Question

An electric heater of resistance 8 Ω draws 15 A from the service mains for two hours. Calculate the rate at which heat is developed in the heater.

Answer 1

Hint: We will solve the question using Joule’s Law of heating. It gives the amount of heat energy developed in a resistor due to a current flowing through it in a time period.
According to Joule’s Law of Heating, heat energy (H) developed in a resistor (R) due to the current (I) flowing through it for a time period (t) is given by \[H={{I}^{2}}Rt\]. Rate at which energy is dissipated or developed by a device is given by $\dfrac{\text{Energy dissipated/produced}}{\text{time}}$

Complete step by step answer:
An electric heater develops heat energy using the principle of Joule’s Law of Heating. According to Joule’s Law of Heating, heat energy (H) developed in a resistor (R) due to the current (I) flowing through it for a time period (t) is given by \[H={{I}^{2}}Rt\]. -(1)
The current (I) drawn by the electric heater is given to be 15 A.
The resistance (R) of the electrical heater is given to be 8 Ω.
The time period (t) of operation is given to be 2 hours or $2\times 3600=7200\text{seconds}$ (Since, we know that $1hour=3600\text{seconds}$).
Hence, using (1), the total heat energy developed in the electrical heater is
$H={{15}^{2}}\times 8\times 7200J=12960000J=1.296\times {{10}^{7}}J$ -(2)
The question asks for the rate at which heat is developed in the heater. Now, the rate at which heat is developed in the electrical heater is also called the power of the heater. The expression for power, by its definition in this case is given by
$\dfrac{\text{Total heat energy developed}}{\text{Time period in which energy developed}}$
$=\dfrac{1.296\times {{10}^{7}}J}{7200\text{seconds}}={1800 \text{Watt}}$
Hence, the rate at which heat is developed in the electrical heater is 1800 Watt.

Note: If instead of the current drawn by the heater, the potential difference across it (V) is given, the heat energy developed is given by$\dfrac{{{V}^{2}}}{R}t$. This comes from the fact $V=IR$, that is Ohm’s Law and using this equation, replacing I in the Joule’s Law of heating equation.
If the current and resistance are uniform for the whole time interval, then the power developed can be directly given by${{I}^{2}}R$. However, this does not work if the current or the resistance changes at any moment in the time interval under consideration. In such cases, the total heat energy developed has to be calculated and then divided by the time interval.