To solve this circuit using Maxwell's loop (mesh) analysis, follow these steps:

Step 1: Define Loop Currents

Let's define three loop currents:

• $I_1$ for the left loop
• $I_2$ for the right loop
• $I_3$ for the bottom loop

Step 2: Apply Kirchhoff’s Voltage Law (KVL)

Each loop equation is formed by summing voltage drops across resistors and equating it to the net voltage sources in the loop.

Loop 1 (Left loop)

• Components: $12V$ battery, $1\Omega$, $5\Omega$, shared $1\Omega$
• KVL equation: $12V - 1(I_1 - I_3) - 5I_1 - 1(I_1 - I_2) = 0$ $12 - 1I_1 + 1I_3 - 5I_1 - 1I_1 + 1I_2 = 0$ $12 - 7I_1 + I_2 + I_3 = 0$

Loop 2 (Right loop)

• Components: $9V$ battery, $1\Omega$, $8\Omega$, shared $1\Omega$
• KVL equation: $-9V + 1(I_2 - I_1) + 8I_2 + 1(I_2 - I_3) = 0$ $-9 + 1I_2 - 1I_1 + 8I_2 + 1I_2 - 1I_3 = 0$ $-9 - I_1 + 10I_2 - I_3 = 0$

Loop 3 (Bottom loop)

• Components: $1\Omega$ (shared), $10\Omega$, shared $1\Omega$
• KVL equation: $1(I_3 - I_1) + 10I_3 + 1(I_3 - I_2) = 0$ $1I_3 - 1I_1 + 10I_3 + 1I_3 - 1I_2 = 0$ $-I_1 - I_2 + 12I_3 = 0$

Step 3: Solve the System of Equations

The equations are:

$7I_1 - I_2 - I_3 = 12$

$I_1 - 10I_2 + I_3 = 9$

$I_1 + I_2 - 12I_3 = 0$

Solve this system using substitution or matrix methods to find $I_1, I_2, I_3$, then calculate the voltage drops across each resistor using Ohm's law:
$V = IR$

Would you like me to solve for the actual currents? Let me know if you need a detailed step-by-step solution!

Related Questions

1. How do Maxwell’s loops differ from Kirchhoff’s Current Law (KCL)?
2. What happens if we use Node Voltage Method instead of Maxwell’s loops?
3. How can mesh analysis be extended to AC circuits?
4. How do we verify the correctness of the computed currents?
5. How does changing a resistor value affect the loop currents?

Tip: When solving loop equations, always check unit consistency (Ohms, Volts, Amperes) to avoid calculation errors.