Question

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

Answer 1

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:
$R=\dfrac{V}{I}$
$W=QV$

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 ($\Omega$).
$I=\dfrac{Q}{t}$ -------- (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-
$R=\dfrac{V}{I}$ ------------- (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)

& W=It\times IR \\\ & \Rightarrow W={{I}^{2}}Rt \\\ \end{aligned}$$ Assuming that all the electrical work is converted into heat energy, then $$W=H$$ Here, $$H$$ is the heat produced in the circuit. $$\therefore H={{I}^{2}}Rt\,joules$$ This relation is called the Joules Law of Heating. **Note:** Using relationships from ohm’s law, the derived formula for heat energy can also be written as- $$H=\dfrac{{{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.