Lorentz force

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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 \cdot(\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 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 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).

[edit] 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. 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.

[edit] Lorentz force in cgs units

The above-mentioned formula uses the SI units (mksA units) which are in general use by technicians and experimentalists and in engineering. In cgs units, which are preferred by some theorists, one has instead

<math>\mathbf{F} = q' \cdot (\mathbf{E'} + \frac{\mathbf{v}}{c} \times \mathbf{B'}).</math>

At the first glance, this looks slightly different, but actually it is completely equivalent, since one has the following cross-relations

<math>q'\,(\,\equiv q_{cgs})=\frac{q}{\sqrt{4\pi \epsilon_0}}</math>,   <math>\mathbf E' =\sqrt{4\pi\epsilon_0}\,\mathbf E</math>, and   <math>\mathbf B' ={\sqrt{4\pi /\mu_0}}\,{\mathbf B}</math>, with the well-known vacuum permittivities. Usually, the primes are left away, which may be somewhat embarrassing, but typically one remains in a given unit system, thus avoiding the problems.

[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 - \frac{|\mathbf{v}|^2}{c^2}}}</math>

is called the Lorentz factor and <math>c</math> 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 <math>p = \gamma mv</math>.

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

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

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

[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.

<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
• Velocity Filter

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 :

• Railguns
• Electrical generators
• Electric motors

[edit] See also

• Electromagnetism
• Gravitomagnetism
• Hendrik Lorentz
• Maxwell's equations
• Formulation of Maxwell's equations in special relativity
• Moving magnet and conductor problem
• Abraham-Lorentz force

[edit] References

• Serway and Jewett (2004). Physics for Scientists and Engineers with Modern Physics. Thomson Brooks/Cole. ISBN 0-534-40846-X. 
• Feynman, Leighton and Sands (2006). The Feynman Lectures on Physics The Definitive Edition Volume II. Pearson Addison Wesley. ISBN 0-8053-9047-2. 

[edit] External links

• Lorentz force (animation)
• Lorentz force (demonstration)bg:Сила на Лоренц

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