Problem 67 - University Physics with Modern Physics

Question

An inductor, a capacitor, and a resistor are all connected in series across an ac source. If the resistance, inductance, and capacitance are all doubled, by what factor does each of the following quantities change? Indicate whether they increase or decrease: (a) the resonance angular frequency; (b) the inductive reactance; (c) the capacitive reactance. (d) Does the impedance double?

Step-by-Step Solution

Answer

(a) Decreases by \( \frac{1}{\sqrt{2}} \), (b) Doubles, (c) Halves, (d) No, impedance doesn't double.

Step 1: Understanding Resonance Angular Frequency

The resonance angular frequency \( \omega_0 \) for an RLC circuit is given by the formula \( \omega_0 = \frac{1}{\sqrt{LC}} \). If both the inductance \( L \) and capacitance \( C \) are doubled, the new resonance frequency becomes \( \omega_0' = \frac{1}{\sqrt{2L \cdot 2C}} = \frac{1}{2} \cdot \frac{1}{\sqrt{LC}} = \frac{1}{2} \omega_0 \). Thus, the resonance angular frequency decreases by a factor of \( \frac{1}{\sqrt{2}} \).

Step 2: Analyzing Inductive Reactance Change

Inductive reactance \( X_L \) is given by \( X_L = \omega L \). If the inductance \( L \) is doubled, then \( X_L' = \omega (2L) = 2X_L \). Hence, the inductive reactance doubles.

Step 3: Analyzing Capacitive Reactance Change

Capacitive reactance \( X_C \) is given by \( X_C = \frac{1}{\omega C} \). If the capacitance \( C \) is doubled, the new reactance becomes \( X_C' = \frac{1}{\omega (2C)} = \frac{1}{2} X_C \). Thus, the capacitive reactance is halved.

Step 4: Evaluating Impedance Change

The impedance \( Z \) in a series RLC circuit is \( Z = \sqrt{R^2 + (X_L - X_C)^2} \). With doubled \( R \), \( L \), and \( C \), the resistance becomes \( 2R \), the inductive reactance becomes \( 2X_L \), and the capacitive reactance changes to \( \frac{1}{2}X_C \). To find out how the impedance changes, it is required to calculate the new \( Z' = \sqrt{(2R)^2 + ((2X_L) - (\frac{1}{2} X_C))^2} \). Although individual reactances and resistances change significantly, the overall formula for \( Z \) must be evaluated to conclude if it doubles. Generally, it does not simply double.

Key Concepts

Resonance Angular Frequency

In an RLC circuit, the resonance angular frequency is crucial as it determines when the circuit oscillates most efficiently. At resonance, the impedance is minimized, and the voltage across the circuit is at its peak. The formula for the resonance angular frequency, denoted as \( \omega_0 \), is given by:\[\omega_0 = \frac{1}{\sqrt{LC}}\]Here, \( L \) stands for inductance and \( C \) for capacitance. When both inductance and capacitance are doubled, the formula changes to:\[\omega_0' = \frac{1}{\sqrt{2L \cdot 2C}} = \frac{1}{\sqrt{4LC}} = \frac{1}{2} \cdot \frac{1}{\sqrt{LC}}\]This result shows that the new resonance angular frequency is half of its original, meaning it has decreased. This decrease implies that the circuit now naturally oscillates at a lower frequency.

Inductive Reactance

Inductive reactance describes how much an inductor resists the change in current. It depends on the angular frequency \( \omega \) and the inductance \( L \). The formula for inductive reactance, \( X_L \), is:\[ X_L = \omega L\]If the inductance is doubled, the new inductive reactance becomes:\[ X_L' = \omega (2L) = 2X_L\]This shows that when the inductance is doubled, the inductive reactance also doubles. Inductors store energy in a magnetic field, and by doubling the inductance, you effectively increase the opposition to current change, leading to higher reactance.

Capacitive Reactance

Capacitive reactance measures the opposition that a capacitor presents to the change of voltage. The formula for capacitive reactance \( X_C \) is as follows:\[X_C = \frac{1}{\omega C}\]Doubling the capacitance changes the reactance to:\[X_C' = \frac{1}{\omega (2C)} = \frac{1}{2} X_C\]This indicates that the capacitive reactance is halved when capacitance is doubled. Capacitors store energy in an electric field, and more capacitance means more storage capacity, which reduces the current's opposition, leading to decreased reactance.

Impedance in RLC Circuit

Impedance in an RLC circuit combines resistance, inductive reactance, and capacitive reactance, showing how much the total circuit resists the electric current. It is represented as \( Z \) and calculated by:\[Z = \sqrt{R^2 + (X_L - X_C)^2}\]Upon doubling \( R \), \( L \), and \( C \), the resistance becomes \( 2R \), inductive reactance becomes \( 2X_L \), and capacitive reactance becomes \( \frac{1}{2}X_C \). The new impedance is:\[Z' = \sqrt{(2R)^2 + ((2X_L) - (\frac{1}{2} X_C))^2}\]Hence, the impedance doesn't just double straightforwardly but needs to be recalculated as a whole to determine the new effect. This highlights the complexity of impedance where changes in individual parameters can affect the overall impedance in non-linear ways.