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## PowerPedia:Lorentz force

Lasted edited by Andrew Munsey, updated on June 14, 2016 at 10:04 pm.

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In physics, the Lorentz force is the Force exerted on a There was an error working with the wiki: Code[10]d There was an error working with the wiki: Code[11] in an Electromagnetic field.

#### Description

A particle will experience forces due to Electric field and the Magnetic field that will alter it path. The Lorentz force is a principle exploited in many devices including:

There was an error working with the wiki: Code[12]s and other circular path There was an error working with the wiki: Code[13]s

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The Lorentz force can act on a current carrying conductor, in this case called There was an error working with the wiki: Code[17], by the interaction of the conduction electrons with the atoms of the conductor material. This force is used in many devices including :

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#### Mathematics

The particle will experience a force due to electric field of q'E, and due to the magnetic field q'v &times B. Combined they give the There was an error working with the wiki: Code[4] force equation (or law):

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

where

:F is the Force (in There was an error working with the wiki: Code[19]s)

:E is the Electric field (in Volts per Meter)

:B is the There was an error working with the wiki: Code[5] per square meter, or equivalently, There was an error working with the wiki: Code[6])

:q is the Electric charge of the particle (in There was an error working with the wiki: Code[20]s)

:v is the instantaneous Velocity of the particle (in meters per There was an error working with the wiki: Code[21])

:and &times is the There was an error working with the wiki: Code[22].

Thus a positively charged particle will be accelerated in the same linear orientation as the E field, but will curve perpendicularly to the B field according to the There was an error working with the wiki: Code[23].

Alternative form

Equivalently, we can express the Lorentz force law in terms of the electric There was an error working with the wiki: Code[24] &rho and There was an error working with the wiki: Code[25] J as

:{F} = \int_V ( \rho {E} + {J} \times {B}) dV

Lorentz force in special relativity

When particle speeds approach the speed of light, the Lorentz force equation must be modified according to There was an error working with the wiki: Code[26]:

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

where

:\gamma \equiv \frac{1}{\sqrt{1 - v^2/c^2}}

is called the There was an error working with the wiki: Code[27] and c is the There was an error working with the wiki: Code[28] in a vacuum.

This expression differs from the expression obtained from the Lorentz force by a factor of \gamma .

The change of energy due to the fields is

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

Covariant form of the Lorentz force

The Lorentz force equation can be written in There was an error working with the wiki: Code[7] in terms of the field strength tensor (cgs units).

: m c { d u^{\alpha} \over { d \tau } } = { {} \over {} }F^{\alpha \beta} q u_{\beta}

where m is the particle There was an error working with the wiki: Code[8], and

: u_{\beta} = \eta_{\beta \alpha } u^{\alpha } = \eta_{\beta \alpha } { d x^{\alpha } \over {d \tau} }

is the There was an error working with the wiki: Code[9] of the particle. Here, \tau is c times the There was an error working with the wiki: Code[29] of the particle and \eta is the There was an error working with the wiki: Code[30] tensor. The fields can be 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 There was an error working with the wiki: Code[31].

#### Related concepts

Electromagnetism

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#### Reference and external articles

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General internet

Lorentz force (animation)

There was an error working with the wiki: Code[1], Wikipedia: The Free Encyclopedia. Wikimedia Foundation.

news://sci.physics Physical laws, properties, etc. Activity: High

news://sci.physics.electromag Electromagnetic theory and applications. Activity: Medium

news://sci.physics.relativity The theory of relativity. Activity: High

Books

Serway and Jewett, "Physics for Scientists and Engineers with Modern Physics" Thomson Brooks/Cole, 2004 ISBN 0-534-40846-X

Andre Koch and Torres Assis, "Weber's Electrodynamics" (Lorentz force laws), Springer, 1994. ISBN 0792331370

H.C. Sharma, Rajesh Kharb, Rajesh Sharma, "Comprehensive Physics for Engineers" (Lorentz force and magnetic induction). Laxmi Publications, 2005.