Joule's laws are a pair of laws concerning the heat produced by a current and the energy dependence of an ideal gas to that of pressure, volume, and temperature, respectively. They are named after James Prescott Joule.

Joule's first law, also known as the Joule effect, is a physical law expressing the relationship between the heat generated by the current flowing through a conductor. Joule studied this phenomenon in the 1840s. It is expressed as:

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where Failed to parse (Missing texvc executable; please see math/README to configure.): Q

is the heat generated by a constant current Failed to parse (Missing texvc executable; please see math/README to configure.): I
flowing through a conductor of electrical resistance Failed to parse (Missing texvc executable; please see math/README to configure.): R

, for a time Failed to parse (Missing texvc executable; please see math/README to configure.): t . When current, resistance and time are expressed in amperes, ohms, and seconds respectively, the unit of Failed to parse (Missing texvc executable; please see math/README to configure.): Q

is the joule. Joule's first law is sometimes called the Joule–Lenz law since it was later independently discovered by Heinrich Lenz. The heating effect of conductors carrying currents is known as Joule heating.

Joule's second law states that the internal energy of an ideal gas is independent of its volume and pressure, depending only on its temperature.

Relation to Ohm's law[link]

In the context of resistive circuits and in light of conservation of energy and electrical potential, Joule's first law and Ohm's law are equivalent and derivable from each other (as explained by James Clerk Maxwell in 1881,^[1] by Mascart in 1883,^[2] and by Oliver Heaviside in 1894^[3]), though they were discovered independently and experimentally, before the notions of conservation of energy and electrical potential were well developed.

Joule's first law states that the rate of heat dissipation in a resistive conductor is proportional to the square of the current through it and to its resistance. That is, the power dissipated in a resistor, in terms of the current through it and its resistance, is:^[4]

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Joule arrived at this result experimentally in 1841, using a calorimeter to measure heat, and a galvanometer to measure current, with a variety of resistive circuits.^[5][6]

The law applies to any circuit that obeys Ohm's law, that is, that conducts a current proportional to the voltage across it, or equivalently, that can be characterized by a resistance. Ohm's law states that for a voltage V across a circuit of resistance R the current will be:^[7][8]

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By substituting this formula for current into one or both factors of current in Joule's law, the power dissipated can be written in the equivalent forms:

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The relation Failed to parse (Missing texvc executable; please see math/README to configure.): P = V \cdot I

is actually more generally applicable than either Joule's law or Ohm's law, as it represents the instantaneous power being applied to a circuit with voltage V across it and current I into it, whether the circuit is resistive or not.[1][9]  In combination with either Ohm's law or Joule's law, it may be used to derive the other.[10]

Since the power dissipated by a resistor is the amount of energy used (electrical work applied) per unit time, the total energy consumed and dissipated in time t is:^[11]

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Hydraulic equivalent[link]

In the energy balance of groundwater flow (see also Darcy's law) an hydraulic equivalent of Joule's law is used ^[12] :

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where:

Failed to parse (Missing texvc executable; please see math/README to configure.): dE/dx

= loss of hydraulic energy (Failed to parse (Missing texvc executable; please see math/README to configure.): E

) due to friction of flow in Failed to parse (Missing texvc executable; please see math/README to configure.): x -direction per unit of time (m/day) – comparable to Failed to parse (Missing texvc executable; please see math/README to configure.): Q/t

Failed to parse (Missing texvc executable; please see math/README to configure.): v_x

= flow velocity in Failed to parse (Missing texvc executable; please see math/README to configure.): x

-direction (m/day) – comparable to Failed to parse (Missing texvc executable; please see math/README to configure.): I

Failed to parse (Missing texvc executable; please see math/README to configure.): K

= hydraulic conductivity of the soil (m/day) – the hydraulic conductivity is inversely proportional to the hydraulic resistance which compares to Failed to parse (Missing texvc executable; please see math/README to configure.): R

See also[link]

• Peltier–Seebeck effect
• Electric heating#Mathematical analysis
• Ohm's Law

Notes[link]