Question

A series circuit contains a \(100.0-\Omega\) resistor, a \(0.500-\mathrm{H}\) inductor, a 0.400 - \(\mu\) F capacitor, and a time-varying source of emf providing \(40.0 \mathrm{~V}\). a) What is the resonant angular frequency of the circuit? b) What current will flow through the circuit at the resonant frequency?

Short answer

Question: Calculate the resonant angular frequency and the current flowing through the circuit at the resonant frequency for a series RLC circuit with an inductor of 0.500 H, a capacitor of 0.400 μF, and a resistance of 100.0 Ω connected to an emf source with a voltage of 40.0 V. Answer: The resonant angular frequency of the circuit is 1432.31 rad/s and the current flowing through the circuit at the resonant frequency is 0.40 A.

Step-by-step solution

Calculate the resonant angular frequency

We need to calculate the resonant angular frequency, \(\omega_{0}\), using the formula \(\omega_{0} = \frac{1}{\sqrt{LC}}\). We are given the values of L (\(0.500 H\)) and C (\(0.400 \times 10^{-6} F\)). So, plug these values into the formula: \(\omega_{0} = \frac{1}{\sqrt{(0.500)(0.400\times10^{-6})}}\) Now, calculate \(\omega_{0}\): \(\omega_{0} = 1432.31 \, rad/s\) The resonant angular frequency of the circuit is \(1432.31 \, rad/s\).

Calculate the current

Next, we have to calculate the current flowing through the circuit at the resonant frequency. We'll use the formula \(I = \frac{V}{Z}\), where \(V = 40.0V\). At resonance, the impedance is equal to the resistance of the circuit, so we will have \(Z = R = 100.0\Omega\). Now, plug these values into the formula: \(I = \frac{40.0V}{100.0\Omega}\) Now, calculate the current: \(I = 0.40 A\) The current flowing through the circuit at the resonant frequency is \(0.40 A\). To summarize: a) The resonant angular frequency of the circuit is \(\omega_{0} =\) \(1432.31 \, rad/s\). b) The current flowing through the circuit at the resonant frequency is \(I = 0.40 A\).

Key concepts

Resonant Frequency

In a series LC circuit, the resonant frequency is a specific frequency at which the circuit naturally oscillates. This happens when the inductive reactance and capacitive reactance in the circuit cancel each other out. At this point, the impedance in the circuit is minimized, and if there is any resistance, it is limited to only that. The circuit can thus operate at maximum current. There are a few critical aspects of resonant frequency to keep in mind:

• The frequency measured is in Hertz (Hz), which determines how many cycles happen in one second.
• Resonant frequency depends on the values of the inductor (L) and the capacitor (C) in the circuit.

To find the resonant frequency, the formula used is \( f_{0} = \frac{1}{2\pi\sqrt{LC}} \). Since angular frequency, \( \omega \), is often used in calculations, it is related to resonant frequency by \( \omega_{0} = 2\pi f_{0} \). This simplifies to \( \omega_{0} = \frac{1}{\sqrt{LC}} \) for the LC circuit. Understanding this concept helps in designing circuits that can handle specific frequencies efficiently without wasting energy.

Impedance in Circuits

Impedance in an electrical circuit is somewhat similar to resistance and is expressed in Ohms (\(\Omega\)). However, while resistance applies only to direct current (DC), impedance is more comprehensive and applies to alternating current (AC). It includes both resistance (real part) and reactance (imaginary part). In an LC circuit:

• Reactance comes from the inductors and capacitors.
• Inductors produce inductive reactance, noted by \( X_L = \omega L \).
• Capacitors produce capacitive reactance, noted by \( X_C = \frac{1}{\omega C} \).

Crucially, at the resonant frequency, the inductive and capacitive reactances are equal but opposite, canceling out each other. Thus, the circuit's impedance at resonance is purely resistive, and it is minimized to just the resistance (R) present in the circuit. The equation \( Z = \sqrt{R^2 + (X_L - X_C)^2} \) gives the impedance in the circuit at any frequency, but simplifies to \( Z = R \) at resonance. This understanding of impedance is essential to predict how circuits will behave at different frequencies and to ensure proper functioning at resonant frequencies.

Angular Frequency

Angular frequency, denoted by \( \omega \), is a measure of how quickly a sinusoidal waveform like a sine wave is oscillating and is related to the regular frequency \( f \) by the formula \( \omega = 2\pi f \). While frequency is the number of cycles per second measured in Hertz, angular frequency gives the cycles in terms of radians per second. This concept is especially useful in understanding the behavior of AC circuits:

• Directly impacts the rate at which the circuit reacts to changes.
• Affects the reactance of inductors and capacitors due to their dependency on \( \omega \).

Angular frequency is critical in the calculation of reactance, which in turn influences the overall impedance of a circuit. Importantly, the resonant angular frequency of a circuit is where the reactance particularly competes effectively with one another (as in LC circuits) causing a significant drop in impedance. Mastering angular frequency brings clarity to how we can predict and manage the energy oscillations in various types of AC circuits.