Ampère's circuital law
The Ampère law ( flux rate, Ampere's law ) is a law of electrodynamics and Maxwell's equations. It was discovered by André- Marie Ampère, and forms for the magnetism of the analogy to the law of induction.
- 2.1 Interpretation of the integral
- 2.2 the magnetic field of the coil
- 2.3 Biot- Savart
Mathematical formulation
Integrals of the form
Law is the line integral of the magnetic field to form a closed curve in relation to the current that flows through the area enclosed by the cam surface.
The integral form of the law is ( assuming constant current density is not required):
In which
Differential form
Equivalently, the differential form
Is the magnetic field strength, which is the magnetic flux density without consideration of paramagnetic and diamagnetic contributions by the medium ( in the vacuum is considered ). Is analogous to the current density (current per unit area ) and the same size regardless of the induced by paramagnetic and diamagnetic current effects. is the rotation operator.
The equivalence of integral and differential form is proved by Stokes' theorem.
Maxwell's extension
James Clerk Maxwell noticed that the so formulated Ampère law when charging a capacitor initially not the case, and concluded the incompleteness of the law. To solve the problem, he developed the concept of the displacement current and placed on a general form which is one of the four Maxwell's equations. It is given in integral form by
Here all sizes as above. is the electric flux density, namely, the electric field strength plus the fields generated by polarization.
Application
Simply put, the Ampère law says the following: An electric current produces a magnetic field proportional him whose direction with that of the river forms a right-handed screw. See also: Right -hand rule.
Interpretation of the integral
The integral formulation
Can be interpreted as follows:
To any shaped head - be it a wire, a metal plate, coil, or even just a very small piece of a larger conductor - you put a mentally (measuring) framework. This framework can be of any shape, such as a rectangle or a circle of any desired size. When a current flows through the conductor, it creates a magnetic field. If you go along the frame and added for each small piece of the frame, the component of the magnetic field in the direction of the small head piece is then obtained when the frame is surrounded, a sum that is the current through the conductor proportional.
Magnetic field of the coil
With direct application of the Ampère law for determining a magnetic field is obtained usually only solutions for simplified cases, for example if it is assumed that the magnetic field of a coil is anywhere along or opposite to the axis of the coil and the inside homogenously, but indefinitely for the coil applies.
You have such a coil having turns each way. You put a rectangular frame through the coil, the upper side is of length in the coil, and its right and left sides are infinitely long. These pages, the magnetic field is perpendicular after the adoption of the component in the direction of the frame is therefore zero. The lower side is infinitely far away, where the magnetic field must be zero. There remains the integral only the upper side, where the component of the magnetic field is exactly parallel. So the following applies:
Whereby there is determined the magnitude of the magnetic field in the coil.
Biot- Savart
Simple cases such as the above are not always commensurate to describe induced currents magnetic fields. In order to treat arbitrary current distributions, the Biot- Savart law provides further statements. It can be derived from the Maxwell's equations, that is, also, that for the non-obvious proof of Maxwell's extension of the Ampère law to the displacement current is necessary.