Ampere's law
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In physics, Ampère's law, discovered by André-Marie Ampère, relates the circulating magnetic field in a closed loop to the electric current passing through the loop. It is the magnetic equivalent of Faraday's law of induction,
Original Ampère's law
In its original form, Ampère's law relates the magnetic field B to its source, the current density J:
- :[\oint_C \frac} \cdot d\mathbf = \int\!\!\!\!\int_S \mathbf \cdot d \mathbf = I_} ]
- where
- :[\oint_C] is the closed line integral around contour (closed curve) [C].
- :[\mathbf] is the magnetic field in Tesla,
- :[d\mathbf] is an infinitesimal element (differential) of the contour [C],
- :[\mathbf] is the current density (in amperes per square meter) through the surface S enclosed by contour C
- :[ d \mathbf \!\ ] is a differential vector element of surface area A, with infinitesimally small magnitude and direction normal to surface S
- :[I_} \!\ ] is the current enclosed by the curve [C], or strictly, the current that penetrates surface [S],
- :[\mu_0 = 4 \pi \times 10^ ] is the permeability of free space (in henries per meter),
Equivalently, the original equation in differential form is
- [\nabla \times \mathbf = \mathbf ]
The magnetic field H in linear media, is related to the magnetic flux density B (in teslas) by
- [ \mathbf \ = \ \mu \mathbf ]
Corrected Ampère's law: the Ampère-Maxwell equation
James Clerk Maxwell noticed a logical inconsistency when applying Ampère's law to a charging or discharging capacitor. If surface [S] passes between the plates of the capacitor, and not through any wires, then [\mathbf = 0] even though [\oint_C \mathbf \cdot d\mathbf\ne 0]. He concluded that this law had to be incomplete. To resolve the problem, he came up with the concept of displacement current and made a generalized version of Ampère's law which was incorporated into Maxwell's equations.The generalized law, as corrected by Maxwell, takes the following integral form:
- [\oint_C \mathbf \cdot d\mathbf = \iint_S \mathbf \cdot d \mathbf + \iint_S \mathbf \cdot d \mathbf]
- [ \mathbf \ = \ \varepsilon \mathbf]
This Ampère-Maxwell law can also be stated in differential form:
- :[\nabla \times \mathbf = \mathbf + \frac}]
With the addition of the displacement current, Maxwell was able to postulate (correctly) that light was a form of electromagnetic wave. See Electromagnetic wave equation for a discussion on this important discovery.
See also
- Maxwell's equations
- Biot-Savart law
- Faraday's law of induction
- Gauss's law
- Electric current
- Vector calculus
- Stokes' theorem
References
External links
- [MISN-0-138 Ampere's Law] (PDF file) by Kirby Morgan for [Project PHYSNET].
- [MISN-0-145 The Ampere-Maxwell Equation; Displacement Current] (PDF file) by J.S. Kovacs for Project PHYSNET.
- [The Ampère's Law Song] (PDF file) by Walter Fox Smith.
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