Joule’s Law of Heating

We know about the heating effect of electric current, when it flows through a circuit due to collision between electrons and atoms of wire. But precisely how much heat is generated during electric current flow through a wire, on what conditions and parameters does it depend?

How can we know about this? To solve this problem, Joule coined a formula which explains this phenomenon accurately. This is known as Joule’s law. This law is explained in detail afterwards.

Joule’s Law of Heating

The heat which is produced due to the flow of electric current within an electric wire, is expressed in Joules. Now the mathematical representation or explanation of Joule’s law is given in the following manner.

i) The amount of heat produced in electric current conducting wire, is proportional to the square of the amount of electric current that is flowing through the circuit, when the electrical resistance of the wire and the time of electric current flow is constant.

i.e. H ∝ i2 (When R & t are constant)

ii) The amount of heat produced is proportional to the electrical resistance of the wire when the electric current in the circuit and the time of electric current flow is constant.
i.e. H ∝ R (when i & t are constant)

iii) Heat generated due to the flow of electric current is proportional to the time of electric current flow, when the resistance and amount of electric current flow is constant.
i.e. H ∝ t (when i & R are constant)
When these three conditions are merged, the resulting formula is like this -

Here ‘H’ is the heat generated in Joules, ‘i’ is the electric current flowing through the circuit in ampere and ‘t’ is in seconds. When any three of these are known the other one can be equated out.

Here, 'J' is a constant, known as Joule's mechanical equivalent of heat. Mechanical equivalent of heat may be defined as the number of work units which, when completely converted into heat, furnishes one unit of heat.

Obviously the value of J will depend on the choice of units for work and heat.
It has been found that

J = 4.2 joules/cal (1 joule = 107 ergs) = 1400 ft. lbs./CHU = 778 ft. lbs/B Th U

It should be noted that the above values are not very accurate but are good enough for general work.
Now according to Joule's law I2Rt = work done in joules electrically when I ampere of electric current are maintained through a resistor of R ohms for t second.

By eliminating I and R in turn in the above expression with the help of Ohm’s law , we get alternative forms as