Lorentz force

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Lorentz_force 

 

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

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 q moving with a velocity v in an electric field E and a magnetic field B experiences a force of

$${\displaystyle \mathbf {F} =q\,\mathbf {E} +q\,\mathbf {v} \times \mathbf {B} }$$

(in SI units). It says that the electromagnetic force on a charge q is a combination of a force in the direction of the electric field E proportional to the magnitude of the field and the quantity of charge, and a force at right angles to 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), 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

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

 

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

 

Charged particles experiencing the Lorentz force.

In many textbook treatments of classical electromagnetism, the Lorentz force law is 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 the functional form:

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

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 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).

Equation

Charged particle

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

The 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):

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

where × is the vector cross product (all boldface quantities are vectors). In terms of Cartesian components, we have:

$${\displaystyle F_{x}=q\left(E_{x}+v_{y}B_{z}-v_{z}B_{y}\right),}$$

$${\displaystyle F_{y}=q\left(E_{y}+v_{z}B_{x}-v_{x}B_{z}\right),}$$

$${\displaystyle F_{z}=q\left(E_{z}+v_{x}B_{y}-v_{y}B_{x}\right).}$$

In general, the electric and magnetic fields are functions of the position and time. Therefore, explicitly, the Lorentz force can be written as:

$${\displaystyle \mathbf {F} \left(\mathbf {r} ,{\dot {\mathbf {r} }},t,q\right)=q\left[\mathbf {E} (\mathbf {r} ,t)+{\dot {\mathbf {r} }}\times \mathbf {B} (\mathbf {r} ,t)\right]}$$

in which r is the position vector of the charged particle, t is time, and the overdot is a time derivative.

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 qE 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

$${\displaystyle \mathbf {v} \cdot \mathbf {F} =q\,\mathbf {v} \cdot \mathbf {E} .}$$

Notice that the magnetic field does not contribute to the power because the magnetic force is always perpendicular to the velocity of the particle.

Continuous charge distribution

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

For a continuous charge distribution in motion, the Lorentz force equation becomes:

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

where ${\displaystyle \mathrm {d} \mathbf {F} }$ is the force on a small piece of the charge distribution with charge ${\displaystyle \mathrm {d} q}$. If both sides of this equation are divided by the volume of this small piece of the charge distribution ${\displaystyle \mathrm {d} V}$, the result is:

$${\displaystyle \mathbf {f} =\rho \left(\mathbf {E} +\mathbf {v} \times \mathbf {B} \right)}$$

where ${\displaystyle \mathbf {f} }$ is the force density (force per unit volume) and ${\displaystyle \rho }$ is the charge density (charge per unit volume). Next, the current density corresponding to the motion of the charge continuum is

$${\displaystyle \mathbf {J} =\rho \mathbf {v} }$$

so the continuous analogue to the equation is

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

The total force is the volume integral over the charge distribution:

$${\displaystyle \mathbf {F} =\iiint \left(\rho \mathbf {E} +\mathbf {J} \times \mathbf {B} \right)\mathrm {d} V.}$$

By eliminating ${\displaystyle \rho }$ and ${\displaystyle \mathbf {J} }$, 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 ${\displaystyle {\boldsymbol {\sigma }}}$, in turn this can be combined with the Poynting vector ${\displaystyle \mathbf {S} }$ to obtain the electromagnetic stress–energy tensor T used in general relativity.

In terms of ${\displaystyle {\boldsymbol {\sigma }}}$ and ${\displaystyle \mathbf {S} }$, another way to write the Lorentz force (per unit volume) is

$${\displaystyle \mathbf {f} =\nabla \cdot {\boldsymbol {\sigma }}-{\dfrac {1}{c^{2}}}{\dfrac {\partial \mathbf {S} }{\partial t}}}$$

where ${\displaystyle c}$ 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

$${\displaystyle \mathbf {J} \cdot \mathbf {E} .}$$

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

$${\displaystyle \mathbf {f} =\left(\rho _{f}-\nabla \cdot \mathbf {P} \right)\mathbf {E} +\left(\mathbf {J} _{f}+\nabla \times \mathbf {M} +{\frac {\partial \mathbf {P} }{\partial t}}\right)\times \mathbf {B} .}$$

where: ${\displaystyle \rho _{f}}$ is the density of free charge; ${\displaystyle \mathbf {P} }$ is the polarization density; ${\displaystyle \mathbf {J} _{f}}$ is the density of free current; and ${\displaystyle \mathbf {M} }$ 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

$${\displaystyle \left(\mathbf {J} _{f}+\nabla \times \mathbf {M} +{\frac {\partial \mathbf {P} }{\partial t}}\right)\cdot \mathbf {E} .}$$

Equation in cgs units

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

$${\displaystyle \mathbf {F} =q_{\mathrm {cgs} }\left(\mathbf {E} _{\mathrm {cgs} }+{\frac {\mathbf {v} }{c}}\times \mathbf {B} _{\mathrm {cgs} }\right).}$$

where c is the speed of light. Although this equation looks slightly different, it is completely equivalent, since one has the following relations:

$${\displaystyle q_{\mathrm {cgs} }={\frac {q_{\mathrm {SI} }}{\sqrt {4\pi \varepsilon _{0}}}},\quad \mathbf {E} _{\mathrm {cgs} }={\sqrt {4\pi \varepsilon _{0}}}\,\mathbf {E} _{\mathrm {SI} },\quad \mathbf {B} _{\mathrm {cgs} }={\sqrt {4\pi /\mu _{0}}}\,{\mathbf {B} _{\mathrm {SI} }},\quad c={\frac {1}{\sqrt {\varepsilon _{0}\mu _{0}}}}.}$$

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

History

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

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

The modern concept of electric and magnetic fields first arose in the theories of Michael Faraday, particularly his idea of lines of force, later to be given full mathematical description by Lord Kelvin and James Clerk Maxwell. From a modern perspective it is possible to identify in Maxwell's 1865 formulation of his field equations a form of the Lorentz force equation in relation to electric currents, although in the time of Maxwell it was not evident how his equations related to the forces on moving charged objects. J. J. Thomson was the first to attempt to derive from Maxwell's field equations the electromagnetic forces on a moving charged object in terms of the object's properties and external fields. Interested in determining the electromagnetic behavior of the charged particles in cathode rays, Thomson published a paper in 1881 wherein he gave the force on the particles due to an external magnetic field as

$${\displaystyle \mathbf {F} ={\frac {q}{2}}\mathbf {v} \times \mathbf {B} .}$$

Thomson derived the correct basic form of the formula, but, because of some miscalculations and an incomplete description of the displacement current, included an incorrect scale-factor of a half in front of the formula. Oliver Heaviside invented the modern vector notation and applied it to Maxwell's field equations; he also (in 1885 and 1889) had fixed the mistakes of Thomson's derivation and arrived at the correct form of the magnetic force on a moving charged object. Finally, in 1895, Hendrik Lorentz  derived the modern form of the formula for the electromagnetic force which includes the contributions to the total force from both the electric and the magnetic fields. Lorentz began by abandoning the Maxwellian descriptions of the ether and conduction. Instead, Lorentz made a distinction between matter and the luminiferous aether and sought to apply the Maxwell equations at a microscopic scale. Using Heaviside's version of the Maxwell equations for a stationary ether and applying Lagrangian mechanics (see below), Lorentz arrived at the correct and complete form of the force law that now bears his name.

Trajectories of particles due to the Lorentz force

Main article: Guiding center

 

Charged particle drifts in a homogeneous magnetic field. (A) No disturbing force (B) With an electric field, E (C) With an independent force, F (e.g. gravity) (D) In an inhomogeneous magnetic field, grad H

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.

Significance of the Lorentz force

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

In real materials the Lorentz force is inadequate to describe the 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).

Force on a current-carrying wire

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

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:

$${\displaystyle \mathbf {F} =I{\boldsymbol {\ell }}\times \mathbf {B} }$$

where ℓ 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 I.

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 ${\displaystyle \mathrm {d} {\boldsymbol {\ell }}}$, then adding up all these forces by integration. Formally, the net force on a stationary, rigid wire carrying a steady current I is

$${\displaystyle \mathbf {F} =I\int \mathrm {d} {\boldsymbol {\ell }}\times \mathbf {B} }$$

This is the net force. In addition, there will usually be torque, plus other effects if the wire is not perfectly rigid.

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

EMF

The magnetic force (qv × B) component of the Lorentz force is responsible for motional electromotive force (or motional EMF), the phenomenon underlying many electrical generators. When a conductor is moved through a magnetic field, the magnetic 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 (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).

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

Lorentz force -image on a wall in Leiden

 

Main article: 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:

$${\displaystyle {\mathcal {E}}=-{\frac {\mathrm {d} \Phi _{B}}{\mathrm {d} t}}}$$

where

$${\displaystyle \Phi _{B}=\iint _{\Sigma (t)}\mathrm {d} \mathbf {A} \cdot \mathbf {B} (\mathbf {r} ,t)}$$

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 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 the internal surface of the wire. The EMF around the closed path ∂Σ(t) is given by:

$${\displaystyle {\mathcal {E}}=\oint _{\partial \Sigma (t)}\mathrm {d} {\boldsymbol {\ell }}\cdot \mathbf {F} /q}$$

where

$${\displaystyle \mathbf {E} =\mathbf {F} /q}$$

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:

$${\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}\,.}$$

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

So we have, the Maxwell Faraday equation:

$${\displaystyle \oint _{\partial \Sigma (t)}\mathrm {d} {\boldsymbol {\ell }}\cdot \mathbf {E} (\mathbf {r} ,\ t)=-\ \iint _{\Sigma (t)}\mathrm {d} \mathbf {A} \cdot {{\mathrm {d} \mathbf {B} (\mathbf {r} ,\ t)} \over \mathrm {d} t}}$$

and the Faraday Law,

$${\displaystyle \oint _{\partial \Sigma (t)}\mathrm {d} {\boldsymbol {\ell }}\cdot \mathbf {F} /q(\mathbf {r} ,\ t)=-{\frac {\mathrm {d} }{\mathrm {d} t}}\iint _{\Sigma (t)}\mathrm {d} \mathbf {A} \cdot \mathbf {B} (\mathbf {r} ,\ t).}$$

The two are equivalent if the wire is not moving. Using the Leibniz integral rule and that div B = 0, results in,

$${\displaystyle \oint _{\partial \Sigma (t)}\mathrm {d} {\boldsymbol {\ell }}\cdot \mathbf {F} /q(\mathbf {r} ,t)=-\iint _{\Sigma (t)}\mathrm {d} \mathbf {A} \cdot {\frac {\partial }{\partial t}}\mathbf {B} (\mathbf {r} ,t)+\oint _{\partial \Sigma (t)}\!\!\!\!\mathbf {v} \times \mathbf {B} \,\mathrm {d} {\boldsymbol {\ell }}}$$

and using the Maxwell Faraday equation,

$${\displaystyle \oint _{\partial \Sigma (t)}\mathrm {d} {\boldsymbol {\ell }}\cdot \mathbf {F} /q(\mathbf {r} ,\ t)=\oint _{\partial \Sigma (t)}\mathrm {d} {\boldsymbol {\ell }}\cdot \mathbf {E} (\mathbf {r} ,\ t)+\oint _{\partial \Sigma (t)}\!\!\!\!\mathbf {v} \times \mathbf {B} (\mathbf {r} ,\ t)\,\mathrm {d} {\boldsymbol {\ell }}}$$

since this is valid for any wire position it implies that,

$${\displaystyle \mathbf {F} =q\,\mathbf {E} (\mathbf {r} ,\ t)+q\,\mathbf {v} \times \mathbf {B} (\mathbf {r} ,\ t).}$$

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 the magnetic field is constant in time or changing. However, there are cases where Faraday's law is either inadequate or difficult to use, and application of the underlying Lorentz force law is necessary. See inapplicability of Faraday's law.

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

Note that the Maxwell Faraday's equation implies that the Electric Field E is non conservative when the Magnetic Field B varies in time, and is not expressible as the gradient of a scalar field, and not subject to the gradient theorem since its rotational is not zero.

Lorentz force in terms of potentials

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

The E and B fields can be replaced by the magnetic vector potential A and (scalar) electrostatic potential ϕ by

$${\displaystyle \mathbf {E} =-\nabla \phi -{\frac {\partial \mathbf {A} }{\partial t}}}$$

$${\displaystyle \mathbf {B} =\nabla \times \mathbf {A} }$$

where ∇ is the gradient, ∇⋅ is the divergence, and ∇× is the curl.

The force becomes

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

Using an identity for the triple product this can be rewritten as,

$${\displaystyle \mathbf {F} =q\left[-\nabla \phi -{\frac {\partial \mathbf {A} }{\partial t}}+\nabla \left(\mathbf {v} \cdot \mathbf {A} \right)-\left(\mathbf {v} \cdot \nabla \right)\mathbf {A} \right],}$$

(Notice that the coordinates and the velocity components should be treated as independent variables, so the del operator acts only on ${\displaystyle \mathbf {A} }$, not on ${\displaystyle \mathbf {v} }$; thus, there is no need of using Feynman's subscript notation in the equation above). Using the chain rule, the total derivative of ${\displaystyle \mathbf {A} }$ is:

$${\displaystyle {\frac {\mathrm {d} \mathbf {A} }{\mathrm {d} t}}={\frac {\partial \mathbf {A} }{\partial t}}+(\mathbf {v} \cdot \nabla )\mathbf {A} }$$

so that the above expression becomes:

$${\displaystyle \mathbf {F} =q\left[-\nabla (\phi -\mathbf {v} \cdot \mathbf {A} )-{\frac {\mathrm {d} \mathbf {A} }{\mathrm {d} t}}\right].}$$

With v = ẋ, we can put the equation into the convenient Euler–Lagrange form

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

where

$${\displaystyle \nabla _{\mathbf {x} }={\hat {x}}{\dfrac {\partial }{\partial x}}+{\hat {y}}{\dfrac {\partial }{\partial y}}+{\hat {z}}{\dfrac {\partial }{\partial z}}}$$

and

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

Lorentz force and analytical mechanics

See also: Momentum

The Lagrangian for a charged particle of mass m and charge q in an electromagnetic field equivalently describes the dynamics of the particle in terms of its energy, rather than the force exerted on it. The classical expression is given by:

$${\displaystyle L={\frac {m}{2}}\mathbf {\dot {r}} \cdot \mathbf {\dot {r}} +q\mathbf {A} \cdot \mathbf {\dot {r}} -q\phi }$$

where A and ϕ are the potential fields as above. The quantity ${\displaystyle V=q(\phi -\mathbf {A} \cdot \mathbf {\dot {r}} )}$ 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.

Derivation of Lorentz force from classical Lagrangian (SI units)

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

The total potential energy is then:

$${\displaystyle V=q\phi -q\mathbf {A} \cdot \mathbf {\dot {r}} }$$

and the kinetic energy is:

$${\displaystyle T={\frac {m}{2}}\mathbf {\dot {r}} \cdot \mathbf {\dot {r}} }$$

hence the Lagrangian:

$${\displaystyle L=T-V={\frac {m}{2}}\mathbf {\dot {r}} \cdot \mathbf {\dot {r}} +q\mathbf {A} \cdot \mathbf {\dot {r}} -q\phi }$$

$${\displaystyle L={\frac {m}{2}}({\dot {x}}^{2}+{\dot {y}}^{2}+{\dot {z}}^{2})+q({\dot {x}}A_{x}+{\dot {y}}A_{y}+{\dot {z}}A_{z})-q\phi }$$

Lagrange's equations are

$${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial {\dot {x}}}}={\frac {\partial L}{\partial x}}}$$

(same for y and z). So calculating the partial derivatives:

$${\displaystyle {\begin{aligned}{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial {\dot {x}}}}&=m{\ddot {x}}+q{\frac {\mathrm {d} A_{x}}{\mathrm {d} t}}\\&=m{\ddot {x}}+{\frac {q}{\mathrm {d} t}}\left({\frac {\partial A_{x}}{\partial t}}dt+{\frac {\partial A_{x}}{\partial x}}dx+{\frac {\partial A_{x}}{\partial y}}dy+{\frac {\partial A_{x}}{\partial z}}dz\right)\\&=m{\ddot {x}}+q\left({\frac {\partial A_{x}}{\partial t}}+{\frac {\partial A_{x}}{\partial x}}{\dot {x}}+{\frac {\partial A_{x}}{\partial y}}{\dot {y}}+{\frac {\partial A_{x}}{\partial z}}{\dot {z}}\right)\\\end{aligned}}}$$

$${\displaystyle {\frac {\partial L}{\partial x}}=-q{\frac {\partial \phi }{\partial x}}+q\left({\frac {\partial A_{x}}{\partial x}}{\dot {x}}+{\frac {\partial A_{y}}{\partial x}}{\dot {y}}+{\frac {\partial A_{z}}{\partial x}}{\dot {z}}\right)}$$

equating and simplifying:

$${\displaystyle m{\ddot {x}}+q\left({\frac {\partial A_{x}}{\partial t}}+{\frac {\partial A_{x}}{\partial x}}{\dot {x}}+{\frac {\partial A_{x}}{\partial y}}{\dot {y}}+{\frac {\partial A_{x}}{\partial z}}{\dot {z}}\right)=-q{\frac {\partial \phi }{\partial x}}+q\left({\frac {\partial A_{x}}{\partial x}}{\dot {x}}+{\frac {\partial A_{y}}{\partial x}}{\dot {y}}+{\frac {\partial A_{z}}{\partial x}}{\dot {z}}\right)}$$

$${\displaystyle {\begin{aligned}F_{x}&=-q\left({\frac {\partial \phi }{\partial x}}+{\frac {\partial A_{x}}{\partial t}}\right)+q\left[{\dot {y}}\left({\frac {\partial A_{y}}{\partial x}}-{\frac {\partial A_{x}}{\partial y}}\right)+{\dot {z}}\left({\frac {\partial A_{z}}{\partial x}}-{\frac {\partial A_{x}}{\partial z}}\right)\right]\\&=qE_{x}+q[{\dot {y}}(\nabla \times \mathbf {A} )_{z}-{\dot {z}}(\nabla \times \mathbf {A} )_{y}]\\&=qE_{x}+q[\mathbf {\dot {r}} \times (\nabla \times \mathbf {A} )]_{x}\\&=qE_{x}+q(\mathbf {\dot {r}} \times \mathbf {B} )_{x}\end{aligned}}}$$

and similarly for the y and z directions. Hence the force equation is:

$${\displaystyle \mathbf {F} =q(\mathbf {E} +\mathbf {\dot {r}} \times \mathbf {B} )}$$

The potential energy depends on the velocity of the particle, so the force is velocity dependent, so it is not conservative.

The relativistic Lagrangian is

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

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

Derivation of Lorentz force from relativistic Lagrangian (SI units)

The equations of motion derived by extremizing the action (see matrix calculus for the notation):

$${\displaystyle {\frac {\mathrm {d} \mathbf {P} }{\mathrm {d} t}}={\frac {\partial L}{\partial \mathbf {r} }}=q{\partial \mathbf {A} \over \partial \mathbf {r} }\cdot {\dot {\mathbf {r} }}-q{\partial \phi \over \partial \mathbf {r} }}$$

$${\displaystyle \mathbf {P} -q\mathbf {A} ={\frac {m{\dot {\mathbf {r} }}}{\sqrt {1-\left({\frac {\dot {\mathbf {r} }}{c}}\right)^{2}}}}}$$

are the same as Hamilton's equations of motion:

$${\displaystyle {\frac {\mathrm {d} \mathbf {r} }{\mathrm {d} t}}={\frac {\partial }{\partial \mathbf {p} }}\left({\sqrt {(\mathbf {P} -q\mathbf {A} )^{2}+(mc^{2})^{2}}}+q\phi \right)}$$

$${\displaystyle {\frac {\mathrm {d} \mathbf {p} }{\mathrm {d} t}}=-{\partial \over \partial \mathbf {r} }\left({\sqrt {(\mathbf {P} -q\mathbf {A} )^{2}+(mc^{2})^{2}}}+q\phi \right)}$$

both are equivalent to the noncanonical form:

$${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\left({m{\dot {\mathbf {r} }} \over {\sqrt {1-\left({\frac {\dot {\mathbf {r} }}{c}}\right)^{2}}}}\right)=q\left(\mathbf {E} +{\dot {\mathbf {r} }}\times \mathbf {B} \right).}$$

This formula is the Lorentz force, representing the rate at which the EM field adds relativistic momentum to the particle.

Relativistic form of the Lorentz force

Covariant form of the Lorentz force

Field tensor

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

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

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

where p^α is the four-momentum, defined as

$${\displaystyle p^{\alpha }=\left(p_{0},p_{1},p_{2},p_{3}\right)=\left(\gamma mc,p_{x},p_{y},p_{z}\right),}$$

τ the proper time of the particle, F^αβ the contravariant electromagnetic tensor

$${\displaystyle F^{\alpha \beta }={\begin{pmatrix}0&-E_{x}/c&-E_{y}/c&-E_{z}/c\\E_{x}/c&0&-B_{z}&B_{y}\\E_{y}/c&B_{z}&0&-B_{x}\\E_{z}/c&-B_{y}&B_{x}&0\end{pmatrix}}}$$

and U is the covariant 4-velocity of the particle, defined as:

$${\displaystyle U_{\beta }=\left(U_{0},U_{1},U_{2},U_{3}\right)=\gamma \left(c,-v_{x},-v_{y},-v_{z}\right),}$$

in which

$${\displaystyle \gamma (v)={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}={\frac {1}{\sqrt {1-{\frac {v_{x}^{2}+v_{y}^{2}+v_{z}^{2}}{c^{2}}}}}}}$$

is the Lorentz factor.

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

$${\displaystyle F'^{\mu \nu }={\Lambda ^{\mu }}_{\alpha }{\Lambda ^{\nu }}_{\beta }F^{\alpha \beta }\,,}$$

where Λ^μ_α is the Lorentz transformation tensor.

Translation to vector notation

The α = 1 component (x-component) of the force is

$${\displaystyle {\frac {\mathrm {d} p^{1}}{\mathrm {d} \tau }}=qU_{\beta }F^{1\beta }=q\left(U_{0}F^{10}+U_{1}F^{11}+U_{2}F^{12}+U_{3}F^{13}\right).}$$

Substituting the components of the covariant electromagnetic tensor F yields

$${\displaystyle {\frac {\mathrm {d} p^{1}}{\mathrm {d} \tau }}=q\left[U_{0}\left({\frac {E_{x}}{c}}\right)+U_{2}(-B_{z})+U_{3}(B_{y})\right].}$$

Using the components of covariant four-velocity yields

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

The calculation for α = 2, 3 (force components in the y and z directions) yields similar results, so collecting the 3 equations into one:

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

and since differentials in coordinate time dt and proper time dτ are related by the Lorentz factor,

$${\displaystyle dt=\gamma (v)\,d\tau ,}$$

so we arrive at

$${\displaystyle {\frac {\mathrm {d} \mathbf {p} }{\mathrm {d} t}}=q\left(\mathbf {E} +\mathbf {v} \times \mathbf {B} \right).}$$

This is precisely the Lorentz force law, however, it is important to note that p is the relativistic expression,

$${\displaystyle \mathbf {p} =\gamma (v)m_{0}\mathbf {v} \,.}$$

Lorentz force in spacetime algebra (STA)

The electric and magnetic fields are dependent on the velocity of an observer, so the relativistic form of the Lorentz force law can best be exhibited starting from a coordinate-independent expression for the electromagnetic and magnetic fields ${\displaystyle {\mathcal {F}}}$, and an arbitrary time-direction, ${\displaystyle \gamma _{0}}$. 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

$${\displaystyle \mathbf {E} =\left({\mathcal {F}}\cdot \gamma _{0}\right)\gamma _{0}}$$

and

$${\displaystyle i\mathbf {B} =\left({\mathcal {F}}\wedge \gamma _{0}\right)\gamma _{0}}$$

${\displaystyle {\mathcal {F}}}$ 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 ${\displaystyle \gamma _{0}}$ pulls a vector (in the space algebra) from the translational part, while the wedge-product creates a trivector (in the space algebra) who is dual to a vector which is the usual magnetic field vector. The relativistic velocity is given by the (time-like) changes in a time-position vector ${\displaystyle v={\dot {x}}}$, where

$${\displaystyle v^{2}=1,}$$

(which shows our choice for the metric) and the velocity is

$${\displaystyle \mathbf {v} =cv\wedge \gamma _{0}/(v\cdot \gamma _{0}).}$$

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

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

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

Lorentz force in general relativity

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

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

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

The equation can also be written as

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

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

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

where ${\displaystyle D}$ is the covariant differential in general relativity (metric, torsion-free).

Applications

The Lorentz force occurs in many devices, including:

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

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

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

Essentialism

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Essentialism

Essentialism is the view that objects have a set of attributes that are necessary to their identity. In early Western thought, Plato's idealism held that all things have such an "essence"—an "idea" or "form". In Categories, Aristotle similarly proposed that all objects have a substance that, as George Lakoff put it, "make the thing what it is, and without which it would be not that kind of thing". The contrary view—non-essentialism—denies the need to posit such an "essence'".

Essentialism has been controversial from its beginning. Plato, in the Parmenides dialogue, depicts Socrates questioning the notion, suggesting that if we accept the idea that every beautiful thing or just action partakes of an essence to be beautiful or just, we must also accept the "existence of separate essences for hair, mud, and dirt". In biology and other natural sciences, essentialism provided the rationale for taxonomy at least until the time of Charles Darwin; the role and importance of essentialism in biology is still a matter of debate.

Historically, beliefs which posit that social identities such as ethnicity, nationality or gender are essential characteristics have in many cases been shown to have destructive or harmful results. It has been argued by some that Essentialist thinking lies at the core of many reductive, discriminatory or extremist ideologies. Psychological essentialism is also correlated with racial prejudice. In medical sciences, essentialism can lead to a reified view of identities—for example assuming that differences in hypertension in Afro-American populations are due to racial differences rather than social causes—leading to fallacious conclusions and potentially unequal treatment. Older social theories were often conceptually essentialist.

In philosophy

An essence characterizes a substance or a form, in the sense of the forms and ideas in Platonic idealism. It is permanent, unalterable, and eternal, and is present in every possible world. Classical humanism has an essentialist conception of the human, in its endorsement of the notion of an eternal and unchangeable human nature. This has been criticized by Kierkegaard, Marx, Heidegger, Sartre, and many other existential and materialist thinkers.

In Plato's philosophy (in particular, the Timaeus and the Philebus), things were said to come into being by the action of a demiurge who works to form chaos into ordered entities. Many definitions of essence hark back to the ancient Greek hylomorphic understanding of the formation of the things. According to that account, the structure and real existence of any thing can be understood by analogy to an artefact produced by a craftsperson. The craftsperson requires hyle (timber or wood) and a model, plan or idea in their own mind, according to which the wood is worked to give it the indicated contour or form (morphe). Aristotle was the first to use the terms hyle and morphe. According to his explanation, all entities have two aspects: "matter" and "form". It is the particular form imposed that gives some matter its identity—its quiddity or "whatness" (i.e., its "what it is").

Plato was one of the first essentialists, postulating the concept of ideal forms—an abstract entity of which individual objects are mere facsimiles. To give an example: the ideal form of a circle is a perfect circle, something that is physically impossible to make manifest; yet the circles we draw and observe clearly have some idea in common—the ideal form. Plato proposed that these ideas are eternal and vastly superior to their manifestations, and that we understand these manifestations in the material world by comparing and relating them to their respective ideal form. Plato's forms are regarded as patriarchs to essentialist dogma simply because they are a case of what is intrinsic and a-contextual of objects—the abstract properties that make them what they are. (For more on forms, read Plato's parable of the cave.)

Karl Popper splits the ambiguous term realism into essentialism and realism. He uses essentialism whenever he means the opposite of nominalism, and realism only as opposed to idealism. Popper himself is a realist as opposed to an idealist, but a methodological nominalist as opposed to an essentialist. For example, statements like "a puppy is a young dog" should be read from right to left, as an answer to "What shall we call a young dog"; never from left to right as an answer to "What is a puppy?"

Metaphysical essentialism

Essentialism, in its broadest sense, is any philosophy that acknowledges the primacy of essence. Unlike existentialism, which posits "being" as the fundamental reality, the essentialist ontology must be approached from a metaphysical perspective. Empirical knowledge is developed from experience of a relational universe whose components and attributes are defined and measured in terms of intellectually constructed laws. Thus, for the scientist, reality is explored as an evolutionary system of diverse entities, the order of which is determined by the principle of causality.

Plato believed that the universe was perfect and that its observed imperfections came from man's limited perception of it. For Plato, there were two realities: the "essential" or ideal and the "perceived". Aristotle (384–322 BC) applied the term essence to that which things in a category have in common and without which they cannot be members of that category (for example, rationality is the essence of man; without rationality a creature cannot be a man). In his critique of Aristotle's philosophy, Bertrand Russell said that his concept of essence transferred to metaphysics what was only a verbal convenience and that it confused the properties of language with the properties of the world. In fact, a thing's "essence" consisted in those defining properties without which we could not use the name for it. Although the concept of essence was "hopelessly muddled" it became part of every philosophy until modern times.

The Egyptian-born philosopher Plotinus (204–270 AD) brought idealism to the Roman Empire as Neoplatonism, and with it the concept that not only do all existents emanate from a "primary essence" but that the mind plays an active role in shaping or ordering the objects of perception, rather than passively receiving empirical data.

Despite the metaphysical basis for the term, academics in science, aesthetics, heuristics, psychology, and gender-based sociological studies have advanced their causes under the banner of essentialism. Possibly the clearest definition for this philosophy was offered by gay/lesbian rights advocate Diana Fuss, who wrote: "Essentialism is most commonly understood as a belief in the real, true essence of things, the invariable and fixed properties which define the 'whatness' of a given entity." Metaphysical essentialism stands diametrically opposed to existential realism in that finite existence is only differentiated appearance, whereas "ultimate reality" is held to be absolute essence.

In psychology

Paul Bloom attempts to explain why people will pay more in an auction for the clothing of celebrities if the clothing is unwashed. He believes the answer to this and many other questions is that people cannot help but think of objects as containing a sort of "essence" that can be influenced.

There is a difference between metaphysical essentialism (see above) and psychological essentialism, the latter referring not to an actual claim about the world but a claim about a way of representing entities in cognitions (Medin, 1989). Influential in this area is Susan Gelman, who has outlined many domains in which children and adults construe classes of entities, particularly biological entities, in essentialist terms—i.e., as if they had an immutable underlying essence which can be used to predict unobserved similarities between members of that class. (Toosi & Ambady, 2011). This causal relationship is unidirectional; an observable feature of an entity does not define the underlying essence (Dar-Nimrod & Heine, 2011).

In developmental psychology

Essentialism has emerged as an important concept in psychology, particularly developmental psychology. Gelman and Kremer (1991) studied the extent to which children from 4–7 years old demonstrate essentialism. Children were able to identify the cause of behaviour in living and non-living objects. Children understood that underlying essences predicted observable behaviours. Participants could correctly describe living objects' behaviour as self-perpetuated and non-living objects as a result of an adult influencing the object's actions. This is a biological way of representing essential features in cognitions. Understanding the underlying causal mechanism for behaviour suggests essentialist thinking (Rangel and Keller, 2011). Younger children were unable to identify causal mechanisms of behaviour whereas older children were able to. This suggests that essentialism is rooted in cognitive development. It can be argued that there is a shift in the way that children represent entities, from not understanding the causal mechanism of the underlying essence to showing sufficient understanding (Demoulin, Leyens & Yzerbyt, 2006). 

There are four key criteria that constitute essentialist thinking. The first facet is the aforementioned individual causal mechanisms (del Rio & Strasser, 2011). The second is innate potential: the assumption that an object will fulfill its predetermined course of development (Kanovsky, 2007). According to this criterion, essences predict developments in entities that will occur throughout its lifespan. The third is immutability (Holtz & Wagner, 2009). Despite altering the superficial appearance of an object it does not remove its essence. Observable changes in features of an entity are not salient enough to alter its essential characteristics. The fourth is inductive potential (Birnbaum, Deeb, Segall, Ben-Aliyahu & Diesendruck, 2010). This suggests that entities may share common features but are essentially different. However similar two beings may be, their characteristics will be at most analogous, differing most importantly in essences.

The implications of psychological essentialism are numerous. Prejudiced individuals have been found to endorse exceptionally essential ways of thinking, suggesting that essentialism may perpetuate exclusion among social groups (Morton, Hornsey & Postmes, 2009). For example, essentialism of nationality has been linked to anti-immigration attitudes(Rad & Ginges, 2018). In multiple studies in India and the United States, Rad & Ginges (2018) showed that in lay view, a person's nationality is considerably fixed at birth, even if that person is adopted and raised by a family of another nationality at day one and never told about their origin. This may be due to an over-extension of an essential-biological mode of thinking stemming from cognitive development. Paul Bloom of Yale University has stated that "one of the most exciting ideas in cognitive science is the theory that people have a default assumption that things, people and events have invisible essences that make them what they are. Experimental psychologists have argued that essentialism underlies our understanding of the physical and social worlds, and developmental and cross-cultural psychologists have proposed that it is instinctive and universal. We are natural-born essentialists." Scholars suggest that the categorical nature of essentialist thinking predicts the use of stereotypes and can be targeted in the application of stereotype prevention (Bastian & Haslam, 2006).

In ethics

Classical essentialists claim that some things are wrong in an absolute sense. For example, murder breaks a universal, objective and natural moral law and not merely an advantageous, socially or ethically constructed one.

Many modern essentialists claim that right and wrong are moral boundaries that are individually constructed; in other words, things that are ethically right or wrong are actions that the individual deems to be beneficial or harmful, respectively. 

In biology

Before evolution was developed as a scientific theory, there existed an essentialist view of biology that posited all species to be unchanging throughout time. The historian Mary P. Winsor has argued that biologists such as Louis Agassiz in the 19th century believed that taxa such as species and genus were fixed, reflecting the mind of the creator. Some religious opponents of evolution continue to maintain this view of biology.

Recent work by historians of systematic biology has, however, cast doubt upon this view of pre-Darwinian thinkers. Winsor, Ron Amundson and Staffan Müller-Wille have each argued that in fact the usual suspects (such as Linnaeus and the Ideal Morphologists) were very far from being essentialists, and it appears that the so-called "essentialism story" (or "myth") in biology is a result of conflating the views expressed by philosophers from Aristotle onwards through to John Stuart Mill and William Whewell in the immediately pre-Darwinian period, using biological examples, with the use of terms in biology like species.

Gender essentialism

Main article: Gender essentialism

In feminist theory and gender studies, gender essentialism is the attribution of fixed essences to men and women—this idea that men and women are fundamentally different continues to be a matter of contention. Women's essence is assumed to be universal and is generally identified with those characteristics viewed as being specifically feminine. These ideas of femininity are usually biologized and are often preoccupied with psychological characteristics, such as nurturance, empathy, support, and non-competitiveness, etc. Feminist theorist Elizabeth Grosz states in her 1995 publication Space, time and perversion: essays on the politics of bodies that essentialism "entails the belief that those characteristics defined as women's essence are shared in common by all women at all times. It implies a limit of the variations and possibilities of change—it is not possible for a subject to act in a manner contrary to her essence. Her essence underlies all the apparent variations differentiating women from each other. Essentialism thus refers to the existence of fixed characteristic, given attributes, and ahistorical functions that limit the possibilities of change and thus of social reorganization."

Gender essentialism is pervasive in popular culture, as illustrated by the #1 New York Times best seller Men Are from Mars, Women Are from Venus, but this essentialism is routinely critiqued in introductory women studies textbooks such as Women: Images & Realities.

Starting in the 1980s, some feminist writers have put forward essentialist theories about gender and science. Evelyn Fox Keller, Sandra Harding,  and Nancy Tuana  argued that the modern scientific enterprise is inherently patriarchal and incompatible with women's nature. Other feminist scholars, such as Ann Hibner Koblitz, Lenore Blum, Mary Gray, Mary Beth Ruskai, and Pnina Abir-Am and Dorinda Outram have criticized those theories for ignoring the diverse nature of scientific research and the tremendous variation in women's experiences in different cultures and historical periods.

In historiography

Essentialism in history as a field of study entails discerning and listing essential cultural characteristics of a particular nation or culture, in the belief that a people or culture can be understood in this way. Sometimes such essentialism leads to claims of a praiseworthy national or cultural identity, or to its opposite, the condemnation of a culture based on presumed essential characteristics. Herodotus, for example, claims that Egyptian culture is essentially feminized and possesses a "softness" which has made Egypt easy to conquer. To what extent Herodotus was an essentialist is a matter of debate; he is also credited with not essentializing the concept of the Athenian identity, or differences between the Greeks and the Persians that are the subject of his Histories.

Essentialism had been operative in colonialism as well as in critiques of colonialism.

Post-colonial theorists such as Edward Said insisted that essentialism was the "defining mode" of "Western" historiography and ethnography until the nineteenth century and even after, according to Touraj Atabaki, manifesting itself in the historiography of the Middle East and Central Asia as Eurocentrism, over-generalization, and reductionism.

Today, most historians, social scientists and humanists reject methodologies associated with essentialism, though some have argued that certain varieties of essentialism may be useful or even necessary.

Language ideology

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Language_ideology 

Language ideology (also known as linguistic ideology or language attitude) is, within anthropology (especially linguistic anthropology), sociolinguistics, and cross-cultural studies, any set of beliefs about languages as they are used in their social worlds. When recognized and explored, language ideologies expose how the speakers' linguistic beliefs are linked to the broader social and cultural systems to which they belong, illustrating how the systems beget such beliefs. By doing so, language ideologies link implicit and explicit assumptions about a language or language in general to their social experience as well as their political and economic interests. Language ideologies are conceptualizations about languages, speakers, and discursive practices. Like other kinds of ideologies, language ideologies are influenced by political and moral interests, and they are shaped in a cultural setting.

Applications and approaches

Definitions

Scholars have noted difficulty in attempting to delimit the scope, meaning, and applications of language ideology. Paul Kroskrity, a linguistic anthropologist, describes language ideology as a "cluster concept, consisting of a number of converging dimensions" with several "partially overlapping but analytically distinguishable layers of significance", and cites that in the existing scholarship on language ideology "there is no particular unity . . . no core literature, and a range of definitions." One of the broadest definitions is offered by Alan Rumsey, who describes language ideologies as "shared bodies of commonsense notions about the nature of language in the world." This definition is seen by Kroskrity as unsatisfactory, however, because "it fails to problematize language ideological variation and therefore promotes an overly homogeneous view of language ideologies within a cultural group." Emphasizing the role of speakers' awareness in influencing language structure, Michael Silverstein defines linguistic ideologies as "sets of beliefs about language articulated by users as a rationalization or justification of perceived language structure and use." Definitions that place greater emphasis on sociocultural factors include Shirley Heath's characterization of language ideologies as "self-evident ideas and objectives a group holds concerning roles of language in the social experiences of members as they contribute to the expression of the group", as well as Judith Irvine's definition of the concept as "the cultural system of ideas about social and linguistic relationships, together with their loading of moral and political interests."

Critical vs. neutral approaches

The basic division in studies of language ideology is between neutral and critical approaches to ideology. In neutral approaches to language ideology, beliefs or ideas about a language are understood to be shaped by the cultural systems in which it is embedded, but no variation within or across these systems is identified. Often, a single ideology will be identified in such cases. Characterizations of language ideology as representative of one community or culture, such as those routinely documented in ethnographic research, are common examples of neutral approaches to language ideology.

Critical approaches to language ideology explore the capacity for language and linguistic ideologies to be used as strategies for maintaining social power and domination. They are described by Kathryn Woolard and Bambi Schieffelin as studies of "some aspects of representation and social cognition, with particular social origins or functional and formal characteristics." Although such studies are often noted for their discussions of language politics and the intersection between language and social class, the crucial difference between these approaches to language ideology and neutral understandings of the concept is that the former emphasize the existence of variability and contradiction both within and amongst ideologies, while the latter approach ideology as a conception on its own terms.

Areas of inquiry

Language use and structure

Many scholars have argued that ideology plays a role in shaping and influencing linguistic structures and speech forms. Michael Silverstein, for example, sees speakers’ awareness of language and their rationalizations of its structure and use as critical factors that often shape the evolution of a language's structure. According to Silverstein, the ideologies speakers possess regarding language mediate the variation that occurs due to their imperfect and limited awareness of linguistic structures, resulting in the regularization of any variation that is rationalized by any sufficiently dominant or culturally widespread ideologies. This is demonstrated by such linguistic changes as the rejection of “he” as the generic pronoun in English, which coincided with the rise of the feminist movement in the second half of the twentieth century. In this instance, the accepted usage of the masculine pronoun as the generic form came to be understood as a linguistic symbol of patriarchal and male-dominated society, and the growing sentiment opposing these conditions motivated some speakers to stop using “he” as the generic pronoun in favor of the construction “he or she.” This rejection of generic “he” was rationalized by the growing desire for gender equality and women's empowerment, which was sufficiently culturally prevalent to regularize the change.

Alan Rumsey also sees linguistic ideologies as playing a role in shaping the structure of a language, describing a circular process of reciprocal influence where a language's structure conditions the ideologies that affect it, which in turn reinforce and expand this structure, altering the language “in the name of making it more like itself.” This process is exemplified by the excessive glottalization of consonants by bilingual speakers of moribund varieties of Xinca, who effectively altered the structure of this language in order to make it more distinct from Spanish. These speakers glottalized consonants in situations in places more competent speakers of Xinca would not because they were less familiar with the phonological rules of the language and also because they wished to distinguish themselves from the socially-dominant Spanish-speakers, who viewed glottalized consonants as “exotic.”

Ethnography of speaking

Studies of "ways of speaking" within specific communities have been recognized as especially productive sites of research in language ideology. They often include a community's own theory of speech as a part of their ethnography, which allows for the documentation of explicit language ideologies on a community-wide level or in “the neutral sense of cultural conceptions.” A study of language socialization practices in Dominica, for example, revealed that local notions of personhood, status, and authority are associated with the strategic usage of Patwa and English in the course of the adult-child interaction. The use of Patwa by children is largely forbidden by adults due to a perception that it inhibits the acquisition of English, thus restricting social mobility, which in turn has imbued Patwa with a significant measure of covert prestige and rendered it a powerful tool for children to utilize in order to defy authority. Thus there are many competing ideologies of Patwa in Dominica: one which encourages a shift away from Patwa usage and another which contributes to its maintenance.

Linguistic ideologies in speech act theory

J. L. Austin and John Searle's speech act theory has been described by several ethnographers, anthropologists, and linguists as being based in a specifically Western linguistic ideology that renders it inapplicable in certain ethnographic contexts. Jef Verschueren characterized speech act theory as privileging “a privatized view of language that emphasizes the psychological state of the speaker while downplaying the social consequences of speech,” while Michael Silverstein argued that the theory's ideas about language “acts” and “forces” are “projections of covert categories typical in the metapragmatic discourse of languages such as English.” Scholars have subsequently used speech act theory to caution against the positioning of linguistic theories as universally applicable, citing that any account of language will reflect the linguistic ideologies held by those who develop it.

Language contact and multilingualism

Several scholars have noted that sites of cultural contact promote the development of new linguistic forms that draw on diverse language varieties and ideologies at an accelerated rate. According to Miki Makihara and Bambi Schieffelin, it becomes necessary during times of cultural contact for speakers to actively negotiate language ideologies and to consciously reflect on language use. This articulation of ideology is essential to prevent misconceptions of meaning and intentions between cultures, and provides a link between sociocultural and linguistic processes in contact situations.

Language policy and standardization

The establishment of a standard language has many implications in the realms of politics and power. Recent examinations of language ideologies have resulted in the conception of “standard” as a matter of ideology rather than fact, raising questions such as “how doctrines of linguistic correctness and incorrectness are rationalized and how they are related to doctrines of the inherent representational power, beauty, and expressiveness of language as a valued mode of action.”.

Language policy

Governmental policies often reflect the tension between two contrasting types of language ideologies: ideologies that conceive of language as a resource, problem, or right and ideologies that conceive of language as pluralistic phenomena. The linguistic policies that emerge in such instances often reflect a compromise between both types of ideologies. According to Blommaert and Verschueren, this compromise is often reinterpreted as a single, unified ideology, evidenced by the many European societies characterized by a language ideological homogenism.

Ideologies of linguistic purism

Purist language ideologies or ideologies of linguistic conservatism can close off languages to nonnative sources of innovation, usually when such sources are perceived as socially or politically threatening to the target language. Among the Tewa, for example, the influence of theocratic institutions and ritualized linguistic forms in other domains of Tewa society have led to a strong resistance to the extensive borrowing and shift that neighboring speech communities have experienced. According to Paul Kroskrity this is due to a "dominant language ideology" through which ceremonial Kiva speech is elevated to a linguistic ideal and the cultural preferences that it embodies, namely regulation by convention, indigenous purism, strict compartmentalization, and linguistic indexing of identity, are recursively projected onto the Tewa language as a whole.

Alexandra Jaffe points out that language purism is often part of “essentializing discourses” that can lead to stigmatizing habitual language practices like code-switching and depict contact-induced linguistic changes as forms of cultural deficiency.

Standard language ideology

Main article: Standard language

As defined by Rosina Lippi-Green, standard language ideology is "a bias toward an abstract, idealized homogeneous language, which is imposed and maintained by dominant institutions and which has as its model the written language, but which is drawn primarily from the spoken language of the upper middle class." According to Lippi-Green, part of this ideology is a belief that standard languages are internally consistent. Linguists generally agree, however, that variation is intrinsic to all spoken language, including standard varieties.

Standard language ideology is strongly connected with the concepts of linguistic purism and prescriptivism. It is also linked with linguicism (linguistic discrimination).

Literacy

Literacy cannot be strictly defined technically, but rather it is a set of practices determined by a community's language ideology. It can be interpreted in many ways that are determined by political, social, and economic forces. According to Kathryn Woolard and Bambi Schieffelin, literacy traditions are closely linked to social control in most societies. The typical European literacy ideology, for example, recognizes literacy solely in an alphabetic capacity.

Kaluli literacy development

In the 1960s, missionaries arrived in Papua New Guinea and exposed the Kaluli to Christianity and modernization, part of which was accomplished through the introduction of literacy. The Kaluli primers that were introduced by the missionaries promoted Westernization, which effectively served to strip the vernacular language of cultural practices and from discourse in church and school. Readers written in the 1970s used derogatory terms to refer to the Kaluli and depicted their practices as inferior, motivating the Kaluli to change their self-perceptions and orient themselves towards Western values. The missionaries' control of these authoritative books and of this new “technology of language literacy” gave them the power to effect culture change and morph the ideology of Kaluli into that of modern Christianity.

Orthography

Orthographic systems always carry historical, cultural, and political meaning that are grounded in ideology. Orthographic debates are focused on political and social issues rather than on linguistic discrepancies, which can make for intense debates characterized by ideologically charged stances and symbolically important decisions.

Classroom practice/second language acquisition

"Language ideologies are not confined merely to ideas or beliefs, but rather is extended to include the very language practices through which our ideas or notions are enacted" (Razfar, 2005). Teachers display their language ideologies in classroom instruction through various practices such as correction or repair, affective alignment, metadiscourse, and narrative (see Razfar & Rumenapp, 2013, p. 289). The study of ideology seeks to uncover the hidden world of students and teachers to shed light on the fundamental forces that shape and give meaning to their actions and interactions.