Question

At a frequency \(\omega_1\) the reactance of a certain capacitor equals that of a certain inductor. (a) If the frequency is changed to \(\omega_2\) = \(2\omega_1\), what is the ratio of the reactance of the inductor to that of the capacitor? Which reactance is larger? (b) If the frequency is changed to \(\omega3\)= \(\omega_1\)/3 , what is the ratio of the reactance of the inductor to that of the capacitor? Which reactance is larger? (c) If the capacitor and inductor are placed in series with a resistor of resistance R to form an \(L-R-C\) series circuit, what will be the resonance angular frequency of the circuit?

Short answer

(a) Ratio: 4, Inductor has larger reactance; (b) Ratio: 1/9, Capacitor has larger reactance; (c) Resonance frequency is \( \omega_1 \).

Step-by-step solution

Determine initial condition at frequency \( \omega_1 \)

The reactance of a capacitor is given by \( X_C = \frac{1}{\omega C} \), and for an inductor it is \( X_L = \omega L \). At \( \omega_1 \), the reactance of both the capacitor and inductor are equal, so: \[ X_C = X_L \Rightarrow \frac{1}{\omega_1 C} = \omega_1 L. \] This gives the initial condition, \( \omega_1^2 = \frac{1}{LC} \).

Analyze reactance at frequency \( \omega_2 = 2\omega_1 \)

When the frequency is changed to \( \omega_2 = 2\omega_1 \), the reactance of the inductor is \( X_L' = \omega_2 L = 2\omega_1 L \), whereas the reactance of the capacitor is \( X_C' = \frac{1}{\omega_2 C} = \frac{1}{2\omega_1 C} \). The ratio of the inductive to capacitive reactance is: \[ \frac{X_L'}{X_C'} = \frac{2\omega_1 L}{\frac{1}{2\omega_1 C}} = 4 \cdot \omega_1^2 LC. \] Given \( \omega_1^2 = \frac{1}{LC} \), the ratio is \( 4 \). The inductive reactance is larger.

Analyze reactance at frequency \( \omega_3 = \frac{\omega_1}{3} \)

When the frequency is changed to \( \omega_3 = \frac{\omega_1}{3} \), the reactance of the inductor is \( X_L'' = \omega_3 L = \frac{\omega_1}{3} L \), and the reactance of the capacitor is \( X_C'' = \frac{1}{\omega_3 C} = 3\frac{1}{\omega_1 C} \). The ratio of the inductive to capacitive reactance is: \[ \frac{X_L''}{X_C''} = \frac{\frac{\omega_1}{3} L}{3\frac{1}{\omega_1 C}} = \frac{1}{9} \cdot \omega_1^2 LC. \] Again, using \( \omega_1^2 = \frac{1}{LC} \), the ratio is \( \frac{1}{9} \). The capacitive reactance is larger.

Calculate the resonance frequency of the L-R-C circuit

For an L-R-C series circuit, the resonance angular frequency is defined as \( \omega = \frac{1}{\sqrt{LC}} \). From the initial condition, we found that \( \omega_1^2 = \frac{1}{LC} \), which shows that at \( \omega = \omega_1 \), the circuit will be at resonance.

Key concepts

Reactance in L-R-C Circuits

Reactance is a key concept in understanding how circuits containing capacitors and inductors behave with changing frequencies. It is important to note that reactance is different from resistance. While resistance is a measure of the opposition to the flow of direct current, reactance deals with alternating currents and revolves around how capacitors and inductors store and release energy.
Reactance is frequency-dependent, which can be illustrated through the formulas for capacitive and inductive reactance:

• Capacitive reactance, given by the formula \( X_C = \frac{1}{\omega C} \), shows that as the frequency \( \omega \) increases, the capacitive reactance decreases.
• Inductive reactance, given by \( X_L = \omega L \), increases with frequency, meaning that as frequency goes up, the inductive reactance also goes up.

Understanding these relationships helps us analyze changes in reactance with changes in frequency, which is essential in step-by-step problems like calculating the ratio of reactance at different frequencies.

Resonance Frequency in L-R-C Circuits

In an L-R-C circuit, resonance is a special condition where the capacitive and inductive reactances are equal in magnitude, but opposite in phase. This means they cancel each other out, resulting in minimal impedance from the reactance, and the circuit behaves like a simple resistor circuit.
Resonance is identified by the resonance angular frequency \( \omega_0 \), which is given by:
\[ \omega_0 = \frac{1}{\sqrt{LC}} \]
This equation shows that the resonance frequency is determined by the values of the inductance \( L \) and capacitance \( C \). At this frequency, the voltage and current in the circuit are in phase, enhancing the circuit's efficiency.
Many practical applications, such as tuning radios and filtering signals, rely on resonance. Understanding the condition for resonance allows engineers and physicists to design systems that maximize the desired outcomes.

The Roles of Capacitors and Inductors

Both capacitors and inductors play crucial roles in L-R-C circuits, each affecting the circuit in unique ways.

• Capacitors: Store energy in an electric field and resist changes in voltage. Their opposition to current flow is measured by capacitive reactance \( X_C = \frac{1}{\omega C} \).
• Inductors: Store energy in a magnetic field and resist changes in current. Inductive reactance \( X_L = \omega L \) quantifies their opposition to current flow.

These components determine how the circuit's reactance changes with frequency. By combining capacitors and inductors, designers can create circuits that selectively filter frequencies, control the timing of signals, or match impedances.
So, understanding the individual characteristics and behavior of capacitors and inductors is critical to mastering complex electrical circuits.