Question 78: Current Electricity

Question

The heat generated in 1 minute between points $A$ and $B$ in the given circuit, when a battery of 9 V with internal resistance of $1 \Omega$ is connected across these points is $\_\_\_\_$ J.

Accepted Answer

A bridge is balanced if the ratio of the resistances in the two arms is equal :
$$ \frac{\mathrm{R}_1}{\mathrm{R}_2}=\frac{\mathrm{R}_3}{\mathrm{R}_4} $$
So, from the given circuit, $\frac{\mathrm{R}_1}{\mathrm{R}_2}=\frac{1}{2}$ and $\frac{\mathrm{R}_3}{\mathrm{R}_4}=\frac{2}{4}=\frac{1}{2} \Rightarrow \frac{\mathrm{R}_1}{\mathrm{R}_2}=\frac{\mathrm{R}_3}{\mathrm{R}_4}$
Since the ratio is equal $\left(\frac{1}{2}=\frac{1}{2}ight)$, the bridge is balanced.
Because the bridge is balanced, no current flows through the central $1 \Omega$ resistor. The circuit now simplifies to two parallel branches :
Here, $\mathrm{R}_1$ and $\mathrm{R}_3$ are in series, $\mathrm{R}_{13}=1+2=3 \Omega$
Also, $\mathrm{R}_2$ and $\mathrm{R}_4$ are in series, $\mathrm{R}_{24}=2+4=6 \Omega$
Now, $\mathrm{R}_{13}$ and $\mathrm{R}_{24}$ are in parallel.
So, the equivalent resistance between A and $\mathrm{B}\left(\mathrm{R}_{\mathrm{AB}}ight)$ is :
$$ \frac{1}{\mathrm{R}_{\mathrm{AB}}}=\frac{1}{3}+\frac{1}{6}=\frac{2+1}{6}=\frac{1}{2} $$
$$\Rightarrow $$ $$ \mathrm{R}_{\mathrm{AB}}=2 \Omega $$
The battery has an emf of 9 V and an internal resistance (r) of $1 \Omega$. The total resistance ( $\mathrm{R}_{	ext {total }}$ ) is:
$$ \mathrm{R}_{	ext {total }}=\mathrm{R}_{\mathrm{AB}}+\mathrm{r}=2+1=3 \Omega $$
Using Ohm's Law for the whole circuit:
$$ \mathrm{I}=\frac{\mathrm{V}}{\mathrm{R}_{	ext {total }}}=\frac{9}{3}=3 	ext { Amperes } $$
So, the power dissipated between points A and B is,
$$\mathrm{P}_{\mathrm{AB}}=\mathrm{I}^2 \cdot \mathrm{R}_{\mathrm{AB}} $$
$$\Rightarrow $$ $$ \mathrm{P}_{\mathrm{AB}}=(3)^2 	imes 2=9 	imes 2=18 	ext { Watts } $$
So, the total heat $(\mathrm{H})$ generated in $\mathrm{t}=1$ minute $=60$ seconds is :
$$ \mathrm{H}=\mathrm{P}_{\mathrm{AB}} 	imes \mathrm{t} $$
$$\Rightarrow $$ $$ \mathrm{H}=18 	imes 60=1080 \mathrm{~J} $$
Therefore, the heat generated in one minute is 1080 J between points A and B .
Hence, the correct answer is 1080.

Current Electricity

The heat generated in 1 minute between points A and B in the given circuit, when a battery of 9 V with internal resistance of 1 \Omega is connected across these points is \_\_\_\_ J.

Integer / Numeric Answer Type