Lorentz Force Law

Lorentz force - Wikipedia, the free encyclopedia
The Lorentz force law has a close relationship with Faraday's law of induction. ... Significance of the Lorentz force. 3 Lorentz force law as the definition of ...
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Magnetic forces
... from the Lorentz Force Law, and specifically from the magnetic force on a moving ... Lorentz Force Law. Both the electric field and magnetic field can be ...
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Lorentz force: Definition from Answers.com
Lorentz force n. The orthogonal force on a charged particle traveling ... Nowadays, Faraday's law is used instead of the Lorentz force in Maxwell's equations. ...
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EcEn 360 Tutorials-Maxwell's Law-Lorentz Force Equation
Lorentz Force Law. Lorentz found that a magnetic field would exert a force on an electric charge ... animation we see a special case of Lorentz's law. ...
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PowerPedia:Lorentz force - PESWiki
... the magnetic field qv × B. Combined they give the Lorentz force equation (or law) ... Equivalently, we can express the Lorentz force law in terms of the electric ...
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Faraday's law of induction - Wikipedia, the free encyclopedia
5.1 Lorentz force law method. 5.2 Faraday's law method ... in agreement with the Lorentz force law calculation. Now let's try a different path. ...
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Lorentz's Force
Lorentz's Force - Inconsistensy of Maxwell's equations ... Lorentz's Equations. Maxwell's Equations. ME in media. Faraday's Law. Lorentz's Force ...
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Page 217
Lorentz Force. The Lorentz Force depends on the velocity, v, the magnetic field, B, and the ... you can change in this applet and observe the deflection of a ...
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Lorentz Force -- from Eric Weisstein's World of Physics
Lorentz Force. A force on moving charges particles in the presence of magnetic B and electric ... Canonical Momentum, Electric Force, Magnetic Field, Magnetic Force ...
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Lorentz Force Law
Lorentz Force Law. table of contents. search. E.4.1 Lorentz Force. This demonstration is an excellent illustration of tne vector cross product. ...
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In physics, the Lorentz force is the force exerted on a electric charge particle in an electromagnetic field. The particle will experience a force due to electric field of qE', and due to the magnetic field qv' × B. Combined they give the Hendrik Lorentz force equation (or law):

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

where F is the force (in newtons) E is the electric field (in volts per meter) B is the magnetic field (in tesla (unit)) q is the electric charge of the particle (in coulombs) v is the instantaneous velocity of the particle (in meters per second) and × is the cross product.

Thus 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 (i.e., if the thumb of the right hand points along v and the index finger along B, then the middle finger points along F).

The Significance of the Lorentz Force The Lorentz force is one of the original eight Maxwell's equations (equation D) and it is the solution to the differential form of Faraday's law of induction. Nowadays, Faraday's law is used instead of the Lorentz force in Maxwell's equations. Faraday's law and the Lorentz force both express the same physics. The discovery of the Lorentz force was before Lorentz's time. It can be seen at equation (77) in Maxwell's 1861 paper On Physical Lines of Force.

Lorentz force in special relativity When particle speeds approach the speed of light, the Lorentz force equation must be modified according to special relativity:

{d \left ( \gamma m \mathbf{v} \right ) \over dt } = \mathbf{F} = q (\mathbf{E} + \mathbf{v} \times \mathbf{B}),

where

\gamma \ \stackrel{\mathrm{def-->{=}\ \frac{1}{\sqrt{1 - \frac{|\mathbf{v}|^2}{c^2-->}

is called the Lorentz factor and c is the speed of light in a vacuum.

This relativistic form is identical to the conventional expression of the Lorentz force if the momentum form of Newton's law, F= dp/dt, is used, and the momentum p is assumed to be p = \gamma mv.

The change of energy due to the electric and magnetic fields, in relativistic form, is simply

{d \left ( \gamma m c^2 \right ) \over dt } = q \mathbf{E} \cdot \mathbf{v} .

The change in energy depends only on the electric field, and not on the magnetic field.

Covariant form of the Lorentz force The Lorentz force equation can be written in Covariant transformation in terms of the field strength tensor.

: \frac{d p^\alpha}{d \tau} = q u_\beta F^{\alpha \beta} where :\tau is c times the proper time of the particle, :q is the Electric charge, :u is the four-velocity of the particle, defined as: :u_\beta = \left(u_0, u_1, u_2, u_3 \right) = \gamma \left(c, v_x, v_y, v_z \right) \,and

:F is the Electromagnetic tensor (or electromagnetic tensor) and is written in terms of fields as:

:F^{\alpha \beta} = \begin{bmatrix} 0 & -E_x/c & -E_y/c & -E_z/c \\E_x/c & 0 & -B_z & B_y \\E_y/c & B_z & 0 & -B_x \\E_z/c & -B_y & B_x & 0\end{bmatrix}.

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

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

where {\Lambda^{\mu-->_{\alpha} is a [Lorentz transformation.

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

Here, \tau is the proper time of the particle. Substituting the components of the electromagnetic tensor F yields : \gamma \frac{d p^1}{d t} = q \left(-u^0 \left(\frac{-E_x}{c} \right) + u^2 (B_z) + u^3 (-B_y) \right) \, Writing the four-velocity in terms of the ordinary velocity yields : \gamma \frac{d p^1}{d t} = q \gamma \left(c \left(\frac{E_x}{c} \right) + v_y B_z - v_z B_y \right) \,

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

The calculation of the \mu = 2 or \mu = 3 is similar yielding : \gamma \frac{d \mathbf{p} }{d t} = \frac{d \mathbf{p} }{d \tau} = q \gamma \left(\mathbf{E} + (\mathbf{v} \times \mathbf{B})\right) \,,

which is the Lorentz force law.

Applications The Lorentz force is a principle exploited in many devices including:

Cyclotrons and other circular path particle accelerators
Homopolar generators
Magnetrons
Magnetoplasmadynamic thrusters
Mass spectrometers
Velocity Filter

The Lorentz force can also act on a current carrying conductor, in this case called Biot-Savart law, by the interaction of the conduction electrons with the atoms of the conductor material. This force is used in many devices including :