Question

Derive the formula for heat produced in a wire carrying current ($$I$$), resistance ($$R$$) in time ($$t$$).

Answer 1

Hint: The work done in a circuit by a battery can be assumed to dissipate as heat energy. Therefore, the work done is heat energy. Using Ohm’s Law and work done we can find a relationship between current ($$I$$), resistance ($$R$$), potential difference ($$V$$) and time ($$t$$).

Formula used:
$$R={\displaystyle \frac{V}{I}}$$
$$W=QV$$

Complete step by step solution:
The rate of flow of electrons in a conductor is called current ($$I$$). Its SI unit is ampere ($$A$$ ).
The property of a material by virtue of which it resists the current flowing through it is called resistance ($$R$$). Its SI unit is ohm ($$\mathrm{\Omega }$$).
$$I={\displaystyle \frac{Q}{t}}$$ -------- (1)
The work done to carry a unit charge from one point to the other is called Potential difference ($$V$$) between the two points.
$$V=V_b-V_a$$ ------------ (2)
According to Ohm’s Law-
$$R={\displaystyle \frac{V}{I}}$$ ------------- (3)
The work done, $$W$$by the battery to move charge $$Q$$in a circuit with potential difference $$V$$applied at its ends is-
$$W=QV$$ ------------ (4)
From eq (1),
$$Q=I\times t$$ Substituting in eq (4)
$$W=(I\times t)V$$ ------------ (5)
From eq (3), we have,
$$V=IR$$ Substituting in eq (5)
$$\begin{array}{rl}& W=It\times IR\\ & \Rightarrow W=I^{2}Rt\end{array}$$
Assuming that all the electrical work is converted into heat energy, then
$$W=H$$
Here, $$H$$ is the heat produced in the circuit.
$$\therefore H=I^{2}Rtjoules$$
This relation is called the Joules Law of Heating.

Note: Using relationships from ohm’s law, the derived formula for heat energy can also be written as-
$$H={\displaystyle \frac{V^2}{R}}t$$ , $$H=VIt$$. Heat is a form of energy, therefore its SI unit is joules. Most of this heat is developed in the resistor and dissipated into the air around its components.