Find the heat produced in external circuit (AB) in one minute.

The correct option is 1181.25 J.

By analyzing the provided circuit diagram:
- The EMF of the battery is $E=9\text{V}$.
- The series resistance (internal resistance) is $r=1\Omega$.
- The external circuit connected between terminals $A$ and $B$ forms a bridge containing five resistors:
  • Upper-left arm: $1\Omega$
  • Lower-left arm: $2\Omega$
  • Central vertical arm: $1\Omega$
  • Upper-right arm: $2\Omega$
  • Lower-right arm: $1\Omega$

Step 1: Find the equivalent resistance of the external circuit $R_{AB}$
Let us apply nodal analysis by assuming a potential difference $V$ across the terminals $A$ and $B$. Let $V_A=V$ and $V_B=0$.
Let $V_C$ be the potential at the upper central junction, and $V_D$ be the potential at the lower central junction.

Applying Kirchhoff's Current Law (KCL) at node $C$:
$\frac{V_C-V}{1}+\frac{V_C-V_D}{1}+\frac{V_C-0}{2}=0$
Multiplying by 2 to clear fractions:
$2(V_C-V)+2(V_C-V_D)+V_C=0$
$5V_C-2V_D=2V$     --- (Equation 1)

Applying KCL at node $D$:
$\frac{V_D-V}{2}+\frac{V_D-V_C}{1}+\frac{V_D-0}{1}=0$
Multiplying by 2 to clear fractions:
$(V_D-V)+2(V_D-V_C)+2V_D=0$
$-2V_C+5V_D=V$     --- (Equation 2)

Solving Equations 1 and 2:
From Equation 2, we have:
$2V_C=5V_D-V\Rightarrow V_C=2.5V_D-0.5V$
Substitute this into Equation 1:
$5(2.5V_D-0.5V)-2V_D=2V$
$12.5V_D-2.5V-2V_D=2V$
$10.5V_D=4.5V\Rightarrow V_D=\frac{3}{7}V$
Now substituting $V_D$ back to find $V_C$:
$V_C=2.5\left(\frac{3}{7}V\right)-0.5V=\frac{4}{7}V$

The total current $I$ entering terminal $A$ is:
$I=\frac{V-V_C}{1}+\frac{V-V_D}{2}$
$I=(V-\frac{4}{7}V)+\frac{V-\frac{3}{7}V}{2}=\frac{3}{7}V+\frac{2}{7}V=\frac{5}{7}V$

Therefore, the equivalent resistance $R_{AB}$ is:
$R_{AB}=\frac{V}{I}=\frac{7}{5}\Omega =1.4\Omega$

Step 2: Calculate the total current in the circuit
The total resistance of the entire circuit is:
$R_{\text{total}}=R_{AB}+r=1.4\Omega +1\Omega =2.4\Omega$
The total current supplied by the $9\text{V}$ battery is:
$I_{\text{total}}=\frac{E}{R_{\text{total}}}=\frac{9}{2.4}\text{A}=3.75\text{A}$

Step 3: Calculate the heat produced in the external circuit (AB) in one minute
The time interval is $t=1\text{minute}=60\text{seconds}$.
The heat produced in $R_{AB}$ is given by Joule's Law:
$H=I_{\text{total}}^2·R_{AB}·t$
$H={\left(3.75\right)}^2·1.4·60$
$H=14.0625·84=1181.25\text{J}$