Magnetic flux density (Индукция магнитного поля)
B – main vector in theory of magnetic field bc we can measure value of this vector in real physical experiment. Measure by force which this vector induces, when we have moving charge particle.
Q – charge, v – velocity.
F
lux:
The Gauss law for the magnetic field:
There are no magnetic charges in the nature.
Biot-Savart’s law (Закон Био-Савара)
Flux density induced by the current sources may be expressed as:
Cross
product
demonstrates that the direction of the flux density will be normal to
the direction of the current and direction from the point or small
volume and the point where we’re going to find the magnetic flux
density.
Important difference between magnetic and electric field. The Coulomb’s law may be applied to very small volume with charges. The Biot-Savart’s law generally saying can’t be applied because point charges or charges which are distributed over the volume exist in reality, but the infinity small current element does not exist by itself. All currents should be closed. If we shall try to calculate the magnetic field which is induced by a part of the coil or part of the current system, we can get absolutely absurd result, which does not satisfy main magnetic laws.
For line current:
Biot-Savart’s law should be applied only to the whole magnetic system!
Ampere’s law (Закон полного тока)
The magnetic field intensity:
– true only for the isotropic media
– magnetic permeability (scalar)
Ampere’s law universal. It describes both isotropic and anisotropic media.
Integral form of the Ampere’s law:
I – is the current crossing the surface limited by the contour.
Интеграл берется по контуру в пространстве. Ток не течёт по контуру, ток проходит сквозь поверхность, ограниченную контуром.
Differential forms:
Scalar magnetic potential (Скалярный магнитный потенциал)
Main relations:
The main sources of the magnetic field are the currents so if currents flow, then the magnetic field is induced. Of course, magnetic field also induced by permanent magnets.
In general case the H – field is not potential:
Currents should be included in the system which contains magnetic fields.
Nevertheless, outside the space with currents:
It is possible to introduce the scalar potential:
Um – magnetic potential. The unit is – A (Ampere)
The cut in the space (Разрез в пространстве)
How to resolve the problem which does not allow to consider the magnetic field as the potential.
The yellow ring is wire with the current. Current induces magnetic field around itself. So if we shall go around the wire we immediately will have a ratio but it is forbidden. If so that we can’t use a scalar magnetic potential at all. In such case we should make the cut in the space.
Black line – cut line. Starts at the infinity, goes around the wire and then go to the infinity.
We shall consider that our problem domain is the space around this line.
Now any closed loop will not contain a current inside. In such case we can use a definition of the scalar magnetic potential.
T
he
cut in the space is necessary to do if we want to consider the
systems with the current.