Ampere's equation: Difference between revisions
imported>Paul Wormer (New page: In physics, more particularly in electrodynamics, '''Ampère's equation''' describes the force between two current-carrying wires. We will list two common cases in two common syste...) |
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In [[physics]], more particularly in [[electrodynamics]], '''Ampère's equation''' describes the force between two current-carrying wires. | 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,<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). pp. 85-88 </ref><ref>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) </ref> 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<ref>J. D. Jackson, ''Classical Electrodynamics'', 2nd edition, John Wiley, New York (1975) pp. 172-173</ref> 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, | |||
:<math> | :<math> | ||
k = \begin{cases} | k = \begin{cases} | ||
{\displaystyle \frac{\mu_0\mu_r}{4\pi}} & \hbox{for SI units}\\ | {\displaystyle \frac{\mu_0\mu_r}{4\pi}} & \hbox{for SI units}\\ | ||
{\displaystyle \frac{1}{c^2}} & \hbox{for Gaussian (rationalized cgs) units} | {\displaystyle \frac{1}{c^2}} & \hbox{for Gaussian (rationalized cgs) units}. | ||
\end{cases} | \end{cases} | ||
</math> | </math> | ||
Here μ<sub>0</sub> is the [[magnetic constant|magnetic permeability]] of the vacuum and μ<sub>''r''</sub> is the relative magnetic permeability. The quantity ''c'' is the velocity of light in the vacuum. | Here μ<sub>0</sub> is the [[magnetic constant|magnetic permeability]] of the vacuum and μ<sub>''r''</sub> is the relative magnetic permeability. The quantity ''c'' is the velocity of light in the vacuum (299 792 458 m s<sup>−1</sup> exactly) . | ||
==Two straight infinite and parallel wires== | ==Two straight infinite and parallel wires== | ||
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F = 2 k \frac{i_1 i_2}{r}.\, | F = 2 k \frac{i_1 i_2}{r}.\, | ||
</math> | </math> | ||
The force is attractive if the currents run in the same direction and repulsive if they flow in opposite direction. | 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 (unit)|ampere]]). If ''r'' = 1 m, F = 2 ⋅ 10<sup>−7</sup> 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 μ<sub>0</sub> = 4π ⋅ 10<sup>−7</sup>. | ||
[[Image:Ampere equation.png|right|thumb|300px|Illustration to the force between two closed current-carrying loops]]. | |||
==Two 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 | 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== | |||
<references /> | |||
[[Category: CZ Live]] | [[Category: CZ Live]] | ||
[[Category: Physics Workgroup]] | [[Category: Physics Workgroup]] | ||
Revision as of 07:44, 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
The force per unit of length between two straight parallel wires a distance r apart 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