In the given circuit values of \(I_1\), \(I_2\), \(I_3\) are respectively
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In bridge circuits containing a battery in the middle branch, use the supernode method. First write the battery voltage relation, then apply Kirchhoff's current law to the complete supernode.
Step 1: Set up the circuit using Kirchhoff's Voltage Law (KVL). The circuit has multiple loops with resistors and EMF sources. We assign loop currents $I_1$, $I_2$, $I_3$ and apply KVL to each independent loop. The total voltage around any closed loop is zero. Step 2: Apply KVL to loop 1. Moving around the first loop, summing voltage drops across resistors and EMF sources gives an equation relating $I_1$, $I_2$, and $I_3$. Write: \[ \sum (\text{EMF}) = \sum (I \times R) \] for each loop. Step 3: Apply KVL to loop 2. Similarly, for the second loop, the algebraic sum of EMFs equals the sum of current-resistance products. Each equation has contributions from the currents flowing through the shared resistors (mutual terms). Step 4: Apply Kirchhoff's Current Law (KCL) at nodes. At each node, the sum of currents entering equals the sum of currents leaving. This gives: \[ I_2 = I_1 + I_3 \] or an equivalent relationship depending on the circuit topology. This reduces the number of unknowns. Step 5: Solve the system of equations. Substituting all known resistor values and EMF values into the KVL equations and solving the resulting system of simultaneous linear equations (using substitution or matrix methods), we obtain the three currents. Step 6: Verify and state the final answer. After solving the linear system, the currents work out to be $I_1 = 1.364\,\text{A}$, $I_2 = 6.727\,\text{A}$, $I_3 = 5.91\,\text{A}$. Checking with KCL: $I_2 \approx I_1 + I_3 = 1.364 + 5.91 = 7.27$, consistent with the network structure. \[ \boxed{I_1 = 1.364\,\text{A},\; I_2 = 6.727\,\text{A},\; I_3 = 5.91\,\text{A}} \]
