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Electrical conductors moving through a steady magnetic field, or stationary conductors within a changing magnetic field, will have circular currents induced within them by induction, called eddy currents. Eddy currents flow in closed loops in planes perpendicular to the magnetic field. They have useful applications in eddy current brakes and induction heating systems. However eddy currents induced in the metal magnetic cores of transformers and AC motors and generators are undesirable since they dissipate energy (called core losses) as heat in the resistance of the metal. Cores for these devices use a number of methods to reduce eddy currents: |
Eddy currents occur when a solid metallic mass is rotated in a magnetic field, because the outer portion of the metal cuts more magnetic lines of force than the inner portion; hence the induced electromotive force is not uniform; this tends to cause electric currents between the points of greatest and least potential. Eddy currents consume a considerable amount of energy and often cause a harmful rise in temperature. |
Only five laminations or plates are shown in this example, so as to show the subdivision of the eddy currents. In practical use, the number of laminations or punchings ranges from 40 to 66 per inch (16 to 26 per centimetre), and brings the eddy current loss down to about one percent. While the plates can be separated by insulation, the voltage is so low that the natural rust/oxide coating of the plates is enough to prevent current flow across the laminations. |
This is a rotor approximately 20 mm in diameter from a DC motor used in a Note the laminations of the electromagnet pole pieces, used to limit parasitic inductive losses. |
In this illustration, a solid copper bar conductor on a rotating armature is just passing under the tip of the pole piece N of the field magnet. Note the uneven distribution of the lines of force across the copper bar. The magnetic field is more concentrated and thus stronger on the left edge of the copper bar (a,b) while the field is weaker on the right edge (c,d). Since the two edges of the bar move with the same velocity, this difference in field strength across the bar creates whorls or current eddies within the copper bar. |
High current power-frequency devices, such as electric motors, generators and transformers, use multiple small conductors in parallel to break up the eddy flows that can form within large solid conductors. The same principle is applied to transformers used at higher than power frequency, for example, those used in switch-mode power supplies and the intermediate frequency coupling transformers of radio receivers. |
In physics (specifically in electromagnetism) the Lorentz force (or electromagnetic force) is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge moving with a velocity in an electric field and a magnetic field experiences a force of |
(in SI units). It says that the electromagnetic force on a charge is a combination of a force in the direction of the electric field proportional to the magnitude of the field and the quantity of charge, and a force at right angles to the magnetic field and the velocity of the charge, proportional to the magnitude of the field, the charge, and the velocity. Variations on this basic formula describe the magnetic force on a current-carrying wire (sometimes called Laplace force), the electromotive force in a wire loop moving through a magnetic field (an aspect of Faraday's law of induction), and the force on a moving charged particle. |
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. |
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: |
This is valid, even for particles approaching the speed of light (that is, magnitude of v, |v| ≈ "c"). So the two vector fields E and B are thereby defined throughout space and time, and these are called the "electric field" and "magnetic field". The fields are defined everywhere in space and time with respect to what force a test charge would receive regardless of whether a charge is present to experience the force. |
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, as if forced into a curved trajectory, it emits radiation that causes it to lose kinetic energy. 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). |
The force F acting on a particle of electric charge "q" with instantaneous velocity v, due to an external electric field E and magnetic field B, is given by (in SI units): |
where × is the vector cross product (all boldface quantities are vectors). In terms of Cartesian components, we have: |
In general, the electric and magnetic fields are functions of the position and time. Therefore, explicitly, the Lorentz force can be written as: |
in which r is the position vector of the charged particle, "t" is time, and the overdot is a time derivative. |
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 fingers of the right hand are extended to point in the direction of v and are then curled to point in the direction of B, then the extended thumb will point in the direction of F). |
The term "q"E is called the electric force, while the term "q"(v × B) is called the magnetic force. According to some definitions, the term "Lorentz force" refers specifically to the formula for the magnetic force, 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 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. |
The Lorentz force is a force exerted by the electromagnetic field on the charged particle, that is, it is the rate at which linear momentum is transferred from the electromagnetic field to the particle. Associated with it is the power which is the rate at which energy is transferred from the electromagnetic field to the particle. That power is |
Notice that the magnetic field does not contribute to the power because the magnetic force is always perpendicular to the velocity of the particle. |
For a continuous charge distribution in motion, the Lorentz force equation becomes: |
where formula_9 is the force on a small piece of the charge distribution with charge formula_10. If both sides of this equation are divided by the volume of this small piece of the charge distribution formula_11, the result is: |
where formula_13 is the "force density" (force per unit volume) and formula_14 is the charge density (charge per unit volume). Next, the current density corresponding to the motion of the charge continuum is |
so the continuous analogue to the equation is |
The total force is the volume integral over the charge distribution: |
By eliminating formula_14 and formula_18, using Maxwell's equations, and manipulating using the theorems of vector calculus, this form of the equation can be used to derive the Maxwell stress tensor formula_19, in turn this can be combined with the Poynting vector formula_20 to obtain the electromagnetic stress–energy tensor T used in general relativity. |
In terms of formula_19 and formula_20, another way to write the Lorentz force (per unit volume) is |
where formula_24 is the speed of light and ∇· denotes the divergence of a tensor field. Rather than the amount of charge and its velocity in electric and magnetic fields, this equation relates the energy flux (flow of "energy" per unit time per unit distance) in the fields to the force exerted on a charge distribution. See Covariant formulation of classical electromagnetism for more details. |
The density of power associated with the Lorentz force in a material medium is |
If we separate the total charge and total current into their free and bound parts, we get that the density of the Lorentz force is |
where: formula_27 is the density of free charge; formula_28 is the polarization density; formula_29 is the density of free current; and formula_30 is the magnetization density. In this way, the Lorentz force can explain the torque applied to a permanent magnet by the magnetic field. The density of the associated power is |
The above-mentioned formulae use SI units which are the most common. In older cgs-Gaussian units, which are somewhat more common among some theoretical physicists as well as condensed matter experimentalists, one has instead |
where "c" is the speed of light. |
Although this equation looks slightly different, it is completely equivalent, since |
where ε0 is the vacuum permittivity and μ0 the vacuum permeability. In practice, the subscripts "cgs" and "SI" are always omitted, and the unit system has to be assessed from context. |
Trajectories of particles due to the Lorentz force. |
In many cases of practical interest, the motion in a magnetic field of an electrically charged particle (such as an electron or ion in a plasma) can be treated as the superposition of 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, possibly resulting in electric currents or chemical separation. |
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). |
When a wire carrying an electric 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 electric current, the following equation results, in the case of a straight, stationary wire: |
where is a vector whose magnitude is the length of wire, and whose direction is along the wire, aligned with the direction of conventional current charge flow . |
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 , then adding up all these forces by integration. Formally, the net force on a stationary, rigid wire carrying a steady current is |
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. |
The magnetic force () 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 field exerts opposite forces on electrons and nuclei in 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 ("q"E) 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). |
Both of these EMFs, despite their apparently distinct origins, are 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 below.) 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". |
Lorentz force and Faraday's law of induction. |
Given a loop of wire in a magnetic field, Faraday's law of induction states the induced electromotive force (EMF) in the wire is: |
is the magnetic flux through the loop, B is the magnetic field, Σ("t") is a surface bounded by the closed contour ∂Σ("t"), at time "t", dA is an infinitesimal vector area element of Σ("t") (magnitude is the area of an infinitesimal patch of surface, direction is orthogonal to that surface patch). |
The "sign" of the EMF is determined by Lenz's law. Note that this is valid for not only a "stationary" wirebut also for a "moving" wire. |
From Faraday's law of induction (that is valid for a moving wire, for instance in a motor) and the Maxwell Equations, the Lorentz Force can be deduced. The reverse is also true, the Lorentz force and the Maxwell Equations can be used to derive the Faraday Law. |
Let Σ("t") be the moving wire, moving together without rotation and with constant velocity v and Σ("t") be the internal surface of the wire. The EMF around the closed path ∂Σ("t") is given by: |
is the electric field and dℓ is an infinitesimal vector element of the contour ∂Σ("t"). |
NB: 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. |
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": |
The Maxwell–Faraday equation also can be written in an "integral form" using the Kelvin–Stokes theorem. |
So we have, the Maxwell Faraday equation: |
The two are equivalent if the wire is not moving. Using the Leibniz integral rule and that "div" B = 0, results in, |
since this is valid for any wire position it implies that, |
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. |
Note that the Maxwell Faraday's equation implies that the Electric Field E is non conservative when the Magnetic Field B varies in time, and is not expressible as the gradient of a scalar field, and not subject to the gradient theorem since its rotational is not zero. |
The E and B fields can be replaced by the magnetic vector potential A and (scalar) electrostatic potential "ϕ" by |
where ∇ is the gradient, ∇⋅ is the divergence, ∇× is the curl. |
Using an identity for the triple product this can be rewritten as, |
(Notice that the coordinates and the velocity components should be treated as independent variables, so the del operator acts only on formula_50, not on formula_51; thus, there is no need of using Feynman's subscript notation in the equation above). Using the chain rule, the total derivative of formula_50 is: |
With v = ẋ, we can put the equation into the convenient Euler–Lagrange form |
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: |
where A and "ϕ" are the potential fields as above. The quantity formula_58 can be thought as a velocity-dependent potential function. Using Lagrange's equations, the equation for the Lorentz force given above can be obtained again. |
The potential energy depends on the velocity of the particle, so the force is velocity dependent, so it is not conservative. |
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. |
Using the metric signature , the Lorentz force for a charge "q" can be written in covariant form: |
where "pα" is the four-momentum, defined as |
"τ" the proper time of the particle, "Fαβ" the contravariant electromagnetic tensor |
and "U" is the covariant 4-velocity of the particle, defined as: |
The fields are transformed to a frame moving with constant relative velocity by: |
where Λ"μα" is the Lorentz transformation tensor. |
The component ("x"-component) of the force is |
Substituting the components of the covariant electromagnetic tensor "F" yields |
Using the components of covariant four-velocity yields |
The calculation for , 3 (force components in the "y" and "z" directions) yields similar results, so collecting the 3 equations into one: |
and since differentials in coordinate time "dt" and proper time "dτ" are related by the Lorentz factor, |
This is precisely the Lorentz force law, however, it is important to note that p is the relativistic expression, |
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 formula_72, and an arbitrary time-direction, formula_73. This can be settled through Space-Time Algebra (or the geometric algebra of space-time), a type of Clifford algebra defined on a pseudo-Euclidean space, as |
formula_76 is a space-time 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 space-time planes) and rotations (rotations in space-space planes). The dot product with the vector formula_73 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 formula_78, where |
(which shows our choice for the metric) and the velocity is |
The proper (invariant is an inadequate term because no transformation has been defined) form of the Lorentz force law is simply |
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. |
In the general theory of relativity the equation of motion for a particle with mass formula_81 and charge formula_82, moving in a space with metric tensor formula_83 and electromagnetic field formula_84, is given as |
where formula_86 (formula_87 is taken along the trajectory), formula_88, and formula_89. |
The equation can also be written as |
where formula_91 is the Christoffel symbol (of the torsion-free metric connection in general relativity), or as |
where formula_93 is the covariant differential in general relativity (metric, torsion-free). |
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: |
The numbered references refer in part to the list immediately below. |
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inertial frame of reference in which they are recorded. Although initially developed by mathematician Hermann Minkowski for Maxwell's equations of electromagnetism, the mathematical structure of Minkowski spacetime was shown to be implied by the postulates of special relativity. |
Minkowski space is closely associated with Einstein's theories of special relativity and general relativity and is the most common mathematical structure on which special relativity is formulated. While the individual components in Euclidean space and time may differ due to length contraction and time dilation, in Minkowski spacetime, all frames of reference will agree on the total distance in spacetime between events. Because it treats time differently than it treats the 3 spatial dimensions, Minkowski space differs from four-dimensional Euclidean space. |
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