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Derive the expression for the heat produced due to a current ‘I’ flowing for a time interval ‘t’ through a resistor ‘R’ having a potential difference ‘V’ across its ends. With which name is the relation known? How much heat will an instrument of 12 W produce in one minute if it is connected to a battery of 12 V?
Since a conductor provides resistance to the flow of current, some work must be done by the current continuously to keep itself flowing.
When an electric charge $Q$ moves against a potential difference $V$, the amount of work done is given as-
$W=Q\times V$ ------------(1)
We know, the electric current, $I$ is given as-
$I=\frac {Q}{t}$
Then, in terms of $Q$, it is given as-
$Q=I\times t$ ------------(2)
By ohm’s law, we know that-
$R=\frac {V}{I}$
Then, in terms of $V$, it is given as-
$V=I\times R$ ------------(3)
Now, putting equation (2) and (3) in eq, (1) we get-
$W=I\times t\times I\times R$
$W=I^{2}Rt$
Assuming that all the electrical work done is converted into heat energy, we get-
Heat produced, $H$ = Work done in the above equation
Thus, $H=I^{2}Rt\ Joules$
This relation is known as Joule’s law of heating.
Given:
Power of instrument, $P=12W$
Voltage, $V=12V$
Time, $t=1minute=60sec$
To find: Heat produced by the instrument, $H$.
Solution:
We know that the formula of electric power is given as-
$P=V\times I$
Substituting the given values we get-
$12=12\times I$
$I=\frac {12}{12}$
Thus, the current flowing in the instrument is 1 Ampere.
From Ohm's law we know that-
$V=I\times R$
Substituting the value of $I$ and $R$ we get-
$12=1\times R$
$R=\frac {12}{1}$
$R=12\Omega$
Thus, the resistance of the instrument is 12 Ohm.
From the formula of heating we know that-
$H=I^{2}Rt$
Substituting the required values we get-
$H=1^{2}\times 12\times 60$
$H=720J$
Thus, the heat produced by the instrument is 720 Joules.