To solve this circuit using mesh analysis, we will follow these steps:

1. Label the mesh currents:

   • Assume the mesh currents as $I_1$ (left loop) and $I_2$ (right loop), flowing clockwise in both loops.

2. Write mesh equations using Kirchhoff's Voltage Law (KVL):

   • Left loop: $V_1 - I_1 R_1 - I_3 R_3 = 0$
   • Right loop: $-I_2 R_2 + V_2 - I_3 R_3 = 0$
   • The current through $R_3$ is $I_3 = I_1 - I_2$.

3. Solve the equations for $I_1$ and $I_2$.

Step 1: Circuit Parameters

• $V_1 = 30 \, \text{V}, V_2 = 10 \, \text{V}$
• $R_1 = 1.2 \, \text{k}\Omega = 1200 \, \Omega$
• $R_2 = 800 \, \Omega$
• $R_3 = 120 \, \Omega$

Step 2: Write Mesh Equations

Left Loop: $V_1 - I_1 R_1 - (I_1 - I_2) R_3 = 0$ Substitute the values: $30 - 1200 I_1 - 120 (I_1 - I_2) = 0$ Simplify: $30 - 1200 I_1 - 120 I_1 + 120 I_2 = 0$ 30 - 1320 I_1 + 120 I_2 = 0 \tag{1}

Right Loop: $-I_2 R_2 + V_2 - (I_1 - I_2) R_3 = 0$ Substitute the values: $-800 I_2 + 10 - 120 (I_1 - I_2) = 0$ Simplify: $-800 I_2 + 10 - 120 I_1 + 120 I_2 = 0$ 10 - 120 I_1 - 680 I_2 = 0 \tag{2}

Step 3: Solve the Equations

From equations (1) and (2):

1. $1320 I_1 - 120 I_2 = 30$
2. $-120 I_1 - 680 I_2 = -10$

Solve this system using substitution or elimination:

1. Multiply equation (1) by 680: $897600 I_1 - 81600 I_2 = 20400$
2. Multiply equation (2) by 1320: $-158400 I_1 - 897600 I_2 = -13200$ Add the equations: $739200 I_1 - 979200 I_2 = 7200$ $I_2 = \frac{7200}{979200} = 0.00735 \, \text{A} \, (\text{or } 7.35 \, \text{mA})$

Substitute $I_2$ into equation (1): $1320 I_1 - 120 (0.00735) = 30$ $1320 I_1 - 0.882 = 30$ $1320 I_1 = 30.882$ $I_1 = \frac{30.882}{1320} = 0.0234 \, \text{A} \, (\text{or } 23.4 \, \text{mA})$

Final Results

• $I_1 = 23.4 \, \text{mA}$
• $I_2 = 7.35 \, \text{mA}$
• $I_3 = I_1 - I_2 = 23.4 \, \text{mA} - 7.35 \, \text{mA} = 16.05 \, \text{mA}$

Let me know if you’d like further clarifications or a breakdown of any step!

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Related Questions:

1. What happens if $R_3$ is removed from the circuit?
2. How does the voltage across $R_3$ change if $V_2$ increases?
3. What is the power dissipated in $R_2$?
4. How does mesh analysis compare to nodal analysis for this circuit?
5. Can this circuit be solved using Thevenin's theorem?

Tip: Always double-check units when converting resistances or currents; small mistakes can propagate through calculations.