# Examination of electromagnetic powers with the example of a Faraday disc dynamo

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## Abstract

This paper studies the mathematical form of electromagnetic powers and their influence on the balance of energy by using the example of a Faraday disc. First, two forms of energy (and balances thereof) are discussed. These employ different forms of powers, which can be distinguished w.r.t.  their physical origins and their interpretations in context with the notions of supply and production. The stationary Faraday disc experiment is modeled following the description by Kovetz (Electromagnetic theory, Oxford University Press, Oxford, 2000). Concepts for formulating the electromagnetic field equations for the rotating disc are discussed, and the corresponding approximate analytical solutions are presented. Based on the obtained electromagnetic fields, the powers of the disc are analyzed for a stationary process. The conversion of mechanical power to heating and electromagnetic powering of an external resistor is explained. The paper concludes with the computation of the time evolution of the angular velocity for a magnetically induced breaking process of the disc.

## Keywords

Faraday disc Rotating body Electromagnetic power Magnetic breaking

## General quantities

$$\xi$$

Radial coordinate in cylindrical coordinates (m)

$$\varphi$$

Azimuthal angle in cylindrical coordinates (1)

z

Axial coordinate in cylindrical coordinates (m)

$$\{\varvec{e}_\xi , \varvec{e}_\varphi , \varvec{e}_z\}$$

Physical basis for cylindrical coordinates (1)

t

Time (s)

## Classical quantities

$$\varvec{v}$$

Velocity field (m/s)

v

Norm of the velocity field (m/s)

c

Speed of light in vacuum (m/s)

$$\gamma$$

Lorentz factor (1)

$$\rho$$

Density of mass (kg/m$$^3$$)

$$\varvec{t}$$

Traction vector (Pa)

$$\varvec{f}$$

Specific body force without electromagnetic influence (m/s$$^2$$)

$$\varvec{q}$$

Heat flux without electromagnetic influence (W/m$$^2$$)

$${\hat{r}}$$

Specific heating without electromagnetic influence (W/kg)

u

Specific internal energy without electromagnetic contribution (J/kg)

## Electromagnetism

$$\mu _0$$

Vacuum permeability (N/A$$^2$$)

$$\epsilon _0$$

Vacuum permittivity (T m/A)

$${\varvec{B}}$$

Magnetic flux density (T)

$${\varvec{E}}$$

Electric field (V/m)

$$\varvec{\mathfrak {E}}$$

Electromotive intensity (V/m)

$$\varvec{H}$$

Potential of total electric current (A/m)

$$\varvec{\mathfrak {H}}$$

Potential of free electric current (A/m)

$${\varvec{M}}$$

Minkowski magnetization (A/m)

$${\varvec{D}}$$

Potential of total electric charge (C/m$$^2$$)

$$\varvec{\mathfrak {D}}$$

Potential of free electric charge (C/m$$^2$$)

$${\varvec{P}}$$

Polarization (C/m$$^2$$)

q

Total electric charge density (C/m$$^3$$)

$$q^\mathrm {f}$$

Density of free electric charge (C/m$$^3$$)

$$q^\mathrm {b}$$

Density of bound electric charge (C/m$$^3$$)

$$\varvec{J}$$

Total electric current density (A/m$$^2$$)

$$\varvec{J}^\mathrm {f}$$

Free electric current density (A/m$$^2$$)

$$\varvec{j}^\mathrm {f}$$

Diffusive part of the free electric current density (A/m$$^2$$)

$$\varvec{J}^\mathrm {b}$$

Bound electric current density (A/m$$^2$$)

V

Electric potential (V)

## Energy and power relations

$$U^\mathrm {cl.}$$

Classical energy without direct electromagnetic contribution (not conserved) (J)

$$U^\mathrm {tot.}$$

Total energy with direct electromagnetic contribution (conserved) (J)

$${\tilde{A}}^\star$$

Mechanical supply power without electromagnetic contribution (supply) (W)

$${\tilde{Q}}^\star$$

Thermal power without electromagnetic contribution (supply) (W)

$${\tilde{E}}^\star$$

Electromagnetic energy flux (supply) (W)

$$A^\star _\text {(EM)}$$

Electromagnetic production part of the mechanical power (W)

$$A^\star$$

Mechanical power with electromagnetic production (W)

$$Q^\star _\text {(EM)}$$

Electromagnetic production part of the thermal power (W)

$$Q^\star$$

Thermal power with electromagnetic production (W)

$$H^\star _\text {(EM)}$$

Combined electromagnetic production power (W)

$$h^\text {(EM)}$$

Density of combined electromagnetic production power (W/m$$^3$$)

$$u^\text {(EM)}$$

Density of electromagnetic energy (J/m$$^3$$)

$$\varvec{q}^\text {(EM)}$$

Electromagnetic energy flux density (W/m$$^2$$)

$$\varvec{g}^\text {(EM)}$$

Density of electromagnetic linear momentum (kg/(m s$$^2$$))

$$\varvec{\sigma }^\text {(EM)}$$

Electromagnetic stress tensor (Pa)

## Faraday disc experiment

H

Height of the disc (m)

$$R_\mathrm {A}$$

Inner radius of the disc (m)

$$R_\mathrm {B}$$

Outer radius of the disc (m)

$${\mathfrak {R}}$$

Strength of the external resistor (W)

$${\mathfrak {R}}^\mathrm {disc}$$

Electrical resistance of the disc (W)

$$B_0$$

Strength of the external magnetic flux density (T)

$$\varvec{B}_0$$

External magnetic flux density (T)

$$\omega _0$$

Stationary angular velocity of the disc (1/s)

$$\sigma$$

Electrical conductivity of the disc (S/m)

$$\kappa$$

Linear coefficient of the disc’s friction torque (N m s)

$$\rho _0$$

Density of mass of the disc (kg/m$$^3$$)

$$\varvec{T}^\text {(mech.)}$$

Transmitted mechanical torque (N m)

$$T^\text {(mech.)}$$

Strength of the transmitted mechanical torque (N m)

$$\varvec{T}^\text {(EM)}$$

Electromagnetic torque acting on the disc (N m)

$$\varvec{T}$$

Total torque acting on the disc (N m)

$${\mathfrak {I}}$$

Electric current strength (A)

$${\mathfrak {U}}$$

Terminal voltage of the disc (V)

$${\mathfrak {U}}^\mathrm {disc}$$

Voltage source of the disc (V)

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## Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

## Authors and Affiliations

1. 1.Institut für Mechanik, Kontinuumsmechanik und MaterialtheorieTechnische Universität BerlinBerlinGermany