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# Joule's first law

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### Joule's first law

Joule heating, also known as ohmic heating and resistive heating, is the process by which the passage of an electric current through a conductor releases heat. The amount of heat released is proportional to the square of the current such that

$Q \propto I^2 \cdot R$

This relationship is known as Joule's first law. The SI unit of energy was subsequently named the joule and given the symbol J. The commonly known unit of power, the watt, is equivalent to one joule per second. Joule heating is independent of the direction of current, unlike heating due to the Peltier effect.

## Background

### History

Resistive heating was first studied by James Prescott Joule in 1841. Joule immersed a length of wire in a fixed mass of water and measured the temperature rise due to a known current flowing through the wire for a 30 minute period. By varying the current and the length of the wire he deduced that the heat produced was proportional to the square of the current multiplied by the electrical resistance of the wire.

### Microscopic description

Joule heating is caused by interactions between the moving particles that form the current (usually, but not always, electrons) and the atomic ions that make up the body of the conductor. Charged particles in an electric circuit are accelerated by an electric field but give up some of their kinetic energy each time they collide with an ion. The increase in the kinetic or vibrational energy of the ions manifests itself as heat and a rise in the temperature of the conductor. Hence energy is transferred from the electrical power supply to the conductor and any materials with which it is in thermal contact.

### Power loss and noise

Joule heating is referred to as ohmic heating or resistive heating because of its relationship to Ohm's Law. It forms the basis for the myriad of practical applications involving electric heating. However, in applications where heating is an unwanted by-product of current use (e.g., load losses in electrical transformers) the diversion of energy is often referred to as resistive loss. The use of high voltages in electric power transmission systems is specifically designed to reduce such losses in cabling by operating with commensurately lower currents. The ring circuits, or ring mains, used in UK homes are another example, where power is delivered to outlets at lower currents, thus reducing Joule heating in the wires. Joule heating does not occur in superconducting materials, as these materials have zero electrical resistance in the superconducting state.

Resistors create electrical noise, called Johnson–Nyquist noise. There is an intimate relationship between Johnson–Nyquist noise and Joule heating, explained by the fluctuation-dissipation theorem.

## Formulas and proof

### Direct current

The most general and fundamental formula for Joule heating is:

$P=VI$

where

• P is the power (energy per unit time) converted from electrical energy to thermal energy,
• I is the current traveling through the resistor or other element,
• V is the voltage drop across the element.

The explanation of this formula (P=VI) is:[1]

(Energy dissipated per unit time) = (Energy dissipated per charge passing through resistor) × (Charge passing through resistor per unit time)

When Ohm's law is also applicable, the formula can be written in other equivalent forms:

$P=IV=I^2R=V^2/R$

where R is the resistance.

### Alternating current

Main article: AC power

When current varies, as it does in AC circuits,

$P\left(t\right)=I\left(t\right)V\left(t\right)$

where t is time and P is the instantaneous power being converted from electrical energy to heat. Far more often, the average power is of more interest than the instantaneous power:

$P_\left\{avg\right\}=I_\left\{rms\right\}V_\left\{rms\right\}=I_\left\{rms\right\}^2R=V_\left\{rms\right\}^2/R$

where "avg" denotes average (mean) over one or more cycles, and "rms" denotes root mean square.

These formulas are valid for an ideal resistor, with zero reactance. If the reactance is nonzero, the formulas are modified:

$P_\left\{avg\right\} = I_\left\{rms\right\}V_\left\{rms\right\}\cos\phi = I_\left\{rms\right\}^2 \operatorname\left\{Re\right\}\left(Z\right) = V_\left\{rms\right\}^2 \operatorname\left\{Re\right\}\left(Y^*\right)$

where $\phi$ is the phase difference between current and voltage, $\operatorname\left\{Re\right\}$ means real part, Z is the complex impedance, and Y* is the complex conjugate of the admittance (equal to 1/Z*).

For more details in the reactive case, see AC power.

### Differential Form

In plasma physics, the Joule heating often needs to be calculated at a particular location in space. The differential form of the Joule heating equation gives the power per unit volume.

$P=\mathbf\left\{J\right\} \cdot \mathbf\left\{E\right\}$

Here, $\mathbf\left\{J\right\}$ is the current density, and $\mathbf\left\{E\right\}$ is the electric field.

## Reason for high-voltage transmission of electricity

In electric power transmission, high voltage is used to reduce Joule heating of the overhead power lines. The valuable electric energy is intended to be used by consumers, not for heating the power lines. Therefore this Joule heating is referred to as a type of transmission loss.

A given quantity of electric power can be transmitted through a transmission line either at low voltage and high current, or with a higher voltage and lower current. Transformers can convert a high transmission voltage to a lower voltage for use by customer loads. Since the power lost in the wires is proportional to the conductor resistance and the square of the current, using low current at high voltage reduces the loss in the conductors due to Joule heating (or alternatively allows smaller conductors to be used for the same relative loss).

## Applications

There are many practical uses of Joule heating. Some of the commonest are as follows.

• Soldering irons and cartridge heaters are very often heated by Joule heating.
• Electric fuses rely on the fact that if enough current flows, enough heat will be generated to melt the fuse wire.
• Thermistors and resistance thermometers are resistors whose resistance changes when the temperature changes. These are sometimes used in conjunction with Joule heating (also called self-heating in this context): If a large current is running through the nonlinear resistor, the resistor's temperature rises and therefore its resistance changes. Therefore, these components can be used in a circuit-protection role similar to fuses, or for feedback in circuits, or for many other purposes. In general, self-heating can turn a resistor into a nonlinear and hysteretic circuit element. For more details see Thermistor#Self-heating effects.

## Less-common applications

• Food processing equipment may make use of Joule heating in food production. In this case, the food material serves as an electrical resistor, and heat is released internally.[2]

## Heating efficiency

Main article: Electric heating

As a heating technology, Joule heating has a coefficient of performance of 1.0, meaning that every 1 watt of electrical power is converted to 1 watt of heat. By comparison, a heat pump can have a coefficient of more than 1.0 since it also absorbs additional heating energy from the environment, moving this thermal energy to where it is needed.

The definition of the efficiency of a heating process requires defining the boundaries of the system to be considered. When heating a building, the overall efficiency is different when considering heating effect per unit of electric energy delivered on the customer's side of the meter, compared to the overall efficiency when also considering the losses in the power plant and transmission of power.

## Hydraulic equivalent

In the energy balance of groundwater flow (see also Darcy's law) a hydraulic equivalent of Joule's law is used:[3]

$\left\{dE \over dx\right\} = \left\{v_x^2 \over K\right\}$

where:

$dE/dx$ = loss of hydraulic energy ($E$) due to friction of flow in $x$-direction per unit of time (m/day) – comparable to $Q/t$
$v_x$ = flow velocity in $x$-direction (m/day) – comparable to $I$
$K$ = hydraulic conductivity of the soil (m/day) – the hydraulic conductivity is inversely proportional to the hydraulic resistance which compares to $R$

## References

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