Question 2.4.2: Find all loop currents in the electric circuit in Figure 2.4...

We use Kirchhoff’s Voltage Law on each of the four loops:

the top left loop: i_{1} + 2(i_{1} – i_{2}) + 4(i_{1} – i_{3}) = 8
the top right loop: 3i_{2} + 2(i_{2} – i_{4}) + 2(i_{2} – i_{1}) = 10
the bottom left loop: 2i_{3} + 4(i_{3} – i_{1}) + 4(i_{3} – i_{4}) = 0
the bottom right loop: 2i_{4} + 4(i_{4} – i_{3}) + 2(i_{4} – i_{2}) = -10
Solving this system of 4 equations in the 4 unknowns I_{1}, I_{2}, I_{3}, I_{4}, we find that the currents are

\left [ \begin{matrix} I_{1} \\ I_{2} \\ I_{3} \\ I_{4} \end{matrix} \right ] = \frac{1}{1069} \left [ \begin{matrix} 2172 \\ 1922 \\ 702 \\ -790 \end{matrix} \right]