Question

Find the current in branch BM in the network shown: 

 

Hint:

Use Kirchhoff's laws to analyze complex circuits: KCL for junctions and KVL for loops.

Answer 1

Applying Kirchhoff’s Laws 
Given that the equivalent resistance across CH is: \[ R_{CH} = 2R \] Equivalent Circuit Diagram
The given circuit is analyzed using Kirchhoff’s Voltage Law (KVL). We apply KVL to two loops in the circuit. Applying Kirchhoff’s Voltage Law (KVL):
In closed loop ABMNA: 
Using KVL in loop ABMNA, we sum the voltage drops: \[ -3IR - 4I_1 R + 16E = 0 \]

 This forms our first equation:  -3IR - 4I_1 R + 16E = 0 

 In closed loop BCHMB: 
Similarly, applying KVL in loop BCHMB: \[ -2R(I - I_1) - 6E + 4I_1 R = 0 \] This forms our second equation:  -2R(I - I_1) - 6E + 4I_1 R = 0 

Solving the Equations
Now, solving equations (1) and (2) simultaneously, we obtain: \[ I_1 = \frac{25E}{13R} \] Thus, the current \( I_1 \) in the circuit is determined.