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2020 | OriginalPaper | Chapter

1. Potential and Current Density Distribution

Author : Jürgen Donnevert

Published in: Maxwell´s Equations

Publisher: Springer Fachmedien Wiesbaden

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Abstract

The subject of the first chapter is the electric field accompanying a current that is constant in time. The terms current density, potential and electric field strength are explained. The gradient of a scalar field and the differential operator Nabla or ∇ are introduced and the formulas for cartesian, cylindrical and spherical coordinates are derived. Kirchhoff's laws for the field of electric current densities are explained and the ohmic law for this field. Finally, the energy required to move a charge in the electric field is calculated.

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next chapter Electrostatics

V = Volt is the unit of measurement defined in the International System of Units (SI) for electrical voltage. It was named after the Italian physicist Alessandro Volta. The capital letter “\( {\text{V}} \)” is used as the unit character.

The volt is a derived SI unit. With the SI base units Watt (W) and Ampere (A), we get

\( 1{\text{V}} = 1\frac{\text{W}}{{{\text{A}}}} = 1\frac{{{\text{N}}\,{\text{m}}}}{{{{{\text{A}}}}\,{\text{s}}}} = 1\frac{{{\text{kg}}\,{\text{m}}^{2} }}{{{{{\text{A}}}}\,{\text{s}}^{3} }} = 1\frac{{{\text{kg}}\frac{{{\text{m}}^{2} }}{{{\text{s}}^{2} }}}}{{{{{\text{{\text{A}}}}}}\,{\text{s}}}} = 1\frac{{{\text{kg}} \cdot \frac{{{\text{m}}^{2} }}{{{\text{s}}^{2} }}}}{{\text{A}} \cdot {\text{s}}} \)

As this definition can hardly be used for calibration purposes as an exact reference, since 1990 the unit Volt is determined by the Josephson effect and the Josephson constant. The unit Ampere (A) is introduced in Sect. 3.2.1.

Historically, the definition of a Volt was derived from the Weston normal element. This element supplies an electrical voltage of exactly 1.01865 V at a temperature of 20°C.

A scalar field is a function that assigns a real number (scalar) to each point in a space.

Ampere (A) unit is one of the four basic units of the SI international system of units. The definition of this unit is discussed in Sect. 3.2.1.

A vector field is a function that assigns a vector to each point in a space.

A scalar is a mathematical quantity that is characterized solely by a numerical value.

Lat. gradus = step.

\( 360^{ \circ } = 2 \cdot \pi \,rad \).

LU = Length unit.

For \( r = 0 \), the \( \alpha \)-component is obviously not defined.

For \( r = 0 \) and \( \vartheta = 0 \) the \( \alpha \)- and \( \vartheta \)-components are obviously not defined.

Gustav Robert Kirchhoff, German physicist, *1824, †1887.

Unit of the specific conductivity\( \left[\upsigma \right] \): \( \frac{{{\text{S}} \cdot {\text{m}}}}{{{\text{mm}}^{2} }} \) but mostly \( \frac{\text{S}}{\text{m}} \) (S = Siemens = \( \frac{1}{\Omega } \)).

Ohm’s law.

Unit \( \left[ \kappa \right]\frac{{\Omega \cdot {\text{mm}}^{2} }}{\text{m}} \) in most case \( \Omega \cdot {\text{m}} \).

The current flows from places with higher potential to places with lower potential.

power \( = \) Energy/time = energy flow, unit of power: W (Watt).

\( P \) is the power dissipation in the conductor.

Unit \( \left[ p \right] = \frac{W}{{m^{3} }} = \frac{V \cdot A}{{m^{3} }} \).

Charles Augustin de Coulomb, French physicist and engineer, *1736, †1806.

The definition of the unit of the current is explained in Sect. 3.2.1. Since the unit Ampere is a base unit, the unit Coulomb is a derived unit.

In physics, often the energy is referred to as eV.

Title: Potential and Current Density Distribution
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_1