# Lorentz force

Electrostatics Electromagnetism Electricity Magnetism Electric charge Coulomb's law Electric field Gauss's law Electric potential Electric dipole moment Ampre's circuital law Magnetic field Magnetic flux Biot-Savart law Magnetic dipole moment Electrical current Lorentz force law Electromotive force (EM) Electromagnetic induction Faraday-Lenz law Displacement current Maxwell's equations (EMF) Electromagnetic field (EM) Electromagnetic radiation Electrical conduction Electrical resistance Capacitance Inductance Impedance Resonant cavities Waveguides Electromagnetic tensor Electromagnetic stress-energy tensor This box:  •  • [ edit]

Lorentz force.

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):

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).

## 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.

## Lorentz force in special relativity

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

where

is called the Lorentz factor and 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 .

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

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 form in terms of the field strength tensor.

:
where
: is c times the proper time of the particle,
:q is the charge,
:u is the 4-velocity of the particle, defined as:
:and

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

:.

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

where is a Lorentz transformation.

### Derivation

The component (x-component) of the force is
:

Here, is the proper time of the particle. Substituting the components of the electromagnetic tensor F yields
:
Writing the four-velocity in terms of the ordinary velocity yields
:

:

The calculation of the or is similar yielding
:,

which is the Lorentz force law.

## Applications

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

## 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.

Electromagnetism is the physics of the electromagnetic field: a field which exerts a force on particles that possess the property of electric charge, and is in turn affected by the presence and motion of those particles.
Electricity (from New Latin ēlectricus, "amberlike") is a general term for a variety of phenomena resulting from the presence and flow of electric charge. This includes many well-known physical phenomena such as lightning, electromagnetic fields and electric currents,
magnetism is one of the phenomena by which materials exert attractive or repulsive forces on other materials. Some well known materials that exhibit easily detectable magnetic properties (called magnets) are nickel, iron and their alloys; however, all materials are influenced to
Electrostatics (also known as static electricity) is the branch of physics that deals with the phenomena arising from what seem to be stationary electric charges. This includes phenomena as simple as the attraction of plastic wrap to your hand after you remove it from a
Flavour in particle physics

Coulomb's law, developed in the 1780s by French physicist Charles Augustin de Coulomb, may be stated as follows:

The magnitude of the electrostatic force between two points electric charges is directly proportional to the product of the magnitudes of each

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The electric field is a vector field with SI units of newtons per coulomb (N C−1
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Electric potential is the potential energy per unit of charge associated with a static (time-invariant) electric field, also called the electrostatic potential, typically measured in volts. It is a scalar quantity.
In physics, the electric dipole moment (or electric dipole for short) is a measure of the polarity of a system of electric charges.

In the simple case of two point charges, one with charge and one with charge , the electric dipole moment is:

Magnetostatics is the study of static magnetic fields. In electrostatics, the charges are stationary, whereas here, the currents are stationary. As it turns out magnetostatics is a good approximation even when the currents are not static as long as the currents do not
magnetic field is a field that permeates space and which exerts a magnetic force on moving electric charges and magnetic dipoles. Magnetic fields surround electric currents, magnetic dipoles, and changing electric fields.
Magnetic flux, represented by the Greek letter Φ (phi), is a measure of quantity of magnetism, taking account of the strength and the extent of a magnetic field.
The Biot-Savart Law is an equation in electromagnetism that describes the magnetic field vector B in terms of the magnitude and direction of the source electric current, the distance from the source electric current, and the magnetic permeability weighting factor.
In physics, the magnetic moment or magnetic dipole moment is a measure of the strength of a magnetic source. In the simplest case of a current loop, the magnetic moment is defined as:
where a
Classical electromagnetism (or classical electrodynamics) is a theory of electromagnetism that was developed over the course of the 19th century, most prominently by James Clerk Maxwell.
Electric current is the flow (movement) of electric charge. The SI unit of electric current is the ampere (A), which is equal to a flow of one coulomb of charge per second.

## Definition

The amount of electric current (measured in amperes) through some surface, e.g.
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