Joule heating

Joule heating, also known as resistive, resistance, or Ohmic heating, is the process by which the passage of an electric current through a conductor produces heat.
Joule's first law, also known as the Joule–Lenz law, states that the power of heating generated by an electrical conductor is proportional to the product of its resistance and the square of the current:
Joule heating affects the whole electric conductor, unlike the Peltier effect which transfers heat from one electrical junction to another.
History.
James Prescott Joule first published in December 1840, an abstract in the "Proceedings of the Royal Society", suggesting that heat could be generated by an electrical current. 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 immersed wire.
In 1841 and 1842, subsequent experiments showed that the amount of heat generated was proportional to the chemical energy used in the voltaic pile that generated the template This led Joule to reject the caloric theory (at that time the dominant theory) in favor of the mechanical theory of heat (according to which heat is another form of energy).
Resistive heating was independently studied by Heinrich Lenz in 1842.
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.
Microscopic description.
Joule heating is caused by interactions between charge carriers (usually electrons) and the body of the conductor (usually atomic ions).
A voltage difference between two points of a conductor creates an electric field that accelerates charge carriers in the direction of the electric field, giving them kinetic energy. When the charged particles collide with ions in the conductor, the particles are scattered; their direction of motion becomes random rather than aligned with the electric field, which constitutes thermal motion. Thus, energy from the electrical field is converted into thermal energy.
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 large number 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 (per wire, by using two paths in parallel), 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.
Direct current.
The most fundamental formula for Joule heating is the generalized power equation:
where
The explanation of this formula (formula_6) is:
Assuming the element behaves as a perfect resistor and that the power is completely converted into heat, the formula can be re-written by substituting Ohm's law, formula_7, into the generalized power equation:
where "R" is the resistance.
Alternating current.
When current varies, as it does in AC circuits,
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:
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:
where formula_12 is 
phase difference between current and voltage, formula_13 means real part, "Z" is the complex impedance, and "Y*" is the complex conjugate of the admittance (equal to 1/"Z*").
Differential form.
Joule heating can also be calculated at a particular location in space. The differential form of the Joule heating equation gives the power per unit volume.
Here, formula_15 is the current density, and formula_16 is the electric field. For a material with a conductivity formula_17, formula_18 and therefore
where formula_20 is the resistivity. This directly resembles the "formula_21" term of the macroscopic form.
In the harmonic case, where all field quantities vary with the angular frequency formula_22 as formula_23, complex valued phasors formula_24 and formula_25 are usually introduced for the current density and the electric field intensity, respectively. The Joule heating then reads 
where formula_27 denotes the complex conjugate.
High-voltage alternating current transmission of electricity.
Overhead power lines transfer electrical energy from electricity producers to consumers. Those power lines have a nonzero resistance and therefore are subject to Joule heating, which causes transmission losses.
The split of power between transmission losses (Joule heating in transmission lines) and load (useful energy delivered to the consumer) can be approximated by a voltage divider. In order to minimize transmission losses, the resistance of the lines has to be as small as possible compared to the load (resistance of consumer appliances). Line resistance is minimized by the use of copper conductors, but the resistance and power supply specifications of consumer appliances are fixed.
Usually, a transformer is placed between the lines and consumption. When a high-voltage, low-intensity current in the primary circuit (before the transformer) is converted into a low-voltage, high-intensity current in the secondary circuit (after the transformer), the equivalent resistance of the secondary circuit becomes higher and transmission losses are reduced in proportion.
During the war of currents, AC installations could use transformers to reduce line losses by Joule heating, at the cost of higher voltage in the transmission lines, compared to DC installations.
Applications.
Joule-heating or resistive-heating is used in multiple devices and industrial process. The part which converts electricity into heat by Joule heating is called a heating element.
There are many practical uses of Joule heating:
Food processing.
Joule heating (Ohmic heating) is a flash pasteurization (also called "high-temperature short-time" (HTST)) aseptic process that runs an alternating current of 50–60 Hz through food. Heat is generated through the electrical resistance of the food. As the product heats up, electrical conductivity increases linearly. A higher electrical current frequency is best as it reduces oxidation and metallic contamination. This heating method is best for foods that contain particulates suspended in a weak salt-containing medium due to their high resistance properties. Ohmic heating allows for a maintained quality of foods due to the uniform heating that decreases deterioration and over-processing of food.
Heating efficiency.
As a heating technology, Joule heating has a coefficient of performance of 1.0, meaning that every joule of electrical energy supplied produces one joule of heat. In contrast, a heat pump can have a coefficient of more than 1.0 since it moves additional thermal energy from the environment to the heated item.
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 a hydraulic equivalent of Joule's law is used:
where: