https://file.aiquickdraw.com/mathbot/file/1726078370348f6bdzmqb.jpg

To solve the problem using mesh analysis, we need to define mesh currents and apply Kirchhoff's Voltage Law (KVL) to each mesh. Let's denote the mesh currents as \( I_1 \), \( I_2 \), and \( I_3 \) as indicated in the diagram.

### Step 1: Define Mesh Currents
- Mesh 1: Current \( I_1 \) flows through the 60Ω, 50Ω, and 20V source.
- Mesh 2: Current \( I_2 \) flows through the 50Ω, 40Ω, and 10Ω resistors.
- Mesh 3: Current \( I_3 \) flows through the 20Ω, 30Ω, and 40Ω resistors.

### Step 2: Write KVL Equations for Each Mesh
For Mesh 1 (left loop):
\[
100I_1 - 50I_2 - 20 = 0
\]
which simplifies to:
\[
100I_1 - 50I_2 = 20 \quad \text{(Equation 1)}
\]

For Mesh 2 (right loop):
\[
-50I_1 + 100I_2 - 40I_3 = 20 + 50
\]
\[
-50I_1 + 100I_2 - 40I_3 = 70 \quad \text{(Equation 2)}
\]

For Mesh 3 (top loop):
\[
40I_2 + 50I_3 + 30I_3 - 20I_1 = 0
\]
which simplifies to:
\[
-40I_2 + 90I_3 - 20I_1 = 0 \quad \text{(Equation 3)}
\]

### Step 3: Solve the System of Equations
Now, let's solve these equations.

1. **Equation 1**: \( 100I_1 - 50I_2 = 20 \)
2. **Equation 2**: \( -50I_1 + 100I_2 - 40I_3 = 70 \)
3. **Equation 3**: \( -20I_1 + 90I_3 - 40I_2 = 0 \)

Let's calculate the currents \( I_1 \), \( I_2 \), and \( I_3 \).The mesh currents are as follows:

- \( I_1 = \frac{463}{475} \, \text{A} \approx 0.9747 \, \text{A} \)
- \( I_2 = \frac{736}{475} \, \text{A} \approx 1.5495 \, \text{A} \)
- \( I_3 = \frac{86}{95} \, \text{A} \approx 0.9053 \, \text{A} \)

### Step 4: Calculate the Current through the 50Ω Resistor
The current through the 50Ω resistor is the difference between \( I_1 \) and \( I_2 \) because it lies between these two meshes.

\[
I_{\text{50Ω}} = I_1 - I_2 = \frac{463}{475} - \frac{736}{475} = \frac{-273}{475} \, \text{A} \approx -0.5747 \, \text{A}
\]

### Final Answer
The current through the 50Ω resistor is approximately \( -0.5747 \, \text{A} \). The negative sign indicates that the actual direction of the current is opposite to the assumed direction in the mesh current \( I_1 \).

Would you like more details or have any questions?

Here are 5 related questions you might find interesting:
1. How does mesh analysis compare to nodal analysis for solving circuit problems?
2. What is the significance of the negative sign in the current calculation?
3. Can mesh analysis be applied to circuits with dependent sources?
4. How do you solve circuits with more than three meshes?
5. What happens if there are multiple sources in different meshes?

**Tip:** When solving mesh equations, double-check the signs for voltage drops and rises to ensure correct equation setup.

Find the current that flows through the 50Ω resistor for the circuit shown using mesh analysis.

Learn how to use mesh analysis and Kirchhoff's Voltage Law (KVL) to calculate the current through a 50Ω resistor in a circuit. This step-by-step solution involves solving a system of equations and applies Ohm's Law. Ideal for undergraduate students.

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