JEE Advance - Physics (2022 - Paper 1 Online - No. 10)
The figure shows a circuit having eight resistances of $1 \Omega$ each, labelled $R_{1}$ to $R_{8}$, and two ideal batteries with voltages $\varepsilon_{1}=12 \mathrm{~V}$ and $\varepsilon_{2}=6 \mathrm{~V}$.
Which of the following statement(s) is(are) correct?
Explanation
Point A and B are at same potential so they can be merged/folded.
For loop 1
$$
\begin{aligned}
& -\frac{1}{2} x-\frac{1}{2}(x+y)-x+6=0 \\\\
&\Rightarrow -2 x-\frac{y}{2}=-6 \\\\
&\Rightarrow -4 x-y=-12 \\\\
&\Rightarrow 4 x+y=12 .......(1)
\end{aligned}
$$
For loop 2
$$
\begin{aligned}
& \frac{1}{2} y+1 y+12+\frac{1}{2}(x+y)=0 \\\\
&\Rightarrow \frac{3}{2} y+\frac{y}{2}+\frac{x}{2}=-12 \\\\
&\Rightarrow 2 y+\frac{x}{2}=-12 \\\\
& \begin{array}{l}
\Rightarrow4 y+x=-24 \\\\
\Rightarrow4 y+x-16 x-4 y
=-24-48
\end{array} \\\\
&\Rightarrow -15 x=-72
\end{aligned}
$$
$$ \Rightarrow $$ $x=\frac{72}{15}$
$$ \Rightarrow $$ $4\left(\frac{72}{15}\right)+y=12$
$$
\begin{aligned}
\Rightarrow y & =12-\frac{288}{15} \\\\
& =\frac{180-288}{15} \\\\
& =\frac{-108}{15}=-7.2 \mathrm{~A}
\end{aligned}
$$
Current in $R_1=7.2 \mathrm{~A}$
Current in $R_2=\frac{x+y}{2}=\left(\frac{72}{15}-\frac{108}{15}\right) \frac{1}{2}$
$$
\begin{aligned}
& =\frac{1}{2}\left(\frac{36}{15}\right)=\frac{2.4}{2} \mathrm{~A} \\\\
& =1.2 \mathrm{~A}
\end{aligned}
$$
Current in $R_3=x=4.8 \mathrm{~A}$
Current in $R_5=\frac{1}{2} x=2.4 \mathrm{~A}$
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