Using Kirchhoff’s Law find ${\mathrm{V}}_{\mathrm{AB}}$

To find V_AB, let us analyze the given circuit step by step and apply Kirchhoff’s Current Law (KCL) and Kirchhoff’s Voltage Law (KVL).

Step 1: Assign Currents

Let the currents in the circuit be such that at junction B:

i₁ + i₂ = i₃

Step 2: Apply KVL to Loop ABFEA

In the loop ABFEA, applying Kirchhoff’s Voltage Law:

V_A + 10 - 6i₁ - 2i₃ = V_A

Simplifying:

6i₁ + 2i₃ = 10

Dividing through by 2:

3i₁ + i₃ = 5 → (1)

Step 3: Apply KVL to Loop ADCBA

For the loop ADCBA, applying KVL:

V_A - 4i₂ - 14 + 6i₁ - 10 = V_A

Simplifying:

6i₁ - 4i₂ = 24

Dividing through by 2:

3i₁ - 2i₂ = 12 → (2)

Step 4: Apply KVL to Loop DEFCD

For the loop DEFCD, applying KVL:

V_B + 4i₂ + 2i₃ + 14 = V_B

Simplifying:

4i₂ + 2i₃ = -14

Dividing through by 2:

2i₂ + i₃ = -7 → (3)

Step 5: Solve the Equations

Using the equations (1), (2), and (3) along with the relationship:

i₁ + i₂ = i₃

We solve to find:

• i₁ = 2 A
• i₂ = -3 A
• i₃ = -1 A

Step 6: Find V_AB

To find V_AB, apply KVL to the segment AB:

V_A + 10 - 6i₁ = V_B

Substitute i₁ = 2 A:

V_A + 10 - 6(2) = V_B

V_A + 10 - 12 = V_B

V_AB = V_A - V_B = 2 V

Conclusion

Using Kirchhoff’s Law, we were able to find V_AB. The final value is:

V_AB = 2 V