Question
Obtain the mathematical expression for the heat generated in a metallic conductor by electric current (Joule’s law).
✓
Answer
If V is the potential difference applied across a metallic conductor of resistance R, the current through the conductor, given by Ohm’s law, is
I = V/R ……(1)
The charge passing through the conductor in time t when the current I flows in the conductor is
Q = It…….(2)
The work done in this process is W = VQ …..(3)
From Eqs. (1), (2) and (3), we have,
$
\begin{aligned}
& W =( IR )( It )= I ^2 Rt = VIt \\
& =V\left(\frac{V}{R}\right) t=\frac{V^2}{R} t
\end{aligned}
$
This work is converted into heat.
When $I$ is expressed in ampere, $R$ in ohm, $t$ in second and $V$ in volt, $W$ is expressed in joule.
In that case,
$
W = I ^2 Rt = VIt =\frac{V^2}{R} t \text { (in joule) }
$
Usually heat energy $( H )$ is expressed in calorie. Using the relation $4.18 J =1 cal$, we have
$
\begin{aligned}
H & =W=\frac{I^2 R t}{4.18}(\text { in cal) } \\
& =\frac{V I t}{4.18}(\text { in cal })=\frac{V^2 t}{4.18 R} \text { (in cal) }
\end{aligned}
$
This is the required equation.
I = V/R ……(1)
The charge passing through the conductor in time t when the current I flows in the conductor is
Q = It…….(2)
The work done in this process is W = VQ …..(3)
From Eqs. (1), (2) and (3), we have,
$
\begin{aligned}
& W =( IR )( It )= I ^2 Rt = VIt \\
& =V\left(\frac{V}{R}\right) t=\frac{V^2}{R} t
\end{aligned}
$
This work is converted into heat.
When $I$ is expressed in ampere, $R$ in ohm, $t$ in second and $V$ in volt, $W$ is expressed in joule.
In that case,
$
W = I ^2 Rt = VIt =\frac{V^2}{R} t \text { (in joule) }
$
Usually heat energy $( H )$ is expressed in calorie. Using the relation $4.18 J =1 cal$, we have
$
\begin{aligned}
H & =W=\frac{I^2 R t}{4.18}(\text { in cal) } \\
& =\frac{V I t}{4.18}(\text { in cal })=\frac{V^2 t}{4.18 R} \text { (in cal) }
\end{aligned}
$
This is the required equation.
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