Lorentz force: Difference between revisions
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In [[physics]] | In [[physics]] the '''Lorentz force''' is the force on an electrically charged particle that moves through a [[magnetic field]] '''B''' and possibly also through an [[electric field]] '''E'''. | ||
The force is named after the Dutch physicist [[Hendrik Antoon Lorentz]], who gave its description in 1892.<ref>H. A. Lorentz, ''La théorie électromagnétique de Maxwell et son application aux corps mouvants'', Archives néerlandaises des Sciences exactes et naturelles, vol. '''25''' p. 363 (1892).</ref> | In the absence of an electric field ('''E''' = 0), the strength of the Lorentz force is proportional to the charge ''q'' of the particle, the speed ''v'' (the size of the velocity '''v''') of the particle, the intensity ''B'' of the magnetic field, and the [[sine]] of the angle between the [[vector]]s '''v''' and '''B'''. The direction of the Lorentz force is given by the [[right hand rule]]: put your right hand along '''v''' with the open palm toward the magnetic field '''B''' (a vector). Stretch the thumb of your right hand, the Lorentz force is along it, pointing from your wrist to the tip of your thumb. | ||
If an electric field '''E''' is also present then the force ''q'' '''E''' must be added vectorially | |||
to the magnetic component of the Lorentz force (the force on the particle when '''E''' = 0). | |||
The force is named after the Dutch physicist [[Hendrik Antoon Lorentz]], who gave its description in 1892.<ref>H. A. Lorentz, ''La théorie électromagnétique de Maxwell et son application aux corps mouvants'' [The electromagnetic theory of Maxwell and its application to moving bodies], Archives néerlandaises des Sciences exactes et naturelles, vol. '''25''' p. 363 (1892).</ref> | |||
==Mathematical description== | ==Mathematical description== | ||
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</math> | </math> | ||
where ''k'' is a constant depending on the units. In [[SI]] units ''k'' = 1; in Gaussian units ''k'' = 1/''c'', where ''c'' is the speed of light in vacuum (299 792 458 m s<sup>−1</sup> exactly). | where ''k'' is a constant depending on the units. In [[SI]] units ''k'' = 1; in Gaussian units ''k'' = 1/''c'', where ''c'' is the speed of light in vacuum (299 792 458 m s<sup>−1</sup> exactly). | ||
The quantity ''q'' is the electric charge of the particle and '''v''' is its velocity. The vector '''B''' is the [[magnetic induction]] (sometimes referred to as the magnetic field). The product of '''v''' and '''B''' is | The quantity ''q'' is the electric charge of the particle and '''v''' is its velocity. The vector '''B''' is the [[magnetic induction]] (sometimes referred to as the magnetic field). The product of '''v''' and '''B''' is the [[vector product]] (a vector with the direction given by the right hand rule mentioned above and of magnitude ''vB''sin α). | ||
The electric field '''E''' is in full generality given by | The electric field '''E''' is in full generality given by | ||
:<math> | :<math> | ||
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\mathbf{E} = - \boldsymbol{\nabla}V. | \mathbf{E} = - \boldsymbol{\nabla}V. | ||
</math> | </math> | ||
It is possible that the electric field '''E''' is absent (zero) and '''B''' is static and non-zero, | It is possible that the electric field '''E''' is absent (zero) and that '''B''' is static and non-zero, | ||
then the Lorentz force is given by, | then the Lorentz force is given by, | ||
:<math> | :<math> | ||
\mathbf{F} = k\,q\, \mathbf{v}\times\mathbf{B} , | \mathbf{F} = k\,q\, \mathbf{v}\times\mathbf{B} , | ||
</math> | </math> | ||
where ''k'' = 1 for SI units and 1/''c'' for Gaussian units. In this form the Lorentz force was given by [[Oliver Heaviside]] in 1889 three years before Lorentz.<ref>E. Whittaker, ''A History of the Theories of Aether and Electricity'', vol. I, 2nd edition, Nelson, London (1951). Reprinted by the American Institute of Physics, (1987). p. 310 </ref> | where ''k'' = 1 for SI units and 1/''c'' for Gaussian units. In this form the Lorentz force was given by [[Oliver Heaviside]] in 1889, three years before Lorentz.<ref>E. Whittaker, ''A History of the Theories of Aether and Electricity'', vol. I, 2nd edition, Nelson, London (1951). Reprinted by the American Institute of Physics, (1987). p. 310 </ref> | ||
== | ==Notes== | ||
<references /> | <references /> | ||
Revision as of 14:24, 6 May 2008
In physics the Lorentz force is the force on an electrically charged particle that moves through a magnetic field B and possibly also through an electric field E.
In the absence of an electric field (E = 0), the strength of the Lorentz force is proportional to the charge q of the particle, the speed v (the size of the velocity v) of the particle, the intensity B of the magnetic field, and the sine of the angle between the vectors v and B. The direction of the Lorentz force is given by the right hand rule: put your right hand along v with the open palm toward the magnetic field B (a vector). Stretch the thumb of your right hand, the Lorentz force is along it, pointing from your wrist to the tip of your thumb.
If an electric field E is also present then the force q E must be added vectorially to the magnetic component of the Lorentz force (the force on the particle when E = 0).
The force is named after the Dutch physicist Hendrik Antoon Lorentz, who gave its description in 1892.[1]
Mathematical description
The Lorentz force F is given by the expression
where k is a constant depending on the units. In SI units k = 1; in Gaussian units k = 1/c, where c is the speed of light in vacuum (299 792 458 m s−1 exactly). The quantity q is the electric charge of the particle and v is its velocity. The vector B is the magnetic induction (sometimes referred to as the magnetic field). The product of v and B is the vector product (a vector with the direction given by the right hand rule mentioned above and of magnitude vBsin α). The electric field E is in full generality given by
where V is a scalar (electric) potential and the (magnetic) vector potential A is connected to B via
The factor k has the same meaning as before. The operator ∇ acting on V gives the gradient of V, while ∇ × A is the curl of A.
If B is static (does not depend on time) then A is also static and
It is possible that the electric field E is absent (zero) and that B is static and non-zero, then the Lorentz force is given by,
where k = 1 for SI units and 1/c for Gaussian units. In this form the Lorentz force was given by Oliver Heaviside in 1889, three years before Lorentz.[2]
Notes
- ↑ H. A. Lorentz, La théorie électromagnétique de Maxwell et son application aux corps mouvants [The electromagnetic theory of Maxwell and its application to moving bodies], Archives néerlandaises des Sciences exactes et naturelles, vol. 25 p. 363 (1892).
- ↑ E. Whittaker, A History of the Theories of Aether and Electricity, vol. I, 2nd edition, Nelson, London (1951). Reprinted by the American Institute of Physics, (1987). p. 310