Lorentz force: Difference between revisions
imported>Paul Wormer |
imported>Paul Wormer |
||
| Line 13: | Line 13: | ||
\mathbf{F} = q ( \mathbf{E} + k \mathbf{v}\times\mathbf{B} ), | \mathbf{F} = q ( \mathbf{E} + k \mathbf{v}\times\mathbf{B} ), | ||
</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]]. | The quantity ''q'' is the electric charge of the particle and '''v''' is its velocity. The vector '''B''' is the [[magnetic induction]]. 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 ''v B'' sin α with α the angle between '''v''' and '''B'''). | ||
The electric field '''E''' is given by | The electric field '''E''' is given by | ||
:<math> | :<math> | ||
| Line 25: | Line 25: | ||
The factor ''k'' has the same meaning as before. | 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'''. | The operator '''∇''' acting on ''V'' gives the [[gradient]] of ''V'', while '''∇''' × '''A''' is the [[curl]] of '''A'''. | ||
Note that the Lorentz force does not depend on the medium; the electric force does not contain the [[electric susceptibility]] ε and the magnetic force does not the contain [[magnetic permeability]] μ. We see here that the electric field '''E''' and the magnetic induction '''B''' are analogous.<ref>In older literature one sometimes finds electromagnetic equations formulated such that '''E''' and '''H''' are analogous.</ref> | |||
If '''B''' is static (does not depend on time) then '''A''' is also static and | If '''B''' is static (does not depend on time) then '''A''' is also static and | ||
Revision as of 08:50, 1 June 2008
In physics the Lorentz force is the force on an electrically charged particle that moves through a magnetic field and an electric field; hence the Lorentz force has a magnetic and an electric component.
The strength of the magnetic component 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 induction, and the sine of the angle between the vectors v and B. The direction of the magnetic component is given by the right hand rule: put your right hand along v with fingers pointing in the direction of v and the open palm toward the vector B. Stretch the thumb of your right hand, then the Lorentz force is along it, pointing from your wrist to the tip of your thumb.
The electric component is equal to q E (charge of the particle times the electric field), which must be added vectorially to the magnetic component of the Lorentz force in order to obtain the total force.
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. 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 v B sin α with α the angle between v and B). The electric field E is 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.
Note that the Lorentz force does not depend on the medium; the electric force does not contain the electric susceptibility ε and the magnetic force does not the contain magnetic permeability μ. We see here that the electric field E and the magnetic induction B are analogous.[2]
If B is static (does not depend on time) then A is also static and
Non-relativistically, the electric field E may be absent (zero) while B is static and non-zero; the Lorentz force is then 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.[3]
In special relativity the Lorentz force transforms as a vector under a Lorentz transformation, giving a linear combination of E and B, because relativistically the fields E and B are components of the same second rank tensor and do not have an independent existence.[4]
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).
- ↑ In older literature one sometimes finds electromagnetic equations formulated such that E and H are analogous.
- ↑ 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. It is of interest to note that James Clerk Maxwell gave the expression for the Lorentz force already in his historic memoir of 1865. (J. Clerk Maxwell, A Dynamical Theory of the Electromagnetic Field, Phil. Trans. Roy. Soc., vol. 155, pp. 459 - 512 (1865) online)
- ↑ J. D. Jackson, Classical Electrodynamics, John Wiley, New York, 2nd ed. (1975), p. 553