Joule’s Law of Heating

When an electric current is passed through a conductor, heat is produced after some time which is called the heating effect of current. This effect is also called as Joule’s heating effect.

Joule’s Law of Heating

Heat produced in a conductor due to the flow of current,

1. Is directly proportional to square of the current,\( H \propto I^2 \)
2. Is directly proportional to the resistance of the conductor, \( H \propto R \)
3. Is directly proportional to time for which the current is passed, \( H \propto t \)

That is,

\begin{align*} H &\propto I^2Rt \\ \text {or,} \: H &= \frac {I^2Rt}{J} \\ \end{align*}

where J is called Joule’s mechanical equivalent of heat. Its value is 4.18 Jcal^-1.

Derivation for the expression of heat developed in a wire

Consider a resistance R between two points A and B in a circuit as shown in the figure. Let I be the steady current passing through it.

In time t the total charge passing through A to B is given\begin{align*} \\ I &= \frac Qt \\ \text {or,} \: Q &= It \dots (i) \\ \end{align*} Let the potential difference between A and B be V. Then \begin{align*} \\ V &= IR \dots (ii) \\ \end{align*}The work done to transfer the charge Q from A to B is given by\begin{align*}\ W &= QV \dots (iii) \\ \text {From} \: (i), \: \text {and} \: (iii) \: \text {we get} \\ W &= It.IR \\ \text {or,} \: W &= I^2 Rt \\ \end{align*}

If resistor is removed, all electrical energy is converted into heat energy\begin{align*} \text {So} \: W &= H \\ H &= I^2Rt \\\end{align*}

This is the expression for heat developed.

Electrical Power

Electrical power is defined as the rate at which the work is done to maintain the steady current in an electric circuit. Electric power of an electric appliance is also defined as the rate at which electrical energy is converted into other forms of energy. Its unit is watt, W is SI-units.

\begin{align*} P &= \frac Wt = \frac {I^2Rt}{t} \\ \therefore P &= I^2 R \\ &= IV \: [V = IR] \\ &= \frac {V^2}{R} \\ \end{align*}

The power of device is said to one watt, if one ampere current flows through it under a p.d. of one volt.

Electrical Energy

It is defined as the total work done to maintain the current in an electrical circuit for a given time.

\begin{align*} \text {Electric energy} = W = VIt = Pt \\ \text {Electrical energy} = \text {Electrical power} \times \text {Time} \\ \text {SI}-\text {units of electrical energy is joule, J} \\ \text {In practice, electrical energy is measured in kilowatt hour.} \\ \text {Electrical energy in kilowatt} = \text {Power in kW} \times \text {Time in hours} \\ \end{align*}

One unit of electric energy is 1kWh. Then when we say that a consumer has consumed 100 units, it means that electrical energy consumption of 100 kWh.

Reference

Manu Kumar Khatry, Manoj Kumar Thapa, et al. Principles of Physics. Kathmandu: Ayam publication PVT LTD, 2010.

S.K. Gautam, J.M. Pradhan. A text Book of Physics. Kathmandu: Surya Publication, 2003.