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2020 | OriginalPaper | Chapter

3. The Stationary Magnetic Field

Author : Jürgen Donnevert

Published in: Maxwell´s Equations

Publisher: Springer Fachmedien Wiesbaden

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Abstract

Chapter 3 deals with temporally constant magnetic fields. A parameter of this field is the magnetic flux density. In the first step, the relationship between magnetic flux density and the force exerted on a conductor through which current flows is derived. Furthermore, it is shown that a voltage will be generated at the ends of a conductor loop moving in a magnetic field in such a way that the concatenated magnetic flux changes in time. The differential form of the law of Ampère is introduced and the vector operator rotation. For Cartesian, cylindrical and spherical coordinate systems the calculation formulas are derived. The magnetic vector potential is introduced and finally the law of Biot-Savart.

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previous chapter Electrostatics

next chapter Time-Varying Electric and Magnetic Fields

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The magnetic flux density is also called magnetic induction.

Also called coil, choke, reactor.

T = Tesla, Nicola, Croatian-American electrical engineer and physicist, *1856, †1943.

Wb, Wilhelm Eduard, German physicist, *1804, †1891.

Hendrik Antoon Lorentz, Dutch mathematician and physicist,*1853, †1928.

In Fig. 3.7, a cross marks the end of the vector of arrow \( \vec{B} \) that points into the drawing plane.

Latin: inducere = introduce.

Permanent magnets are manufactured by pressing crystalline powder into shape under the influence of a strong magnetic field. The crystals align with their preferred magnetization axis in the direction of the magnetic field. Then, the presslings are sintered at a temperature of more than 1000° C, whereby the magnetic field is lost. After the magnets have cooled down, the magnetic field is restored by a sufficiently strong magnetization pulse. The magnetic field lines made visible by iron filings in Fig. 3.8 close within the magnetic material.

Hall, Edwin H., American physicist, *1855, †1938.

André-Marie Ampère, French physicist and mathematician, *1775, †1836.

Rogowski, W., German electrical engineer, *1881, †1947.

The cause of the time-varying, concatenated magnetic flux \( {\text{d}}\Phi _\text{con} /\text{d}t \) in Eq. (3.40) is the current, which increases with time during the switch-on process. The time-varying, concatenated magnetic flux in Eq. (3.22) is caused by the changing cross section of the conductor loop. In order for the induced voltage to arise, it is indifferent whereby the time-varying, concatenated flux arises. In detail, time-varying magnetic fields are dealt with in Sect. 4.1.

The sign of the voltage \( v_{12} \) is irrelevant in this case.

From 20 May 2019, a new definition of the Ampere unit in the International System of Units (SI) applies. The new definition of the unit Ampere is based on the precisely defined value of the elementary charge \( e \). It was possible to change the definition, as individual charge can now be counted well. According to the new definition, the unit \( 1 \,{\text{A}} \) is present if \( 1{,}602 176 634 \cdot 10^{9} \) elementary charges flow through the conductor during 1 s. As a result of this definition, the field constants \( \mu_{0} ,\,\varepsilon_{0} \) and the characteristic impedance of vacuum are now derived variables subject to uncertainty (Source: Wikipedia).

The surface element can therefore be approximated by a rectangle.

For displacement current, see Sect. 4.2.

Sir Stokes, Gabriel, British physicist and mathematician, *1819, †1903.

For the vector potential, the same designation \( \vec{A} \) is used as for the surface vector \( \vec{A} \). From the context, it is always clear what is the meaning of \( \vec{A} \). There will be no confusion.

See [3] p. 197.

Biot, Jean-Baptiste, French physicist and astronomer, *1774, †1862

Savart, Félix, French physicist, *1791, †1841.

Title: The Stationary Magnetic Field
Author: Jürgen Donnevert
Publisher: Springer Fachmedien Wiesbaden
Book: Maxwell´s Equations
Print ISBN: 978-3-658-29375-8

Electronic ISBN: 978-3-658-29376-5

Copyright Year: 2020
DOI: https://doi.org/10.1007/978-3-658-29376-5_3