Biot–Savart law

The Biot- Savart law establishes a connection between the magnetic field strength H and the electric current density J, and allows the calculation of spatial magnetic field strength distributions based on the knowledge of the spatial current distributions. Here, the bill is treated as a relationship between the magnetic flux density B and electric current density J.

Note: In special cases, these are, for example, magnetic linear and isotropic materials or even in a vacuum exists between the magnetic flux density and the field strength of the relation B = μ H with a constant proportionality factor μ, the magnetic conductivity or permeability. In the general case (eg, magnets ), however, may be a function of the magnetic field strength or the spatial orientation of the magnetic permeability μ, which can result in significantly more complex and no longer analytically representable may contexts.

It was named this law after the two French mathematicians Jean -Baptiste Biot and Félix Savart. It is next to the Ampère law is one of the basic laws of magnetostatics, a section of the electrodynamics, dar.

Formulation

A conductor of infinitesimal length dl at r ', which is traversed by a current I is generated at the point r, the magnetic flux density dB (using the cross-product ):

The total magnetic flux density is obtained by summing all existing infinitesimal pieces, ie by integrating. The resulting path integral can be using by

Forming in a volume integral, where J is the electrical current density. Thus we obtain the integral form of the biot - Savart law:

Similar these two formulas ( with currents instead charges ) the Coulomb's law, which describes the shape of the electric field as a function of a charge distribution.

In the above two formulas has been neglected here is that the conductors have a finite cross section. In many real applications, however, this is compared to the extent of the magnetic field is actually of no importance. Another inaccuracy is that propagates the contribution of a charge at one place to the magnetic field at a different location with the speed of light. The corresponding retardation effect is not taken into account in the Biot- Savart law. It is therefore only for stationary flows strictly valid for point charges and to a good approximation, provided that their speed is small compared to the speed of light.

Derivation of the Maxwell equations

Below retardation effects are neglected and considered the time-constant case in the form of magnetostatics. From the Maxwell equations then follows the Poisson equation for the vector potential:

With the solution:

This yields for the magnetic flux density:

By means of the formulas for the application of the operator to a red product of scalar function and the vector function, and

Follows the final result when one considers that in the integral only acts on the variable r and not for R '. Frequently, it is advantageous to calculate the vector potential and the magnetic flux density thereof.

The same conclusion is by using the Helmholtz decomposition and the Maxwell equations for the static case.

Application

Circular conductor loop

The magnitude of the magnetic field strength of a circular conductor loop can be closed by means of the specified Biot - Savart law to the symmetry axis perpendicular to the conductor loop:

Here, the radius of the in the yz-plane conductor loop and the distance of the observation point from the center of the conductor loop. The field is directed in the x direction.

By substitution:

One obtains

In case the field of the conductor loop can be treated as a dipole field: for example, points on the x-axis, it shows large distances (large x) a - a function:

With the magnetic ( dipole ) moment ( current × area of the loop ).

Straight line cable

The calculation of the flux density B at a point P can be applied the following formula:

When applied to the angle obtained

Was being used

Is the unit vector perpendicular to the plane in which P and the conductors are ( direction according to the rules of the cross product ).

For the magnetic field of a straight, infinitely long waveguide along the z - axis results, the above line integral of

Wherein the vertical distance to the Z -axis and the unit vector with respect to the angle of the associated cylinder coordinates. The magnetic field thus forming concentric circles around the conductor and decreases in inverse proportion to the distance from the conductor. In vector form we obtain:

Loop coil

After the round coil, the coil frame is ( with N turns ) the variant most commonly used. The formula for the magnetic field in the center can be derived from the formula for the line conductor by treating the straight portions of the coil as a line head.

With

For the magnetic field on the x- axis at a large distance from the coil, results in:

So again a dependency as in the dipole. With magnetic moment written: