic Electromotive force ( emf, \mathcal{E} is a term used to characterize electrical devices such as Voltaic cells thermoelectric devices electrical Lenz's law (ˈlɛntsɨz ˌlɔː gives the direction of the induced Electromotive force (emf and current resulting from Electromagnetic induction.

Using the Lorentz force law, the EMF around a closed path ∂Σ is given by:[12][13]

\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 dℓ 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,
dℓ is an infinitesimal vector element of the contour ∂Σ,
v is the velocity of the infinitesimal contour element dℓ,
B is the magnetic field. Infinitesimals (from a 17th century Modern Latin coinage infinitesimus, originally referring to the " Infinite[[ th]]" member of a series have In Physics, velocity is defined as the rate of change of Position. In Physics, a magnetic field is a Vector field that permeates space and which can exert a magnetic force on moving Electric charges
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. In Mathematics, two Vectors are orthogonal if they are Perpendicular, i
Both dℓ 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. For the related yet different principle relating to electromagnetic coils see Right hand grip rule. In Differential geometry, Stokes' theorem is a statement about the integration of Differential forms which generalizes several Theorems from

The surface integral at the right-hand side of this equation is the explicit expression for the magnetic flux ΦB through Σ. In Mathematics, a surface integral is a Definite integral taken over a Surface (which may be a curved set in Space) it can be thought Magnetic flux, represented by the Greek letter Φ ( Phi) is a measure of quantity of Magnetism, taking into account the strength and the extent of a Magnetic Thus, incorporating the Lorentz law in Faraday's equation, we find:[14] [15]

\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:[16]

\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. Faraday's law of induction describes an important basic law of electromagnetism which is involved in the working of Transformers Inductors and many forms of In Classical electromagnetism, Maxwell's equations are a set of four Partial differential equations that describe the properties of the electric In Differential geometry, Stokes' theorem is a statement about the integration of Differential forms which generalizes several Theorems from In Mathematics, Leibniz's rule for Differentiation under the integral sign, named after Gottfried Leibniz, tells us that if we have an Integral [17] 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. This article describes the Faraday paradox in electromagnetism

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[18] 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(\nabla\times\mathbf{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}\cdot\mathbf{A})-(\mathbf{v}\cdot\nabla)\mathbf{A} )

where

A is the magnetic vector potential
φ is the electrostatic potential
The symbols \nabla,(\nabla\times),(\nabla\cdot) denote gradient, curl, and divergence, respectively. At a point in space the electric potential is the Potential energy per unit of charge that is associated with a static (time-invariant Electric field The magnetic potential provides a mathematical way to define a Magnetic field in Classical electromagnetism. In Mathematics, in the area of Vector calculus, Helmholtz's theorem, also known as the fundamental theorem of vector calculus, states that any sufficiently This article is about mathematics See Lawson criterion for the use of the term triple product in relation to Nuclear fusion. The magnetic potential provides a mathematical way to define a Magnetic field in Classical electromagnetism. At a point in space the electric potential is the Potential energy per unit of charge that is associated with a static (time-invariant Electric field In Vector calculus, the gradient of a Scalar field is a Vector field which points in the direction of the greatest rate of increase of the scalar cURL is a Command line tool for transferring files with URL syntax. In Vector calculus, the divergence is an Operator that measures the magnitude of a Vector field &rsquos source or sink at a given point the

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. The centimetre-gram-second system ( CGS) is a system of physical units. Although this equation looks slightly different, it is completely equivalent, since one has the following relations:

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

where ε0 and μ0 are the vacuum permittivity and vacuum permeability, respectively. Vacuum permittivity, referred to by international standards organizations as the electric constant, and denoted by the symbol ε0 is a fundamental Physical The vacuum permeability, referred to by international standards organizations as the magnetic constant, and denoted by the symbol μ 0 (also 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

Main article: Formulation of Maxwell's equations in special relativity

Newton's law of motion can be written in covariant form in terms of the field strength tensor. The covariant formulation of Classical electromagnetism refers to ways of writing the laws of classical electromagnetism (in particular Maxwell's equations In Physics, a covariant transformation is a rule (specified below that describes how certain physical entities change under a change of Coordinate system.

\frac{d p^\alpha}{d \tau} = q u_\beta F^{\alpha \beta}
where
τ 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} . In relativity, proper time is Time measured by a single Clock between events that occur at the same place as the clock Electric charge is a fundamental conserved property of some Subatomic particles which determines their Electromagnetic interaction. In Physics, in particular in Special relativity and General relativity, the four-velocity of an object is a Four-vector (vector in four-dimensional The electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is

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. In Physics, the Lorentz transformation converts between two different observers' measurements of space and time where one observer is in constant motion with respect to 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:[19]

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 μ = 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} = q\left(-u^0 F^{10} + u^1 F^{11} + u^2 F^{12} + u^3 F^{13} \right) .\,

Here, τ is the proper time of the particle. In relativity, proper time is Time measured by a single Clock between events that occur at the same place as the clock 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 μ = 2 or μ = 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. In Physics, in particular in Special relativity and General relativity, the four-velocity of an object is a Four-vector (vector in four-dimensional

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). A wire is a single usually cylindrical, elongated string of drawn Metal. Electric current is the flow (movement of Electric charge. The SI unit of electric current is the Ampere. 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. In Mathematics, the cross product is a Binary operation on two vectors in a three-dimensional Euclidean space that results in another vector which Electric current is the flow (movement of Electric charge. The SI unit of electric current is the Ampere.

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. Infinitesimals (from a 17th century Modern Latin coinage infinitesimus, originally referring to the " Infinite[[ th]]" member of a series have The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space Formally, the net force on a stationary, rigid wire carrying a current I is

\mathbf{F} = I\oint d\boldsymbol{\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. A torque (τ in Physics, also called a moment (of force is a pseudo- vector that measures the tendency of a force to rotate an object about )

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. The force of attraction or repulsion between two current-carrying wires (see Figure 1 is often called Ampère's force law. For more information, see the article: Ampère's force law. The force of attraction or repulsion between two current-carrying wires (see Figure 1 is often called 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. Electromotive force ( emf, \mathcal{E} is a term used to characterize electrical devices such as Voltaic cells thermoelectric devices electrical In Electricity generation, an electrical generator is a device that converts Mechanical energy to Electrical energy, generally using Electromagnetic When a conductor is moved through a magnetic field, the magnetic force tries to push electrons through the wire, and this creates the EMF. In Science and engineering, a conductor is a material which contains movable Electric charges. The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J 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). Faraday's law of induction describes an important basic law of electromagnetism which is involved in the working of Transformers Inductors and many forms of In Classical electromagnetism, Maxwell's equations are a set of four Partial differential equations that describe the properties of the electric [20]

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. This article describes the Faraday paradox in electromagnetism

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. Magnetic flux, represented by the Greek letter Φ ( Phi) is a measure of quantity of Magnetism, taking into account the strength and the extent of a Magnetic (This is Faraday's law of induction, see above. In Physics, the Lorentz force is the Force on a Point charge due to Electromagnetic fields It is given by the following equation ) Einstein's theory of special relativity was partially motivated by the desire to better understand this link between the two effects. Special relativity (SR (also known as the special theory of relativity or STR) is the Physical theory of Measurement in Inertial [20] 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. In Physics, an inertial frame of reference is a Frame of reference which belongs to a set of frames in which Physical laws hold in the same and simplest In Vector calculus a solenoidal vector field (also known as an incompressible vector field) is a Vector field v with Divergence zero [21]

General references

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

Numbered footnotes and references

  1. ^ a b c See Jackson page 2. The book lists the four modern Maxwell's equations, and then states, "Also essential for consideration of charged particle motion is the Lorentz force equation, F = q ( E+ v × B ), which gives the force acting on a point charge q in the presence of electromagnetic fields. "
  2. ^ These definitions use Helmholtz's theorem. In Mathematics, in the area of Vector calculus, Helmholtz's theorem, also known as the fundamental theorem of vector calculus, states that any sufficiently Because divB = 0 (Gauss's law for magnetism), Helmholtz's theorem proves that we can define a vector field A (called the magnetic potential) such that B = ∇ × A. The magnetic potential provides a mathematical way to define a Magnetic field in Classical electromagnetism. From the Maxwell-Faraday equation, ∇ × E = −∂t B so ∇ × [ E + ∂t A ] = 0. In Classical electromagnetism, Maxwell's equations are a set of four Partial differential equations that describe the properties of the electric Applying Helmholtz's theorem again to E + ∂t A, which has zero curl, we find that we can define a scalar field ɸ (called the electric potential) with E + ∂t A = −∇ɸ. At a point in space the electric potential is the Potential energy per unit of charge that is associated with a static (time-invariant Electric field The equation for B automatically satisfies ∇•B = 0, that is, demonstrates that B is a solenoidal vector field. In Vector calculus a solenoidal vector field (also known as an incompressible vector field) is a Vector field v with Divergence zero Also, the equation for E shows that it can have two different components: a conservative or irrotational vector field component (which originates in electric charges) and a nonconservative or curl component (which originates in the Maxwell-Faraday equation). In Vector calculus a conservative vector field is a Vector field which is the Gradient of a Scalar potential. In Classical electromagnetism, Maxwell's equations are a set of four Partial differential equations that describe the properties of the electric For more details, see magnetic potential and electric potential. The magnetic potential provides a mathematical way to define a Magnetic field in Classical electromagnetism. At a point in space the electric potential is the Potential energy per unit of charge that is associated with a static (time-invariant Electric field
  3. ^ See Griffiths page 204.
  4. ^ For example, see the website of the "Lorentz Institute": [1], or Griffiths.
  5. ^ Darrigol, Olivier (2000), Electrodynamics from Ampère to Einstein, Oxford, [England]: Oxford University Press, p. André-Marie Ampère (20 January 1775 &ndash 10 June 1836 was a French Physicist and Mathematician who is generally credited as one of the main discoverers Albert Einstein ( German: ˈalbɐt ˈaɪ̯nʃtaɪ̯n; English: ˈælbɝt ˈaɪnstaɪn (14 March 1879 – 18 April 1955 was a German -born theoretical 327, ISBN 0-198-50593-0, <http://books.google.com/books?id=ysMf2pAid94C&printsec=frontcover&dq=Electrodynamics&lr=&as_brr=0&sig=RuQLjK14uJPu7mYCC0vRr0SdQFw#PPA327,M1> 
  6. ^ "While the wire is subject to either volta-electric or magneto-electric induction it appears to be in a peculiar state, for it resists the formation of an electrical current in it. … I have … ventured to designate it as the electro-tonic state. " Quoted by Maxwell from Faraday, Trans. Cam. Phil. Soc., p. 51, v. 10 (1864)
  7. ^ At the experimental level in classical electromagnetism, E and B are the fundamental, measurable, physical fields. See, for example, Griffiths page 417, or Jackson page 239. However, in quantum field theory, the potentials A and φ play a fundamental role. See, for example, Srednicki, Chapter 58, p. 351 ff. and R Littlejohn on quantization of the electromagnetic field; Physics 221B notes–quantizationPhysics 221B notes–interaction However, the fields themselves can be related to electromotive force (in the modern definition) only by addition of the Lorentz force. Maxwell did not formulate the equations with a separate Lorentz force equation.
  8. ^ See Griffiths page 417, or Jackson page 239.
  9. ^ See Griffiths page 326, which states that Maxwell's equations, "together with the [Lorentz] force law. . . summarize the entire theoretical content of classical electrodynamics".
  10. ^ That is, a first-principles approach might be approximated to make calculation possible without complications that are not very significant to the results. For example, a metallic boundary might be approximated as having infinite conductivity. A statistical mechanical model of a plasma might approximate the treatment of collisions with boundaries and between particles.
  11. ^ See, for example, Jackson p777-8.
  12. ^ Landau, L. D. , Lifshit︠s︡, E. M. , & Pitaevskiĭ, L. P. (1984). Electrodynamics of continuous media; Volume 8 Course of theoretical physics, Second Edition, Oxford: Butterworth-Heinemann, §63 (§49 pp. 205-207 in 1960 edition). ISBN 0750626348.  
  13. ^ M N O Sadiku (2007). Elements of elctromagnetics, Fourth Edition, NY/Oxford: Oxford University Press, p. 391. ISBN 0-19-530048-3.  
  14. ^ If the boundary deforms, so velocity varies with location, the velocity v is the velocity at the location of dℓ. See Rothwell Edward J Rothwell, Michael J Cloud (2001). Electromagnetics. Boca Raton, Fla: CRC Press, p. 56. ISBN 084931397X.  
  15. ^ Jackson JD. Eqs. 5.141 & 5.142, p. 211. ISBN 0-471-30932-X.  
  16. ^ Roger F Harrington (2003). Introduction to electromagnetic engineering. Mineola, NY: Dover Publications, p. 56. ISBN 0486432416.  
  17. ^ If the surface deforms, the Leibniz integral rule is more complicated. A mathematical demonstration of this result for deformable surfaces has not been located.
  18. ^ That is, a field that is not conservative, not expressible as the gradient of a scalar field, and not subject to the gradient theorem. In Vector calculus a conservative vector field is a Vector field which is the Gradient of a Scalar potential. In Mathematics and Physics, a scalar field associates a scalar value which can be either mathematical in definition or physical, to every point The gradient theorem, sometimes also known as the fundamental theorem of calculus for line integrals, says that a Line integral through a Gradient field
  19. ^ DJ Griffiths (1999). Introduction to electrodynamics. Saddle River NJ: Pearson/Addison-Wesley, p. 541. ISBN 0-13-805326-X.  
  20. ^ a b See Griffiths pages 301–3.
  21. ^ Tai L. Chow (2006). Electromagnetic theory. Sudbury MA: Jones and Bartlett, p. 395. ISBN 0-7637-3827-1.  

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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Dictionary

Lorentz force

-noun

  1. (physics) the force exerted on a charged particle in an electromagnetic field
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