Calculate the current through the 2 ohm resistor using KVL.
Understand the Problem
The question asks to calculate the current through a 2-ohm resistor within a circuit using Kirchhoff's Voltage Law (KVL). KVL states that the sum of all voltage drops in a closed loop is zero. Applying KVL to the provided circuit diagram will help determining the current.
Answer
$\frac{50}{13} \text{ A}$
Answer for screen readers
The current through the 2-ohm resistor is $\frac{50}{13} \text{ A} \approx 3.85 \text{ A}$.
Steps to Solve
- Apply KVL to Loop 1
Starting from the 100V source and moving clockwise: $100 - 4I_1 - 2(I_1 - I_2) = 0$. Simplifying gives: $100 - 4I_1 - 2I_1 + 2I_2 = 0$ $6I_1 - 2I_2 = 100$ $3I_1 - I_2 = 50$
- Apply KVL to Loop 2
Starting from the 50V source and moving counter-clockwise: $50 - 3I_2 - 2(I_2 - I_1) = 0$. Simplifying gives: $50 - 3I_2 - 2I_2 + 2I_1 = 0$ $2I_1 - 5I_2 = -50$
- Solve the system of equations
We now have two equations: $3I_1 - I_2 = 50$ $2I_1 - 5I_2 = -50$
Multiply the first equation by -5: $-15I_1 + 5I_2 = -250$ Add this to the second equation: $2I_1 - 5I_2 = -50$ $-13I_1 = -300$ $I_1 = \frac{300}{13} \approx 23.08 \text{ A}$
- Substitute $I_1$ back into one of the equations to solve for $I_2$
Using the first equation: $3I_1 - I_2 = 50$ $3(\frac{300}{13}) - I_2 = 50$ $\frac{900}{13} - I_2 = 50$ $I_2 = \frac{900}{13} - 50 = \frac{900 - 650}{13} = \frac{250}{13} \approx 19.23 \text{ A}$
- Calculate the current through the 2-ohm resistor
The current through the 2-ohm resistor is $I_1 - I_2 = \frac{300}{13} - \frac{250}{13} = \frac{50}{13} \approx 3.85 \text{ A}$
The current through the 2-ohm resistor is $\frac{50}{13} \text{ A} \approx 3.85 \text{ A}$.
More Information
The current flows from left to right through the 2-ohm resistor.
Tips
A common mistake would be to incorrectly set up the KVL equations, especially regarding the signs of the voltage drops and rises. Another mistake could be in solving the system of linear equations.
AI-generated content may contain errors. Please verify critical information
