Chapter 31: Problem 6
A capacitance \(C\) and an inductance \(L\) are operated at the same angular frequency. (a) At what angular frequency will they have the same reactance? (b) If \(L\) = 5.00 mH and \(C\) = 3.50 \(\mu\)F, what is the numerical value of the angular frequency in part (a), and what is the reactance of each element?
Short Answer
Step by step solution
Understanding Reactance Equality
Solving for Angular Frequency
Calculating Angular Frequency
Computing the Angular Frequency Numerically
Calculating Reactance
Confirming Reactance Equality
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Angular Frequency
In the context of inductors and capacitors, angular frequency is a critical component because it affects the reactance of these components. For inductors, the reactance (\( X_L \)) is directly proportional to angular frequency, as given by the formula \( X_L = \omega L \). On the other hand, the reactance of capacitors (\( X_C \)) is inversely related, described by \( X_C = \frac{1}{\omega C} \).
If you want the reactance of both an inductor and a capacitor to be equal, you can set these two equations equal to each other, leading to \( \omega = \frac{1}{\sqrt{LC}} \). By doing so, you define a special angular frequency at which the reactance of both elements is identical.
Inductance
A key factor about inductors is their reactance, often called inductive reactance, which signifies the opposition to current changes at a particular frequency. This opposition, expressed as \( X_L = \omega L \), means that as the angular frequency \( \omega \) increases, so does the reactance. Therefore, the reactance of an inductor is dependent on both operation frequency and its inductance value.
- Higher inductance results in higher reactance.
- Increased frequency leads to increased reactance.
Capacitance
The reactance of a capacitor is inversely proportional to both the angular frequency and the capacitance, as illustrated by the formula \( X_C = \frac{1}{\omega C} \). This relationship implies that the reactance of a capacitor decreases with an increase in angular frequency. Hence, capacitors allow more current to pass at higher frequencies.
- Lower capacitance results in higher reactance.
- Higher frequency reduces reactance, allowing easier current flow.