What is the impedance of an L-R circuit? (A) R² + ω²L² (B) R + ωL
Understand the Problem
The question is asking about the impedance of an L-R circuit, providing multiple choice options for the answer. We need to determine the correct expression for impedance based on the circuit components.
Answer
The impedance of an L-R circuit is $Z = \sqrt{R^2 + (\omega L)^2}$.
Answer for screen readers
The impedance of an L-R circuit is given by:
$$ Z = \sqrt{R^2 + (\omega L)^2} $$
Thus, the best choice from the options provided, given that they aren’t exact, would relate closely to option (A).
Steps to Solve
- Understanding Impedance in an L-R Circuit
The impedance $Z$ of a circuit containing a resistor ($R$) and an inductor ($L$) in series can be calculated using the formula:
$$ Z = \sqrt{R^2 + (\omega L)^2} $$
where $\omega$ is the angular frequency of the source.
- Identifying Components in the Formula
We see that the impedance involves both the resistive and inductive components. The inductor contributes to the imaginary part of the impedance, while the resistor contributes to the real part.
- Comparing Options
We need to check which of the provided options matches the impedance formula we derived.
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Option (A) is $R^2 + \omega^2 L^2$, which represents a quantity under a square root that describes impedance.
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Option (C) is $R + \omega L$, which combines the two components linearly and does not account for the square root or the proper impedance relationship.
- Concluding the Correct Expression
Since the correct formula is $Z = \sqrt{R^2 + (\omega L)^2}$, we see that option (A) is related in terms of the individual components, but we must remember that the actual impedance includes the square root.
The impedance of an L-R circuit is given by:
$$ Z = \sqrt{R^2 + (\omega L)^2} $$
Thus, the best choice from the options provided, given that they aren’t exact, would relate closely to option (A).
More Information
The impedance of an L-R circuit describes how the circuit resists alternating current (AC). The presence of the inductor introduces phase differences between voltage and current, which is why the impedance is a complex quantity that incorporates both resistance and reactance.
Tips
- Choosing the wrong formula: Sometimes students might confuse the linear addition of components instead of using the square root for calculating impedance.
- Ignoring the square root: Forgetting to apply the square root when interpreting the impedance formula can lead to incorrect answers.
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