Question

. Derive the expression for the heat produced due to a current 'l' 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?​

Answer 1

By Ohm's Law, the voltage $\displaystyle\sf {V,}$ current $\displaystyle\sf {I}$ and resistance $\displaystyle\sf {R}$ are connected as,

• $\displaystyle\sf {V=IR\quad\quad\dots(i)}$

• $\displaystyle\sf {I=\dfrac {V}{R}\quad\quad\dots(ii)}$

• $\displaystyle\sf {R=\dfrac {V}{I}\quad\quad\dots(iii)}$

And we know current is flow of charge per unit time.

• $\displaystyle\sf {q=It\quad\quad\dots(iv)}$

We know work done due to charge is given by,

$\displaystyle\sf{\longrightarrow W=Vq}$

From (iv),

$\displaystyle\sf{\longrightarrow W=VIt}$

This work done is liberated as heat. Thus,

$\displaystyle\sf{\longrightarrow\underline {\underline {H=VIt}}\quad\quad\dots(1)}$

From (ii),

$\displaystyle\sf{\longrightarrow H=V\cdot\dfrac {V}{R}\cdot t}$

$\displaystyle\sf{\longrightarrow\underline {\underline {H=\dfrac {V^2t}{R}}}\quad\quad\dots(2)}$

From (i),

$\displaystyle\sf{\longrightarrow H=\dfrac {(IR)^2t}{R}}$

$\displaystyle\sf{\longrightarrow H=\dfrac {I^2R^2t}{R}}$

$\displaystyle\sf{\longrightarrow\underline {\underline {H=I^2Rt}}\quad\quad\dots(3)}$

This relation is known as Joule's Law.

It states that,

(1) the heat produced in a electric circuit is directly proportional to the voltage, current and time taken.

(2) the heat produced in a electric circuit is directly proportional to square of voltage, time taken and is inversely proportional to resistance.

(3) the heat produced in a electric circuit is directly proportional to square of current, resistance and time taken.

We're asked to find how much heat is produced by an instrument of power $\displaystyle\sf {12\ W}$ in one minute if it is connected to a battery of $\displaystyle\sf {12\ V.}$

We know power is amount of heat produced per unit time.

$\displaystyle\sf{\longrightarrow P=\dfrac {H}{t}}$

$\displaystyle\sf{\longrightarrow H=Pt}$

Here,

• $\displaystyle\sf {P=12\ W}$

• $\displaystyle\sf {t=1\ min=60\ s}$

Hence heat produced is,

$\displaystyle\sf{\longrightarrow H=Pt}$

$\displaystyle\sf{\longrightarrow H=12\times 60\ J}$

$\displaystyle\sf{\longrightarrow\underline {\underline {H=720\ J}}}$

Hence 720 J is the answer.