Ampere's equation: Difference between revisions
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Let <math>\scriptstyle i_1</math> and <math>\scriptstyle i_2</math> be electric currents constant in time. They run in separate loops, see figure on the right, where all quantities are defined. The total force between two loops is given by the double path integral over the loops | Let <math>\scriptstyle i_1</math> and <math>\scriptstyle i_2</math> be electric currents constant in time. They run in separate loops, see figure on the right, where all quantities are defined. The total force between two loops is given by the double path integral over the loops | ||
:<math> | :<math> | ||
\mathbf{F}_{12} = -k i_1 i_2 \oint_{l_1}\oint_{l_2} \frac{ (d\mathbf{l}_1 \cdot d\mathbf{l}_2)\, \mathbf{r}_{12} }{|\mathbf{r}_{12}|^3} | \mathbf{F}_{12} = -k i_1 i_2 \oint_{l_1}\oint_{l_2} \frac{ (d\mathbf{l}_1 \cdot d\mathbf{l}_2)\, \mathbf{r}_{12} }{|\mathbf{r}_{12}|^3}. | ||
</math> | </math> | ||
==Reference== | ==Reference== | ||
<references /> | <references /> | ||
Revision as of 07:20, 19 February 2008
In physics, more particularly in electrodynamics, Ampère's equation describes the force between two infinitesimal elements of current-carrying wires. The equation is named for the early nineteenth century French physicist and mathematician André-Marie Ampère.
Rather than giving the infinitesimal equation, which is not without problems,[1][2] we will describe two common integrated cases: two straight wires and two closed loops. The equations for these systems are generally accepted and are in full agreement with experiment.
Equations[3] will be given in two common systems of electromagnetic units (SI and rationalized Gaussian) and to that end we define the constant k as follows,
Here μ0 is the magnetic permeability of the vacuum and μr is the relative magnetic permeability. The quantity c is the velocity of light in the vacuum (299 792 458 m s−1 exactly) .
Two straight, infinite, and parallel wires
Consider two wires, one carrying a current , the other Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle i_2} . Both currents are constant in time; the wires are infinite, straight, and parallel. If the wires are a distance r apart, the force (per unit of length) between them is,
The force is attractive if the currents run in the same direction and repulsive if they flow in opposite direction. This equation is used to define the SI unit of current A (ampere). If r = 1 m, F = 2 ⋅ 10−7 N (per meter of wire in vacuum), and the currents are equal, then the currents are equal to 1 A. Note that in SI units this implies that μ0 = 4π ⋅ 10−7.
.
Two loops
Let and be electric currents constant in time. They run in separate loops, see figure on the right, where all quantities are defined. The total force between two loops is given by the double path integral over the loops
Reference
- ↑ 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). pp. 85-88
- ↑ C. Christodoulides, Comparison of the Ampère and Biot-Savart magnetostatic force laws in their line-current-element forms, American Journal of Physics, vol. 56, pp. 357-362 (1988)
- ↑ J. D. Jackson, Classical Electrodynamics, 2nd edition, John Wiley, New York (1975) pp. 172-173