Lorentz force

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Image:Lorentz force.svg

In physics, the Lorentz force is the force exerted on a charged 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 Lorentz force equation (or law):

<math>\mathbf{F} = q (\mathbf{E} + \mathbf{v} \times \mathbf{B}),</math>

where

F is the force (in newtons)

E is the electric field (in volts per meter)

B is the magnetic field (in webers per square meter, or equivalently, teslas)

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 the B field according to the right-hand rule.

1 Lorentz force in special relativity
2 Covariant form of the Lorentz force
- 2.1 Derivation
3 Applications
4 See also
5 Reference
6 External links

[edit] Lorentz force in special relativity

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

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

where

<math>\gamma \ \stackrel{\mathrm{def}}{=}\ \frac{1}{\sqrt{1 - v^2/c^2}}</math>

is called the Lorentz factor and <math>c</math> is the speed of light in a vacuum.

This expression differs from the expression obtained from the Lorentz force by a factor of <math>\gamma </math>.

The change of energy due to the fields is

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

[edit] Covariant form of the Lorentz force

Main article: Formulation of Maxwell's equations in special relativity

The Lorentz force equation can be written in covariant form in terms of the field strength tensor (cgs units).

<math> \frac{d p^\alpha}{d \tau} = q u_\beta F^{\alpha \beta} </math>

where

<math>\tau </math> is c times the proper time of the particle,

q is the charge,

u is the 4-velocity of the particle, defined as:

<math>u_\beta = \left(u_0, u_1, u_2, u_3 \right) = \gamma \left(c, v_x, v_y, v_z \right) \,</math>and

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

<math>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} </math>.

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

<math> \acute{F}^{\mu \nu} = {\Lambda^{\mu}}_{\alpha} {\Lambda^{\nu}}_{\beta} F^{\alpha \beta}

,</math>

where <math> {\Lambda^{\mu}}_{\alpha}

</math> is a Lorentz transformation.

[edit] Derivation

The <math>\mu =1</math> component (x-component) of the force is

<math> \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) .\,</math>

Here, <math> \tau </math> is the proper time of the particle. Substituting the components of the electromagnetic tensor F yields

<math> \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) \,</math>

Writing the four-velocity in terms of the ordinary velocity yields

<math> \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) \,</math>

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

The calculation of the <math>\mu = 2</math> or <math>\mu = 3</math> is similar yielding

<math> \gamma \frac{d \mathbf{p} }{d t} = \frac{d \mathbf{p} }{d \tau} = q \gamma \left(\mathbf{E} + (\mathbf{v} \times \mathbf{B}\right) \,</math>,

which is the Lorentz force law.

[edit] 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

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