Misplaced Pages

#514485

147-439: A magnetic field (sometimes called B-field ) is a physical field that describes the magnetic influence on moving electric charges , electric currents , and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to the magnetic field. A permanent magnet 's magnetic field pulls on ferromagnetic materials such as iron , and attracts or repels other magnets. In addition,

294-407: A field is a physical quantity , represented by a scalar , vector , or tensor , that has a value for each point in space and time . An example of a scalar field is a weather map, with the surface temperature described by assigning a number to each point on the map. A surface wind map, assigning an arrow to each point on a map that describes the wind speed and direction at that point,

441-473: A magnetic monopole is a hypothetical particle (or class of particles) that physically has only one magnetic pole (either a north pole or a south pole). In other words, it would possess a "magnetic charge" analogous to an electric charge. Magnetic field lines would start or end on magnetic monopoles, so if they exist, they would give exceptions to the rule that magnetic field lines neither start nor end. Some theories (such as Grand Unified Theories ) have predicted

588-512: 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. While the modern Maxwell's equations describe how electrically charged particles and currents or moving charged particles give rise to electric and magnetic fields,

735-698: A stationary wire – but 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

882-524: A classical field are usually specified by the Lagrangian density in terms of the field components; the dynamics can be obtained by using the action principle . It is possible to construct simple fields without any prior knowledge of physics using only mathematics from multivariable calculus , potential theory and partial differential equations (PDEs). For example, scalar PDEs might consider quantities such as amplitude, density and pressure fields for

1029-530: A classical field theory should, at least in principle, permit a recasting in quantum mechanical terms; success yields the corresponding quantum field theory . For example, quantizing classical electrodynamics gives quantum electrodynamics . Quantum electrodynamics is arguably the most successful scientific theory; experimental data confirm its predictions to a higher precision (to more significant digits ) than any other theory. The two other fundamental quantum field theories are quantum chromodynamics and

1176-476: A construction of the dynamics of a field, i.e. a specification of how a field changes with time or with respect to other independent physical variables on which the field depends. Usually this is done by writing a Lagrangian or a Hamiltonian of the field, and treating it as a classical or quantum mechanical system with an infinite number of degrees of freedom . The resulting field theories are referred to as classical or quantum field theories. The dynamics of

1323-430: A distance (although they set it aside because of the ongoing utility of the field concept for research in general relativity and quantum electrodynamics ). There are several examples of classical fields . Classical field theories remain useful wherever quantum properties do not arise, and can be active areas of research. Elasticity of materials, fluid dynamics and Maxwell's equations are cases in point. Some of

1470-481: A field cannot be a scalar field somewhere and a vector field somewhere else. For example, the Newtonian gravitational field is a vector field: specifying its value at a point in spacetime requires three numbers, the components of the gravitational field vector at that point. Moreover, within each category (scalar, vector, tensor), a field can be either a classical field or a quantum field , depending on whether it

1617-417: A field line produce synchrotron radiation that is detectable in radio waves . The finest precision for a magnetic field measurement was attained by Gravity Probe B at 5 aT ( 5 × 10 T ). The field can be visualized by a set of magnetic field lines , that follow the direction of the field at each point. The lines can be constructed by measuring the strength and direction of the magnetic field at

SECTION 10

#1732790967515

1764-428: A formal definition for a field in 1851. The independent nature of the field became more apparent with James Clerk Maxwell 's discovery that waves in these fields, called electromagnetic waves , propagated at a finite speed. Consequently, the forces on charges and currents no longer just depended on the positions and velocities of other charges and currents at the same time, but also on their positions and velocities in

1911-490: A height field. Fluid dynamics has fields of pressure , density , and flow rate that are connected by conservation laws for energy and momentum. The mass continuity equation is a continuity equation , representing the conservation of mass ∂ ρ ∂ t + ∇ ⋅ ( ρ u ) = 0 {\displaystyle {\frac {\partial \rho }{\partial t}}+\nabla \cdot (\rho \mathbf {u} )=0} and

2058-640: A homogeneous field: F = I ℓ × B , {\displaystyle \mathbf {F} =I{\boldsymbol {\ell }}\times \mathbf {B} ,} where 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 conventional current I . If the wire is not straight, the force on it can be computed by applying this formula to each infinitesimal segment of wire d ℓ {\displaystyle \mathrm {d} {\boldsymbol {\ell }}} , then adding up all these forces by integration . This results in

2205-409: A large number of points (or at every point in space). Then, mark each location with an arrow (called a vector ) pointing in the direction of the local magnetic field with its magnitude proportional to the strength of the magnetic field. Connecting these arrows then forms a set of magnetic field lines. The direction of the magnetic field at any point is parallel to the direction of nearby field lines, and

2352-420: A magnetic H -field is produced by fictitious magnetic charges that are spread over the surface of each pole. These magnetic charges are in fact related to the magnetization field M . The H -field, therefore, is analogous to the electric field E , which starts at a positive electric charge and ends at a negative electric charge. Near the north pole, therefore, all H -field lines point away from

2499-503: A magnetic field may vary with location, it is described mathematically by a function assigning a vector to each point of space, called a vector field (more precisely, a pseudovector field). In electromagnetics , the term magnetic field is used for two distinct but closely related vector fields denoted by the symbols B and H . In the International System of Units , the unit of B , magnetic flux density,

2646-486: 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

2793-421: A magnetized material, the quantities on each side of this equation differ by the magnetization field of the material. Magnetic fields are produced by moving electric charges and the intrinsic magnetic moments of elementary particles associated with a fundamental quantum property, their spin . Magnetic fields and electric fields are interrelated and are both components of the electromagnetic force , one of

2940-429: A magnetized object is divided in half, a new pole appears on the surface of each piece, so each has a pair of complementary poles. The magnetic pole model does not account for magnetism that is produced by electric currents, nor the inherent connection between angular momentum and magnetism. The pole model usually treats magnetic charge as a mathematical abstraction, rather than a physical property of particles. However,

3087-457: 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

SECTION 20

#1732790967515

3234-477: 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 F = q 2 v × B . {\displaystyle \mathbf {F} ={\frac {q}{2}}\mathbf {v} \times \mathbf {B} .} Thomson derived

3381-406: A nonuniform magnetic field exerts minuscule forces on "nonmagnetic" materials by three other magnetic effects: paramagnetism , diamagnetism , and antiferromagnetism , although these forces are usually so small they can only be detected by laboratory equipment. Magnetic fields surround magnetized materials, electric currents, and electric fields varying in time. Since both strength and direction of

3528-401: A north and a south pole. The magnetic field of permanent magnets can be quite complicated, especially near the magnet. The magnetic field of a small straight magnet is proportional to the magnet's strength (called its magnetic dipole moment m ). The equations are non-trivial and depend on the distance from the magnet and the orientation of the magnet. For simple magnets, m points in

3675-506: A particle makes a field, and a field acts on another particle, and the field has such familiar properties as energy content and momentum, just as particles can have." In practice, the strength of most fields diminishes with distance, eventually becoming undetectable. For instance the strength of many relevant classical fields, such as the gravitational field in Newton's theory of gravity or the electrostatic field in classical electromagnetism,

3822-451: A particle of charge q moving with a velocity v in an electric field E and a magnetic field B experiences a force (in SI units ) of F = q ( E + v × B ) . {\displaystyle \mathbf {F} =q\left(\mathbf {E} +\mathbf {v} \times \mathbf {B} \right).} It says that the electromagnetic force on a charge q

3969-412: A set of differential equations which directly relate E and B to ρ and J . Alternatively, one can describe the system in terms of its scalar and vector potentials V and A . A set of integral equations known as retarded potentials allow one to calculate V and A from ρ and J , and from there the electric and magnetic fields are determined via the relations At the end of the 19th century,

4116-417: A simplified physical model of an isolated closed system is set . They are also subject to the inverse-square law . For electromagnetic waves, there are optical fields , and terms such as near- and far-field limits for diffraction. In practice though, the field theories of optics are superseded by the electromagnetic field theory of Maxwell Gravity waves are waves in the surface of water, defined by

4263-500: A small distance vector d , such that m = q m   d . The magnetic pole model predicts correctly the field H both inside and outside magnetic materials, in particular the fact that H is opposite to the magnetization field M inside a permanent magnet. Since it is based on the fictitious idea of a magnetic charge density , the pole model has limitations. Magnetic poles cannot exist apart from each other as electric charges can, but always come in north–south pairs. If

4410-452: A small magnet is proportional both to the applied magnetic field and to the magnetic moment m of the magnet: τ = m × B = μ 0 m × H , {\displaystyle {\boldsymbol {\tau }}=\mathbf {m} \times \mathbf {B} =\mu _{0}\mathbf {m} \times \mathbf {H} ,\,} where × represents the vector cross product . This equation includes all of

4557-407: A torque proportional to the distance (perpendicular to the force) between them. With the definition of m as the pole strength times the distance between the poles, this leads to τ = μ 0 m H sin  θ , where μ 0 is a constant called the vacuum permeability , measuring 4π × 10 V · s /( A · m ) and θ is the angle between H and m . Mathematically, the torque τ on

Magnetic field - Misplaced Pages Continue

4704-467: 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 in

4851-425: Is The electric field is conservative , and hence can be described by a scalar potential, V ( r ): A steady current I flowing along a path ℓ will create a field B, that exerts a force on nearby moving charged particles that is quantitatively different from the electric field force described above. The force exerted by I on a nearby charge q with velocity v is where B ( r ) is the magnetic field , which

4998-406: 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 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

5145-491: Is tesla (symbol: T). The Gaussian-cgs unit of B is the gauss (symbol: G). (The conversion is 1 T ≘ 10000 G.) One nanotesla corresponds to 1 gamma (symbol: γ). The magnetic H field is defined: H ≡ 1 μ 0 B − M {\displaystyle \mathbf {H} \equiv {\frac {1}{\mu _{0}}}\mathbf {B} -\mathbf {M} } where μ 0 {\displaystyle \mu _{0}}

5292-453: Is a combination of (1) a force in the direction of the electric field E (proportional to the magnitude of the field and the quantity of charge), and (2) a force at right angles to both the magnetic field B and the velocity v 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 ),

5439-522: 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 v ⋅ F = q v ⋅ E . {\displaystyle \mathbf {v} \cdot \mathbf {F} =q\,\mathbf {v} \cdot \mathbf {E} .} Notice that

5586-557: Is a specific example of a general rule that magnets are attracted (or repulsed depending on the orientation of the magnet) into regions of higher magnetic field. Any non-uniform magnetic field, whether caused by permanent magnets or electric currents, exerts a force on a small magnet in this way. The details of the Amperian loop model are different and more complicated but yield the same result: that magnetic dipoles are attracted/repelled into regions of higher magnetic field. Mathematically,

5733-420: Is an example of a vector field , i.e. a 1-dimensional (rank-1) tensor field. Field theories, mathematical descriptions of how field values change in space and time, are ubiquitous in physics. For instance, the electric field is another rank-1 tensor field, while electrodynamics can be formulated in terms of two interacting vector fields at each point in spacetime, or as a single-rank 2-tensor field. In

5880-411: Is associated with a gravitational field g which describes its influence on other bodies with mass. The gravitational field of M at a point r in space corresponds to the ratio between force F that M exerts on a small or negligible test mass m located at r and the test mass itself: Stipulating that m is much smaller than M ensures that the presence of m has a negligible influence on

6027-410: Is characterized by numbers or quantum operators respectively. In this theory an equivalent representation of field is a field particle , for instance a boson . To Isaac Newton , his law of universal gravitation simply expressed the gravitational force that acted between any pair of massive objects. When looking at the motion of many bodies all interacting with each other, such as the planets in

Magnetic field - Misplaced Pages Continue

6174-494: 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

6321-522: Is determined from I by the Biot–Savart law : The magnetic field is not conservative in general, and hence cannot usually be written in terms of a scalar potential. However, it can be written in terms of a vector potential , A ( r ): In general, in the presence of both a charge density ρ( r , t ) and current density J ( r , t ), there will be both an electric and a magnetic field, and both will vary in time. They are determined by Maxwell's equations ,

6468-656: Is formulated in the terms of jet manifolds ( covariant classical field theory ). In modern physics , the most often studied fields are those that model the four fundamental forces which one day may lead to the Unified Field Theory . A convenient way of classifying a field (classical or quantum) is by the symmetries it possesses. Physical symmetries are usually of two types: Fields are often classified by their behaviour under transformations of spacetime . The terms used in this classification are: Lorentz force law The Lorentz force law states that

6615-1156: Is given by ( SI definition of quantities ): F = q ( E + v × B ) {\displaystyle \mathbf {F} =q\left(\mathbf {E} +\mathbf {v} \times \mathbf {B} \right)} where × is the vector cross product (all boldface quantities are vectors). In terms of Cartesian components, we have: F x = q ( E x + v y B z − v z B y ) , F y = q ( E y + v z B x − v x B z ) , F z = q ( E z + v x B y − v y B x ) . {\displaystyle {\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,

6762-402: Is inversely proportional to the square of the distance from the source (i.e. they follow Gauss's law ). A field can be classified as a scalar field, a vector field, a spinor field or a tensor field according to whether the represented physical quantity is a scalar , a vector , a spinor , or a tensor , respectively. A field has a consistent tensorial character wherever it is defined: i.e.

6909-417: Is no net force on that magnet since the force is opposite for opposite poles. If, however, the magnetic field of the first magnet is nonuniform (such as the H near one of its poles), each pole of the second magnet sees a different field and is subject to a different force. This difference in the two forces moves the magnet in the direction of increasing magnetic field and may also cause a net torque. This

7056-466: Is not the case. Ampère also formulated a force law . Based on this law, Gauss concluded that the electromagnetic force between two point charges depends not only on the distance but also on the relative velocity. The Weber force is a central force and complies with Newton's third law . This demonstrates not only the conservation of momentum but also that the conservation of energy and the conservation of angular momentum apply. Weber electrodynamics

7203-465: Is only a quasistatic approximation , i.e. it should not be used for higher velocities and accelerations. However, the Weber force illustrates that the Lorentz force can be traced back to central forces between numerous point-like charge carriers. 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 ,

7350-422: Is particularly true for magnetic fields, such as those due to electric currents, that are not generated by magnetic materials. A realistic model of magnetism is more complicated than either of these models; neither model fully explains why materials are magnetic. The monopole model has no experimental support. The Amperian loop model explains some, but not all of a material's magnetic moment. The model predicts that

7497-712: Is possible to approach their quantum counterparts from a purely mathematical view using similar techniques as before. The equations governing the quantum fields are in fact PDEs (specifically, relativistic wave equations (RWEs)). Thus one can speak of Yang–Mills , Dirac , Klein–Gordon and Schrödinger fields as being solutions to their respective equations. A possible problem is that these RWEs can deal with complicated mathematical objects with exotic algebraic properties (e.g. spinors are not tensors , so may need calculus for spinor fields ), but these in theory can still be subjected to analytical methods given appropriate mathematical generalization . Field theory usually refers to

SECTION 50

#1732790967515

7644-449: Is strictly only valid for magnets of zero size, but is often a good approximation for not too large magnets. The magnetic force on larger magnets is determined by dividing them into smaller regions each having their own m then summing up the forces on each of these very small regions . If two like poles of two separate magnets are brought near each other, and one of the magnets is allowed to turn, it promptly rotates to align itself with

7791-418: Is that of maximum increase of m · B . The dot product m · B = mB cos( θ ) , where m and B represent the magnitude of the m and B vectors and θ is the angle between them. If m is in the same direction as B then the dot product is positive and the gradient points "uphill" pulling the magnet into regions of higher B -field (more strictly larger m · B ). This equation

7938-587: Is the force density (force per unit volume) and ρ {\displaystyle \rho } is the charge density (charge per unit volume). Next, the current density corresponding to the motion of the charge continuum is J = ρ v {\displaystyle \mathbf {J} =\rho \mathbf {v} } so the continuous analogue to the equation is f = ρ E + J × B {\displaystyle \mathbf {f} =\rho \mathbf {E} +\mathbf {J} \times \mathbf {B} } The total force

8085-409: 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 , d A 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

8232-426: 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

8379-962: Is the speed of light . Although this equation looks slightly different, it is equivalent, since one has the following relations: q G = q S I 4 π ε 0 , E G = 4 π ε 0 E S I , B G = 4 π / μ 0 B S I , c = 1 ε 0 μ 0 . {\displaystyle 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 ε 0

8526-546: Is the tesla (in SI base units: kilogram per second squared per ampere), which is equivalent to newton per meter per ampere. The unit of H , magnetic field strength, is ampere per meter (A/m). B and H differ in how they take the medium and/or magnetization into account. In vacuum , the two fields are related through the vacuum permeability , B / μ 0 = H {\displaystyle \mathbf {B} /\mu _{0}=\mathbf {H} } ; in

8673-905: Is the vacuum permeability , and M is the magnetization vector . In a vacuum, B and H are proportional to each other. Inside a material they are different (see H and B inside and outside magnetic materials ). The SI unit of the H -field is the ampere per metre (A/m), and the CGS unit is the oersted (Oe). An instrument used to measure the local magnetic field is known as a magnetometer . Important classes of magnetometers include using induction magnetometers (or search-coil magnetometers) which measure only varying magnetic fields, rotating coil magnetometers , Hall effect magnetometers, NMR magnetometers , SQUID magnetometers , and fluxgate magnetometers . The magnetic fields of distant astronomical objects are measured through their effects on local charged particles. For instance, electrons spiraling around

8820-514: Is the vacuum permittivity and μ 0 the vacuum permeability . In practice, the subscripts "G" and "SI" are omitted, and the used convention (and unit) must be determined from context. 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

8967-502: Is the volume integral over the charge distribution: F = ∫ ( ρ E + J × B ) d V . {\displaystyle \mathbf {F} =\int \left(\rho \mathbf {E} +\mathbf {J} \times \mathbf {B} \right)\mathrm {d} V.} By eliminating ρ {\displaystyle \rho } and J {\displaystyle \mathbf {J} } , using Maxwell's equations , and manipulating using

SECTION 60

#1732790967515

9114-416: Is the density of free charge; P {\displaystyle \mathbf {P} } is the polarization density ; J f {\displaystyle \mathbf {J} _{f}} is the density of free current; and M {\displaystyle \mathbf {M} } is the magnetization density. In this way, the Lorentz force can explain the torque applied to a permanent magnet by

9261-400: Is the electric field and d ℓ is an infinitesimal vector element of the contour ∂Σ( t ) . NB: Both d ℓ and d A 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

9408-586: Is the force on a small piece of the charge distribution with charge d q {\displaystyle \mathrm {d} q} . If both sides of this equation are divided by the volume of this small piece of the charge distribution d V {\displaystyle \mathrm {d} V} , the result is: f = ρ ( E + v × B ) {\displaystyle \mathbf {f} =\rho \left(\mathbf {E} +\mathbf {v} \times \mathbf {B} \right)} where f {\displaystyle \mathbf {f} }

9555-407: 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

9702-475: Is valid for any wire position it implies that, F = q E ( r , t ) + q v × B ( r , t ) . {\displaystyle \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

9849-475: The Barnett effect or magnetization by rotation . Rotating the loop faster (in the same direction) increases the current and therefore the magnetic moment, for example. Specifying the force between two small magnets is quite complicated because it depends on the strength and orientation of both magnets and their distance and direction relative to each other. The force is particularly sensitive to rotations of

9996-1128: The Maxwell–Faraday equation : ∇ × E = − ∂ B ∂ t . {\displaystyle \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 . So we have, the Maxwell Faraday equation: ∮ ∂ Σ ( t ) d ℓ ⋅ E ( r ,   t ) = −   ∫ Σ ( t ) d A ⋅ d B ( r , t ) d t {\displaystyle \oint _{\partial \Sigma (t)}\mathrm {d} {\boldsymbol {\ell }}\cdot \mathbf {E} (\mathbf {r} ,\ t)=-\ \int _{\Sigma (t)}\mathrm {d} \mathbf {A} \cdot {\frac {\mathrm {d} \mathbf {B} (\mathbf {r} ,\,t)}{\mathrm {d} t}}} and

10143-650: The Navier–Stokes equations represent the conservation of momentum in the fluid, found from Newton's laws applied to the fluid, ∂ ∂ t ( ρ u ) + ∇ ⋅ ( ρ u ⊗ u + p I ) = ∇ ⋅ τ + ρ b {\displaystyle {\frac {\partial }{\partial t}}(\rho \mathbf {u} )+\nabla \cdot (\rho \mathbf {u} \otimes \mathbf {u} +p\mathbf {I} )=\nabla \cdot {\boldsymbol {\tau }}+\rho \mathbf {b} } if

10290-484: The Solar System , dealing with the force between each pair of bodies separately rapidly becomes computationally inconvenient. In the eighteenth century, a new quantity was devised to simplify the bookkeeping of all these gravitational forces. This quantity, the gravitational field , gave at each point in space the total gravitational acceleration which would be felt by a small object at that point. This did not change

10437-416: The cross product . The direction of force on the charge can be determined by a mnemonic known as the right-hand rule (see the figure). Using the right hand, pointing the thumb in the direction of the current, and the fingers in the direction of the magnetic field, the resulting force on the charge points outwards from the palm. The force on a negatively charged particle is in the opposite direction. If both

10584-548: The electromagnetic field was understood as a collection of two vector fields in space. Nowadays, one recognizes this as a single antisymmetric 2nd-rank tensor field in spacetime. Einstein's theory of gravity, called general relativity , is another example of a field theory. Here the principal field is the metric tensor , a symmetric 2nd-rank tensor field in spacetime . This replaces Newton's law of universal gravitation . Waves can be constructed as physical fields, due to their finite propagation speed and causal nature when

10731-435: 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

10878-583: The electroweak theory . In quantum chromodynamics, the color field lines are coupled at short distances by gluons , which are polarized by the field and line up with it. This effect increases within a short distance (around 1 fm from the vicinity of the quarks) making the color force increase within a short distance, confining the quarks within hadrons . As the field lines are pulled together tightly by gluons, they do not "bow" outwards as much as an electric field between electric charges. These three quantum field theories can all be derived as special cases of

11025-525: The spontaneous emission of a photon , the quantum of the electromagnetic field. This was soon followed by the realization (following the work of Pascual Jordan , Eugene Wigner , Werner Heisenberg , and Wolfgang Pauli ) that all particles, including electrons and protons , could be understood as the quanta of some quantum field, elevating fields to the status of the most fundamental objects in nature. That said, John Wheeler and Richard Feynman seriously considered Newton's pre-field concept of action at

11172-493: The temperature gradient is a vector field defined as ∇ T {\displaystyle \nabla T} . In thermal conduction , the temperature field appears in Fourier's law, where q is the heat flux field and k the thermal conductivity. Temperature and pressure gradients are also important for meteorology. It is now believed that quantum mechanics should underlie all physical phenomena, so that

11319-477: 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. Coulomb's law is only valid for point charges at rest. In fact, the electromagnetic force between two point charges depends not only on the distance but also on the relative velocity . For small relative velocities and very small accelerations, instead of

11466-580: The "magnetic field" written B and H . While both the best names for these fields and exact interpretation of what these fields represent has been the subject of long running debate, there is wide agreement about how the underlying physics work. Historically, the term "magnetic field" was reserved for H while using other terms for B , but many recent textbooks use the term "magnetic field" to describe B as well as or in place of H . There are many alternative names for both (see sidebars). The magnetic field vector B at any point can be defined as

11613-597: The "number" of field lines through a surface. These concepts can be quickly "translated" to their mathematical form. For example, the number of field lines through a given surface is the surface integral of the magnetic field. Various phenomena "display" magnetic field lines as though the field lines were physical phenomena. For example, iron filings placed in a magnetic field form lines that correspond to "field lines". Magnetic field "lines" are also visually displayed in polar auroras , in which plasma particle dipole interactions create visible streaks of light that line up with

11760-503: 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 ⋅ d A differential element will change because of the different angle between B and d A , 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

11907-791: The Coulomb force, the Weber force can be applied. The sum of the Weber forces of all charge carriers in a closed DC loop on a single test charge produces – regardless of the shape of the current loop – the Lorentz force. The interpretation of magnetism by means of a modified Coulomb law was first proposed by Carl Friedrich Gauss . In 1835, Gauss assumed that each segment of a DC loop contains an equal number of negative and positive point charges that move at different speeds. If Coulomb's law were completely correct, no force should act between any two short segments of such current loops. However, around 1825, André-Marie Ampère demonstrated experimentally that this

12054-709: The Faraday Law, ∮ ∂ Σ ( t ) d ℓ ⋅ F / q ( r ,   t ) = − d d t ∫ Σ ( t ) d A ⋅ B ( r ,   t ) . {\displaystyle \oint _{\partial \Sigma (t)}\mathrm {d} {\boldsymbol {\ell }}\cdot \mathbf {F} /q(\mathbf {r} ,\ t)=-{\frac {\mathrm {d} }{\mathrm {d} t}}\int _{\Sigma (t)}\mathrm {d} \mathbf {A} \cdot \mathbf {B} (\mathbf {r} ,\ t).} The two are equivalent if

12201-535: The Lorentz equation is from the theory of electrostatics , and says that a particle of charge q in an electric field E experiences an electric force: F electric = q E . {\displaystyle \mathbf {F} _{\text{electric}}=q\mathbf {E} .} The second term is the magnetic force: F magnetic = q ( v × B ) . {\displaystyle \mathbf {F} _{\text{magnetic}}=q(\mathbf {v} \times \mathbf {B} ).} Using

12348-400: The Lorentz force (per unit volume) is f = ∇ ⋅ σ − 1 c 2 ∂ S ∂ t {\displaystyle \mathbf {f} =\nabla \cdot {\boldsymbol {\sigma }}-{\dfrac {1}{c^{2}}}{\dfrac {\partial \mathbf {S} }{\partial t}}} where c {\displaystyle c}

12495-877: The Lorentz force in a material medium is J ⋅ E . {\displaystyle \mathbf {J} \cdot \mathbf {E} .} 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 f = ( ρ f − ∇ ⋅ P ) E + ( J f + ∇ × M + ∂ P ∂ t ) × B . {\displaystyle \mathbf {f} =\left(\rho _{f}-\nabla \cdot \mathbf {P} \right)\mathbf {E} +\left(\mathbf {J} _{f}+\nabla \times \mathbf {M} +{\frac {\partial \mathbf {P} }{\partial t}}\right)\times \mathbf {B} .} where: ρ f {\displaystyle \rho _{f}}

12642-482: 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

12789-887: The Maxwell Faraday equation, ∮ ∂ Σ ( t ) d ℓ ⋅ F / q ( r ,   t ) = ∮ ∂ Σ ( t ) d ℓ ⋅ E ( r ,   t ) + ∮ ∂ Σ ( t ) v × B ( r ,   t ) d ℓ {\displaystyle \oint _{\partial \Sigma (t)}\mathrm {d} {\boldsymbol {\ell }}\cdot \mathbf {F} /q(\mathbf {r} ,\ t)=\oint _{\partial \Sigma (t)}\mathrm {d} {\boldsymbol {\ell }}\cdot \mathbf {E} (\mathbf {r} ,\ t)+\oint _{\partial \Sigma (t)}\!\!\!\!\mathbf {v} \times \mathbf {B} (\mathbf {r} ,\ t)\,\mathrm {d} {\boldsymbol {\ell }}} since this

12936-406: 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. In many cases of practical interest, the motion in a magnetic field of an electrically charged particle (such as an electron or ion in

13083-522: The area of the loop and depends on the direction of the current using the right-hand rule. An ideal magnetic dipole is modeled as a real magnetic dipole whose area a has been reduced to zero and its current I increased to infinity such that the product m = Ia is finite. This model clarifies the connection between angular momentum and magnetic moment, which is the basis of the Einstein–de Haas effect rotation by magnetization and its inverse,

13230-469: The behavior of M . According to Newton's law of universal gravitation , F ( r ) is given by where r ^ {\displaystyle {\hat {\mathbf {r} }}} is a unit vector lying along the line joining M and m and pointing from M to m . Therefore, the gravitational field of M is The experimental observation that inertial mass and gravitational mass are equal to an unprecedented level of accuracy leads to

13377-498: 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 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

13524-502: The charge carriers in a material through the Hall effect . The Earth produces its own magnetic field , which shields the Earth's ozone layer from the solar wind and is important in navigation using a compass . The force on an electric charge depends on its location, speed, and direction; two vector fields are used to describe this force. The first is the electric field , which describes

13671-525: The components of the 3x3 infinitesimal strain and L i j k l {\displaystyle L_{ijkl}} is the elasticity tensor , a fourth-rank tensor with 81 components (usually 21 independent components). Assuming that the temperature T is an intensive quantity , i.e., a single-valued, continuous and differentiable function of three-dimensional space (a scalar field ), i.e., that T = T ( r ) {\displaystyle T=T(\mathbf {r} )} , then

13818-518: The contribution of the magnetic force. 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 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

13965-738: 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 F = q G ( E G + v c × B G ) , {\displaystyle \mathbf {F} =q_{\mathrm {G} }\left(\mathbf {E} _{\mathrm {G} }+{\frac {\mathbf {v} }{c}}\times \mathbf {B} _{\mathrm {G} }\right),} where c

14112-415: 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

14259-401: The definition of the cross product, the magnetic force can also be written as a scalar equation: F magnetic = q v B sin ⁡ ( θ ) {\displaystyle F_{\text{magnetic}}=qvB\sin(\theta )} where F magnetic , v , and B are the scalar magnitude of their respective vectors, and θ is the angle between the velocity of

14406-505: The density ρ , pressure p , deviatoric stress tensor τ of the fluid, as well as external body forces b , are all given. The flow velocity u is the vector field to solve for. Linear elasticity is defined in terms of constitutive equations between tensor fields, where σ i j {\displaystyle \sigma _{ij}} are the components of the 3x3 Cauchy stress tensor , ε i j {\displaystyle \varepsilon _{ij}}

14553-424: 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

14700-428: The direction of a line drawn from the south to the north pole of the magnet. Flipping a bar magnet is equivalent to rotating its m by 180 degrees. The magnetic field of larger magnets can be obtained by modeling them as a collection of a large number of small magnets called dipoles each having their own m . The magnetic field produced by the magnet then is the net magnetic field of these dipoles; any net force on

14847-431: The early stages, André-Marie Ampère and Charles-Augustin de Coulomb could manage with Newton-style laws that expressed the forces between pairs of electric charges or electric currents . However, it became much more natural to take the field approach and express these laws in terms of electric and magnetic fields ; in 1845 Michael Faraday became the first to coin the term "magnetic field". And Lord Kelvin provided

14994-652: The electric and magnetic fields are functions of the position and time. Therefore, explicitly, the Lorentz force can be written as: F ( r ( t ) , r ˙ ( t ) , t , q ) = q [ E ( r , t ) + r ˙ ( t ) × B ( r , t ) ] {\displaystyle \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

15141-414: 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

15288-408: The existence of magnetic monopoles, but so far, none have been observed. In the model developed by Ampere , the elementary magnetic dipole that makes up all magnets is a sufficiently small Amperian loop with current I and loop area A . The dipole moment of this loop is m = IA . These magnetic dipoles produce a magnetic B -field. The magnetic field of a magnetic dipole is depicted in

15435-410: 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

15582-411: The figure. From outside, the ideal magnetic dipole is identical to that of an ideal electric dipole of the same strength. Unlike the electric dipole, a magnetic dipole is properly modeled as a current loop having a current I and an area a . Such a current loop has a magnetic moment of m = I a , {\displaystyle m=Ia,} where the direction of m is perpendicular to

15729-522: The first. In this example, the magnetic field of the stationary magnet creates a magnetic torque on the magnet that is free to rotate. This magnetic torque τ tends to align a magnet's poles with the magnetic field lines. A compass, therefore, turns to align itself with Earth's magnetic field. In terms of the pole model, two equal and opposite magnetic charges experiencing the same H also experience equal and opposite forces. Since these equal and opposite forces are in different locations, this produces

15876-549: The force acting on a stationary charge and gives the component of the force that is independent of motion. The magnetic field, in contrast, describes the component of the force that is proportional to both the speed and direction of charged particles. The field is defined by the Lorentz force law and is, at each instant, perpendicular to both the motion of the charge and the force it experiences. There are two different, but closely related vector fields which are both sometimes called

16023-517: The force and torques between two magnets as due to magnetic poles repelling or attracting each other in the same manner as the Coulomb force between electric charges. At the microscopic level, this model contradicts the experimental evidence, and the pole model of magnetism is no longer the typical way to introduce the concept. However, it is still sometimes used as a macroscopic model for ferromagnetism due to its mathematical simplicity. In this model,

16170-406: The force on a small magnet having a magnetic moment m due to a magnetic field B is: F = ∇ ( m ⋅ B ) , {\displaystyle \mathbf {F} ={\boldsymbol {\nabla }}\left(\mathbf {m} \cdot \mathbf {B} \right),} where the gradient ∇ is the change of the quantity m · B per unit distance and the direction

16317-472: The force on the particle when its velocity is v ; repeat with v in some other direction. Now find a B that makes the Lorentz force law fit all these results—that is the magnetic field at the place in question. The B field can also be defined by the torque on a magnetic dipole, m . τ = m × B {\displaystyle {\boldsymbol {\tau }}=\mathbf {m} \times \mathbf {B} } The SI unit of B

16464-538: 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

16611-414: The four fundamental forces of nature. Magnetic fields are used throughout modern technology, particularly in electrical engineering and electromechanics . Rotating magnetic fields are used in both electric motors and generators . The interaction of magnetic fields in electric devices such as transformers is conceptualized and investigated as magnetic circuits . Magnetic forces give information about

16758-433: The functional form : F = q ( E + v × B ) {\displaystyle \mathbf {F} =q(\mathbf {E} +\mathbf {v} \times \mathbf {B} )} 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

16905-426: The identity that gravitational field strength is identical to the acceleration experienced by a particle. This is the starting point of the equivalence principle , which leads to general relativity . Because the gravitational force F is conservative , the gravitational field g can be rewritten in terms of the gradient of a scalar function, the gravitational potential Φ( r ): Michael Faraday first realized

17052-406: The importance of a field as a physical quantity, during his investigations into magnetism . He realized that electric and magnetic fields are not only fields of force which dictate the motion of particles, but also have an independent physical reality because they carry energy. These ideas eventually led to the creation, by James Clerk Maxwell , of the first unified field theory in physics with

17199-524: The induced electromotive force (EMF) in the wire is: E = − d Φ B d t {\displaystyle {\mathcal {E}}=-{\frac {\mathrm {d} \Phi _{B}}{\mathrm {d} t}}} where Φ B = ∫ Σ ( t ) d A ⋅ B ( r , t ) {\displaystyle \Phi _{B}=\int _{\Sigma (t)}\mathrm {d} \mathbf {A} \cdot \mathbf {B} (\mathbf {r} ,t)}

17346-475: The internal surface of the wire. The EMF around the closed path ∂Σ( t ) is given by: E = ∮ ∂ Σ ( t ) d ℓ ⋅ F / q {\displaystyle {\mathcal {E}}=\oint _{\partial \Sigma (t)}\!\!\mathrm {d} {\boldsymbol {\ell }}\cdot \mathbf {F} /q} where E = F / q {\displaystyle \mathbf {E} =\mathbf {F} /q}

17493-403: The introduction of equations for the electromagnetic field . The modern versions of these equations are called Maxwell's equations . A charged test particle with charge q experiences a force F based solely on its charge. We can similarly describe the electric field E so that F = q E . Using this and Coulomb's law tells us that the electric field due to a single charged particle

17640-432: The local density of field lines can be made proportional to its strength. Magnetic field lines are like streamlines in fluid flow , in that they represent a continuous distribution, and a different resolution would show more or fewer lines. An advantage of using magnetic field lines as a representation is that many laws of magnetism (and electromagnetism) can be stated completely and concisely using simple concepts such as

17787-763: The local direction of Earth's magnetic field. Field lines can be used as a qualitative tool to visualize magnetic forces. In ferromagnetic substances like iron and in plasmas, magnetic forces can be understood by imagining that the field lines exert a tension , (like a rubber band) along their length, and a pressure perpendicular to their length on neighboring field lines. "Unlike" poles of magnets attract because they are linked by many field lines; "like" poles repel because their field lines do not meet, but run parallel, pushing on each other. Permanent magnets are objects that produce their own persistent magnetic fields. They are made of ferromagnetic materials, such as iron and nickel , that have been magnetized, and they have both

17934-465: The magnet is a result of adding up the forces on the individual dipoles. There are two simplified models for the nature of these dipoles: the magnetic pole model and the Amperian loop model . These two models produce two different magnetic fields, H and B . Outside a material, though, the two are identical (to a multiplicative constant) so that in many cases the distinction can be ignored. This

18081-537: 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: d F = d q ( E + v × B ) {\displaystyle \mathrm {d} \mathbf {F} =\mathrm {d} q\left(\mathbf {E} +\mathbf {v} \times \mathbf {B} \right)} where d F {\displaystyle \mathrm {d} \mathbf {F} }

18228-424: 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

18375-419: The magnetic field. The density of the associated power is ( J f + ∇ × M + ∂ P ∂ t ) ⋅ E . {\displaystyle \left(\mathbf {J} _{f}+\nabla \times \mathbf {M} +{\frac {\partial \mathbf {P} }{\partial t}}\right)\cdot \mathbf {E} .} The above-mentioned formulae use

18522-501: 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

18669-412: The magnets due to magnetic torque. The force on each magnet depends on its magnetic moment and the magnetic field of the other. To understand the force between magnets, it is useful to examine the magnetic pole model given above. In this model, the H -field of one magnet pushes and pulls on both poles of a second magnet. If this H -field is the same at both poles of the second magnet then there

18816-486: The modern framework of the quantum field theory , even without referring to a test particle, a field occupies space, contains energy, and its presence precludes a classical "true vacuum". This has led physicists to consider electromagnetic fields to be a physical entity, making the field concept a supporting paradigm of the edifice of modern physics. Richard Feynman said, "The fact that the electromagnetic field can possess momentum and energy makes it very real, and [...]

18963-449: The motion of electrons within an atom are connected to those electrons' orbital magnetic dipole moment , and these orbital moments do contribute to the magnetism seen at the macroscopic level. However, the motion of electrons is not classical, and the spin magnetic moment of electrons (which is not explained by either model) is also a significant contribution to the total moment of magnets. Historically, early physics textbooks would model

19110-444: The north pole (whether inside the magnet or out) while near the south pole all H -field lines point toward the south pole (whether inside the magnet or out). Too, a north pole feels a force in the direction of the H -field while the force on the south pole is opposite to the H -field. In the magnetic pole model, the elementary magnetic dipole m is formed by two opposite magnetic poles of pole strength q m separated by

19257-440: The particle and the magnetic field. The vector B is defined as the vector field necessary to make the Lorentz force law correctly describe the motion of a charged particle. In other words, [T]he command, "Measure the direction and magnitude of the vector B at such and such a place," calls for the following operations: Take a particle of known charge q . Measure the force on q at rest, to determine E . Then measure

19404-402: The past. Maxwell, at first, did not adopt the modern concept of a field as a fundamental quantity that could independently exist. Instead, he supposed that the electromagnetic field expressed the deformation of some underlying medium—the luminiferous aether —much like the tension in a rubber membrane. If that were the case, the observed velocity of the electromagnetic waves should depend upon

19551-406: The physics in any way: it did not matter if all the gravitational forces on an object were calculated individually and then added together, or if all the contributions were first added together as a gravitational field and then applied to an object. The development of the independent concept of a field truly began in the nineteenth century with the development of the theory of electromagnetism . In

19698-460: The qualitative information included above. There is no torque on a magnet if m is in the same direction as the magnetic field, since the cross product is zero for two vectors that are in the same direction. Further, all other orientations feel a torque that twists them toward the direction of magnetic field. Currents of electric charges both generate a magnetic field and feel a force due to magnetic B-fields. Physical field In science ,

19845-426: The same for all observers. By doing away with the need for a background medium, this development opened the way for physicists to start thinking about fields as truly independent entities. In the late 1920s, the new rules of quantum mechanics were first applied to the electromagnetic field. In 1927, Paul Dirac used quantum fields to successfully explain how the decay of an atom to a lower quantum state led to

19992-621: The same formal expression, but ℓ should now be understood as the vector connecting the end points of the curved wire with direction from starting to end point of conventional current. Usually, there will also be a net torque . If, in addition, the magnetic field is inhomogeneous, the net force on a stationary rigid wire carrying a steady current I is given by integration along the wire, F = I ∫ d ℓ × B . {\displaystyle \mathbf {F} =I\int \mathrm {d} {\boldsymbol {\ell }}\times \mathbf {B} .} One application of this

20139-431: The simplest physical fields are vector force fields. Historically, the first time that fields were taken seriously was with Faraday's lines of force when describing the electric field . The gravitational field was then similarly described. A classical field theory describing gravity is Newtonian gravitation , which describes the gravitational force as a mutual interaction between two masses . Any body with mass M

20286-589: The so-called standard model of particle physics . General relativity , the Einsteinian field theory of gravity, has yet to be successfully quantized. However an extension, thermal field theory , deals with quantum field theory at finite temperatures , something seldom considered in quantum field theory. In BRST theory one deals with odd fields, e.g. Faddeev–Popov ghosts . There are different descriptions of odd classical fields both on graded manifolds and supermanifolds . As above with classical fields, it

20433-453: The speed and the charge are reversed then the direction of the force remains the same. For that reason a magnetic field measurement (by itself) cannot distinguish whether there is a positive charge moving to the right or a negative charge moving to the left. (Both of these cases produce the same current.) On the other hand, a magnetic field combined with an electric field can distinguish between these, see Hall effect below. The first term in

20580-541: The term tensor , derived from the Latin word for stretch), complex fluid flows or anisotropic diffusion , which are framed as matrix-tensor PDEs, and then require matrices or tensor fields, hence matrix or tensor calculus . The scalars (and hence the vectors, matrices and tensors) can be real or complex as both are fields in the abstract-algebraic/ ring-theoretic sense. In a general setting, classical fields are described by sections of fiber bundles and their dynamics

20727-686: The theorems of vector calculus , this form of the equation can be used to derive the Maxwell stress tensor σ {\displaystyle {\boldsymbol {\sigma }}} , in turn this can be combined with the Poynting vector S {\displaystyle \mathbf {S} } to obtain the electromagnetic stress–energy tensor T used in general relativity . In terms of σ {\displaystyle {\boldsymbol {\sigma }}} and S {\displaystyle \mathbf {S} } , another way to write

20874-546: 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

21021-439: The vector that, when used in the Lorentz force law , correctly predicts the force on a charged particle at that point: F = q E + q ( v × B ) {\displaystyle \mathbf {F} =q\mathbf {E} +q(\mathbf {v} \times \mathbf {B} )} Here F is the force on the particle, q is the particle's electric charge , v , is the particle's velocity , and × denotes

21168-459: The velocity of the observer with respect to the aether. Despite much effort, no experimental evidence of such an effect was ever found; the situation was resolved by the introduction of the special theory of relativity by Albert Einstein in 1905. This theory changed the way the viewpoints of moving observers were related to each other. They became related to each other in such a way that velocity of electromagnetic waves in Maxwell's theory would be

21315-667: The wave equation and fluid dynamics ; temperature/concentration fields for the heat / diffusion equations . Outside of physics proper (e.g., radiometry and computer graphics), there are even light fields . All these previous examples are scalar fields . Similarly for vectors, there are vector PDEs for displacement, velocity and vorticity fields in (applied mathematical) fluid dynamics, but vector calculus may now be needed in addition, being calculus for vector fields (as are these three quantities, and those for vector PDEs in general). More generally problems in continuum mechanics may involve for example, directional elasticity (from which comes

21462-971: The wire is not moving. Using the Leibniz integral rule and that div B = 0 , results in, ∮ ∂ Σ ( t ) d ℓ ⋅ F / q ( r , t ) = − ∫ Σ ( t ) d A ⋅ ∂ ∂ t B ( r , t ) + ∮ ∂ Σ ( t ) v × B d ℓ {\displaystyle \oint _{\partial \Sigma (t)}\mathrm {d} {\boldsymbol {\ell }}\cdot \mathbf {F} /q(\mathbf {r} ,t)=-\int _{\Sigma (t)}\mathrm {d} \mathbf {A} \cdot {\frac {\partial }{\partial t}}\mathbf {B} (\mathbf {r} ,t)+\oint _{\partial \Sigma (t)}\!\!\!\!\mathbf {v} \times \mathbf {B} \,\mathrm {d} {\boldsymbol {\ell }}} and using

21609-492: 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 . Given a loop of wire in a magnetic field , Faraday's law of induction states

#514485