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:
\[R=\dfrac{V}{I}\]
\[W=QV\]

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)
\[\begin{align}
  & W=It\times IR \\
 & \Rightarrow W={{I}^{2}}Rt \\
\end{align}\]
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.