# Electricity heating law

The electricity heat law (also First Joule's law according to James Prescott Joule or Joule-Lenz law according to Joule and Emil Lenz ) states that an electrical current in an electrical conductor generates the thermal energy  through continuous conversion of electrical energy that is taken from the conductor ${\ displaystyle Q _ {\ mathrm {W}}}$ ${\ displaystyle E _ {\ mathrm {el}}}$

${\ displaystyle Q _ {\ mathrm {W}} = E _ {\ mathrm {el}} = P \ cdot t}$

with the electrical power and the duration , - or with variable power ${\ displaystyle P}$${\ displaystyle t}$

${\ displaystyle Q _ {\ mathrm {W}} = E _ {\ mathrm {el}} = \ int _ {0} ^ {t} P \ mathrm {d} \ tau}$

The cause of the warming when a current flows is described in Electrical resistance # The electrical resistance in the particle model .

The terms Joule heat and electricity heat are not used uniformly, partly in the sense of energy, partly of power.

## Electricity heat in an electrical line

A current is preferably carried in an electrical line . The electrical power is always an active power in connection with heat generation . It results from the current strength  and the voltage dropping along the conductor as a result of the conductor current (the formula symbols apply to direct quantities as well as to the rms values of alternating quantities) ${\ displaystyle I}$ ${\ displaystyle U}$

As a result of a current flow, the
double helix is heated to the point of red heat
${\ displaystyle P = U \ cdot I}$

Since the voltage is created by the ohmic resistance of the conductor, Ohm's law applies${\ displaystyle R}$

${\ displaystyle U = R \ cdot I}$

This means that the heating (e.g. in an electrical line, a transformer or a heating resistor ) increases with the square of the current strength

${\ displaystyle Q _ {\ mathrm {W}} = I ^ {2} \ cdot R \ cdot t = {\ frac {U ^ {2}} {R}} \ cdot t \ ;.}$

If the generation of the heat is desired, the heat is referred to as electric heat , otherwise as electricity heat loss or ohmic loss.

The thermal energy primarily leads to a heating of the conductor by a temperature difference

${\ displaystyle \ Delta \ vartheta = {\ frac {Q _ {\ mathrm {W}}} {C _ {\ vartheta}}}}$

with the heat capacity . At constant power, it increases linearly with time. This means that the temperature also increases linearly with time until another process is superimposed. ${\ displaystyle C _ {\ vartheta}}$${\ displaystyle Q _ {\ mathrm {W}}}$

Since the conductor becomes warmer than its surroundings, it transfers thermal energy through thermal conduction , thermal radiation or convection . If the energy supply is constant and constant, a state of equilibrium is established at an increased temperature, in which the emitted heat flow ( heat per time span, i.e. a thermal power ) equals the consumed electrical power: ${\ displaystyle {\ dot {Q}} _ {W}}$

${\ displaystyle {\ dot {Q}} _ {\ mathrm {W}} = {\ frac {\ Delta Q _ {\ mathrm {W}}} {\ Delta t}} = P \ ,.}$

A temperature difference arises in the case of a surface involved in heat transport and a heat transfer coefficient ${\ displaystyle A}$ ${\ displaystyle \ alpha}$

${\ displaystyle \ Delta \ vartheta = {\ frac {P} {\ alpha \, A}} \ ,.}$

In general, bodies have such a thermal inertia that when the current is stationary, the temperature difference is established as a constant variable, even when heated by alternating current . A temperature or brightness fluctuation with twice the frequency of the alternating current can only be observed with a very small ratio of mass to surface, as in the case of the double helix shown.

## Current heat in the electric flow field

If a conductive substance distributed over a larger volume is flowed through by a current, a current of strength flows through a surface element${\ displaystyle \ mathrm {d} A}$

${\ displaystyle \ mathrm {d} I = J \; \ mathrm {d} A}$,

on its way along a path element a tension ${\ displaystyle \ mathrm {d} s}$

${\ displaystyle \ mathrm {d} U = E \; \ mathrm {d} s = \ rho \, J \ mathrm {d} s}$

falls, producing heat. It stands for the electrical current density, for the electrical field strength, for Ohm's law, for the specific electrical resistance (reciprocal value of electrical conductivity ). ${\ displaystyle J}$${\ displaystyle E}$${\ displaystyle E = \ rho \; J}$${\ displaystyle \ rho}$ ${\ displaystyle \ sigma}$

The loss of electric power results in the volume element to ${\ displaystyle \ mathrm {d} V = \ mathrm {d} A \ cdot \ mathrm {d} s}$

${\ displaystyle \ mathrm {d} P = \ mathrm {d} U \ cdot \ mathrm {d} I = \ rho \; J ^ {2} \ mathrm {d} V}$ .

Metallic conductors have a specific electrical resistance that is largely independent of current (but temperature-dependent). In semiconductors is not constant. In superconductors is , there arises no current heat. ${\ displaystyle \ rho}$${\ displaystyle \ rho = 0}$

The total amount of current heat loss in a current-carrying conductor is generally calculated from the volume integral

${\ displaystyle P = \ iiint _ {V} \ rho \ J ^ {2} \ mathrm {d} V}$ .

If is constant, this factor can be taken before the integral. In a homogeneous conductor, for example in a long wire through which a direct current flows, the current distribution is independent of the location, so that for such an object through which an integral current flows, the power loss is based on the macroscopic formula given above ${\ displaystyle \ rho}$

${\ displaystyle P = R \; I ^ {2}}$

leads. In the case of a more complicated geometric design with non-uniform current distribution, this must be done, for. B. calculated using the finite element method in order to determine the power loss and the macroscopic resistance of the conductor.

A current-dependent resistance can be found in materials with non-constant resistivity . The calculation of the electricity heat loss through is then valid in this way. ${\ displaystyle R (I)}$${\ displaystyle P = R (I) \ cdot I ^ {2}}$

## literature

• Dieter Meschede (Ed.): Gerthsen Physik . 22., completely reworked. Edition. Springer, Berlin a. a. 2004, ISBN 3-540-02622-3 , pp. 321 .

## Individual evidence

1. Ludwig Bergmann, Clemens Schaefer: Textbook of Experimental Physics, Volume II, Electricity and Magnetism . de Gruyter, 1971, p. 150
2. Dieter Zastrow: Electrical engineering: A basic textbook . Vieweg + Teubner, 2010, p. 59
3. Ulrich Harten: Physics: An introduction for engineers and natural scientists . Springer, 2014, p. 186
4. Andreas Binder: Electrical machines and drives: Fundamentals, operating behavior . Springer, 2012, p. 430
5. Günther Lehner: Electromagnetic field theory for engineers and physicists . Springer, 2010, p. 111
6. ^ Wilhelm Raith: Electromagnetism . de Gruyter, 2006, p. 109