Problem 25 - Physics

Question

When a resistor is connected by itself to an ac generator, the average power delivered to the resistor is \(1.000 \mathrm{~W}\). When a capacitor is added in series with the resistor, the power delivered is \(0.500 \mathrm{~W}\). When an inductor is added in series with the resistor (without the capacitor), the power delivered is \(0.250 \mathrm{~W}\). Determine the power delivered when both the capacitor and the inductor are added in series with the resistor. Section 23.4 Resonance in Electric Circuits

Step-by-Step Solution

Answer

The power delivered with both capacitor and inductor in series is 1.000 W.

Step 1: Understanding the Power Formula

In an AC circuit, the power delivered to a component is given by \( P = V_{rms} I_{rms} \cos \phi \), where \( \phi \) is the phase angle. For a purely resistive circuit, \( \cos \phi = 1 \), meaning all power is dissipated as heat.

Step 2: Power with Resistor Only

When the resistor is connected by itself, the entire power is dissipated as heat in the resistor. Here, \( P_R = 1.000 \mathrm{~W} \). This indicates \( \cos \phi = 1 \), and the entire power component is resistive.

Step 3: Power with Resistor and Capacitor

Adding a capacitor in series causes a phase difference that reduces \( \cos \phi \). Power is now \( P_{RC} = 0.500 \mathrm{~W} \). This reduction indicates reactive power is present, reducing the average power.

Step 4: Power with Resistor and Inductor

Similarly, the addition of an inductor in series with the resistor reduces the power further, \( P_{RL} = 0.250 \mathrm{~W} \). This indicates an increased phase shift compared to when only the resistor is used.

Step 5: Analyzing the RLC Circuit

When both the capacitor and inductor are added in series, they can produce resonance. In resonance, inductive and capacitive reactances cancel each other, making the circuit purely resistive again. Thus, \( \cos \phi = 1 \), leading back to maximal power delivery.

Step 6: Calculate Power at Resonance

Since at resonance the circuit acts purely resistive, the power delivered returns to the value with only the resistor: \( P_{RLC} = 1.000 \mathrm{~W} \). Hence, with resonance, the reactive components have no effect, and the net power returns to the resistive power level.

Key Concepts

Power in AC Circuits

In alternating current (AC) circuits, calculating the power is slightly different than in direct current (DC) circuits due to the alternating nature of the current and voltage, which can lead to varying phase angles. The power in AC circuits is computed using the formula:\[ P = V_{rms} I_{rms} \cos \phi \]where \( P \) is the average power, \( V_{rms} \) is the root mean square voltage, \( I_{rms} \) is the root mean square current, and \( \phi \) is the phase angle between current and voltage.
When only resistors are involved in the circuit, the phase angle \( \phi \) is zero, making\( \cos \phi = 1 \). Thus, all power supplied by the source is dissipated (or consumed) by the resistor and converted to heat.

• When reactive components, like inductors or capacitors, are added to the circuit, the situation changes.
• These components introduce a phase shift, reducing the effective power by causing \( \cos \phi \) to decrease from 1.
• This is why, in a mixed resistor-capacitor or resistor-inductor circuit, the average power delivered is less than when only a resistor is present.

Phase Angle in AC Circuits

The phase angle \( \phi \) is a critical concept in AC circuits, representing the lag or lead between the voltage and current waveforms. In circuits consisting solely of resistors, the current is in phase with the voltage, meaning the phase angle is zero (\( \phi = 0 \)). This makes \( \cos \phi = 1 \), and hence, optimal power transfer occurs.
However, when capacitors or inductors are introduced,

• Capacitors cause the current to lead the voltage, resulting in a negative phase angle.
• Inductors, on the other hand, make the current lag behind the voltage, creating a positive phase angle.

These results mean that \( \cos \phi \) decreases, and therefore, the actual power available for doing work gets reduced. By understanding and adjusting the phase angle, engineers can optimize power usage in AC systems, which is crucial for efficient electrical grid management and any AC driven equipment.

Inductive and Capacitive Reactance

Inductive and capacitive reactance are important aspects of AC circuits. These factors describe how inductors and capacitors resist changes in current, leading to shifts in current and voltage relationships. Reactance is measured in ohms (\( \Omega \)) and affects the circuit impedance.

• Inductive reactance \( X_L \) is given by \( X_L = 2\pi f L \), where \( f \) is the frequency in hertz and \( L \) is the inductance in henries.
• Capacitive reactance \( X_C \) is given by \( X_C = \frac{1}{2\pi f C} \), where \( C \) is the capacitance in farads.

In circuits with both inductors and capacitors, these reactances can either add up or cancel each other out, depending on the circuit configuration. At resonance, where \( X_L = X_C \), inductive and capacitive reactances cancel each other, leaving only the resistive component. The impedance is minimized, maximizing the power factor and enabling maximal power delivery, as the AC circuit behaves like a purely resistive circuit. Understanding these concepts helps in designing circuits for specific functionalities such as filters and oscillators.