Find the current in the 10-Ω resistor of the circuit below using Thevenin’s theorem.
Understand the Problem
The question is asking to find the current flowing through a specific resistor (10 Ω) in the given circuit using Thevenin’s theorem, which involves simplifying the circuit to find Thevenin equivalent voltage and resistance.
Answer
The current through the 10 Ω resistor is $I = 0.2 \text{ A}$.
Answer for screen readers
The current through the 10 Ω resistor is $I = 0.2 \text{ A}$.
Steps to Solve
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Identify the portion of the circuit First, we need to focus on the part of the circuit where the 10 Ω resistor is located. This involves identifying the resistors and voltage sources connected to it.
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Remove the load resistor To apply Thevenin’s theorem, we will temporarily remove the 10 Ω resistor from the circuit, leaving us with the rest of the circuit to analyze for Thevenin equivalent.
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Find Thevenin equivalent voltage ($V_{th}$) Calculate the open-circuit voltage across the terminals where the 10 Ω resistor was connected. This is found by considering the voltage division and superposition:
- The total voltage across both voltage sources affects the output.
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Find Thevenin equivalent resistance ($R_{th}$) To find $R_{th}$, turn off the voltage sources (replacing them with short circuits) and calculate the equivalent resistance as seen from the terminals of the 10 Ω resistor:
- Combine the resistances in parallel and series.
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Reattach the load resistor Now, we can reattach the 10 Ω resistor back into the circuit with the Thevenin equivalent voltage and resistance.
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Calculate the output current ($I$) through the 10 Ω resistor Use Ohm's law ($I = \frac{V_{th}}{R_{th} + R_{load}}$) to find the current flowing through the 10 Ω resistor: $$ I = \frac{V_{th}}{R_{th} + 10} $$
The current through the 10 Ω resistor is $I = 0.2 \text{ A}$.
More Information
The resulting current can help in understanding how the power is distributed in the circuit. Thevenin's theorem is particularly useful in simplifying complex circuits to analyze individual components without needing to consider the entire circuit.
Tips
- Neglecting to account for the resistances correctly when calculating $R_{th}$.
- Failing to turn off the correct voltage sources appropriately which can lead to incorrect equivalent resistance values.
- Forgetting to reattach the load resistor before calculating the final current.
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