Relation To Ohm's Law
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, by Mascart in 1883, and by Oliver Heaviside in 1894), 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:
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.
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:
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:
The relation 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. In combination with either Ohm's law or Joule's law, it may be used to derive the other.
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:
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