Question

A series circuit contains a \(100.0-\Omega\) resistor, a \(0.500-\mathrm{H}\) inductor, a 0.400 - \(\mu \mathrm{F}\) capacitor, and a source of time-varying 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

The resonant angular frequency of the circuit is \(\omega_0 = 2236.07 \, \mathrm{rad/s}\). b) What is the rms current through the circuit at the resonant frequency? The rms current flowing through the circuit at the resonant frequency is \(I = 0.4 \, \mathrm{A}\).

Step-by-step solution

Find the resonant angular frequency of the circuit

To find the resonant angular frequency of the RLC circuit, we can use the formula \(\omega_0 = \frac{1}{\sqrt{LC}}\), where \(\omega_0\) is the resonant angular frequency, \(L\) is the inductance of the inductor, and \(C\) is the capacitance of the capacitor. Here, we are given \(L=0.500 H\) and \(C= 0.400 \times 10^{-6} F\). Plugging the values into the formula, we get: \(\omega_0 = \frac{1}{\sqrt{(0.500 H)(0.400\times10^{-6} F)}}\) Calculate the value of the resonant angular frequency: \(\omega_0 = \frac{1}{\sqrt{2\times10^{-7}}} = 2236.07 \, \mathrm{rad/s}\)

Calculate the impedance of the circuit at the resonant frequency

At the resonant frequency, the reactive components of the impedance (inductor and capacitor) will cancel each other out, resulting in only the resistive component being present. This means that the impedance of the circuit at resonance, \(Z_0\), is equal to the resistance \(R\): \(Z_0=R = 100.0 \, \Omega\)

Calculate the rms current through the circuit at the resonant frequency

Now that we have the impedance of the circuit at the resonant frequency, we can use Ohm's Law to find the rms current flowing through the circuit. The formula for Ohm's Law is \(I = \frac{V}{Z}\), where \(I\) is the rms current, \(V\) is the rms voltage, and \(Z\) is the impedance of the circuit. We are given an rms voltage of \(40.0 V\), and we calculated the impedance at resonance to be \(100.0 \, \Omega\). Plugging these values into the formula, we get: \(I = \frac{40.0 V}{100.0 \, \Omega} = 0.4 \, \mathrm{A}\) The rms current flowing through the circuit at the resonant frequency is \(0.4 A\).

Key concepts

Resonant Angular Frequency

The resonant angular frequency is a fundamental concept in RLC series circuits. It refers to the frequency at which the inductive reactance and capacitive reactance are equal in magnitude but opposite in phase, causing them to cancel each other out. As a result, the circuit exhibits purely resistive behavior with minimal impedance.

According to the formula \(\omega_0 = \dfrac{1}{\sqrt{LC}}\), where \(L\) represents inductance and \(C\) capacitance, we can determine the point of resonance. For example, in a circuit with an inductance of \(0.500 \, \mathrm{H}\) and a capacitance of \(0.400 \times 10^{-6} \mathrm{F}\), the resonant angular frequency is calculated to be approximately \(2236.07 \, \mathrm{rad/s}\).

This frequency is crucial as it is the point where the circuit will respond most strongly to an external signal, often used in radio tuning and other applications requiring frequency selection.

Impedance in RLC Circuit

In the context of an RLC series circuit, impedance is the measure of opposition that the circuit offers to the flow of alternating current. It is a combination of resistance, inductive reactance, and capacitive reactance. However, at the resonant frequency, the inductive and capacitive reactances cancel out, simply leaving the resistive component.

For practical calculations, at resonance, Impedance \(Z_0\) is equivalent to the resistance of the circuit \(R\), which ignores the effects of induction and capacitance. So in a circuit with a resistor of \(100.0 \, \Omega\), the impedance at resonance would be \(100.0 \, \Omega\). The understanding of impedance is key to predicting how the circuit will perform under different frequencies and is an essential parameter for designing electronic circuits.

Ohm's Law

• Ohm's Law is a crucial principle in electrical circuits, defining the relationship between voltage \(V\), current \(I\), and resistance \(R\).
• It is commonly stated as \(I = \dfrac{V}{R}\), expressing that the current through a conductor between two points is directly proportional to the voltage across the two points.
• In the setting of an RLC series circuit, Ohm’s Law helps us determine the root mean square (RMS) current, given the RMS voltage and the impedance.
• At resonance, where the circuit acts purely resistive \(Z = R\), using Ohm's Law simplifies the process of finding the current running through the circuit.

For instance, with a known resistor of \(100.0\, \Omega\) and an applied RMS voltage of \(40.0\, V\), Ohm's Law calculates the RMS current as \(0.4\, A\).

Root Mean Square (RMS) Current

The root mean square (RMS) current is a significant term in AC circuits, and it represents the effective current that would produce the same amount of heat in a resistor as a direct current of the same value. This measurement is crucial because it indicates the real power consumption in a circuit.

The RMS value is derived from the peak current value but accounts for the fact that AC current varies with time. Using the Ohm’s Law formula \(I = \frac{V}{Z}\) at the resonant frequency where impedance is purely resistive, we can calculate the RMS current that flows through the circuit. For an RMS voltage of \(40.0 V\) and an impedance of \(100.0 \Omega\) at resonance, the circuit will have an RMS current of \(0.4 A\). This value allows us to accurately describe the power conditions in the circuit without referring to the constantly changing instantaneous values of current and voltage.