Problem 33 - Physics

Question

The average power used by a stereo speaker is \(55 \mathrm{~W}\). Assuming that the speaker can be treated as a \(4.0-\Omega\) resistance, find the peak value of the ac voltage applied to the speaker.

Step-by-Step Solution

Answer

The peak voltage is approximately 20.95 V.

Step 1: Understanding Power in AC Circuits

In AC circuits, we often use root mean square (RMS) values to simplify calculations. For a given power (P) and resistance (R), we use the formula for the average power: \( P = \frac{V_{rms}^2}{R} \). Here, \( R = 4.0 \Omega \) and \( P = 55 \mathrm{~W} \).

Step 2: Calculating RMS Voltage

Rearrange the formula to solve for \( V_{rms} \): \( V_{rms} = \sqrt{P \times R} \). Substitute the known values: \( V_{rms} = \sqrt{55 \times 4.0} \).

Step 3: Evaluating RMS Voltage

Calculate \( V_{rms} \): \( V_{rms} = \sqrt{220} \approx 14.83 \mathrm{~V} \).

Step 4: Finding Peak Voltage

The peak voltage \( V_{peak} \) is related to the RMS voltage by the equation \( V_{peak} = V_{rms} \times \sqrt{2} \). Substitute \( V_{rms} = 14.83 \mathrm{~V} \): \( V_{peak} = 14.83 \times \sqrt{2} \).

Step 5: Calculate Peak Voltage

Perform the calculation: \( V_{peak} = 14.83 \times 1.414 \approx 20.95 \mathrm{~V} \).

Key Concepts

Average Power in AC Circuits

In AC circuits, the concept of average power is essential to determine how much energy is used over a complete cycle. Unlike DC circuits where power is constant, AC power varies with time. Hence, we talk about the 'average power' which helps in practical analysis.

The formula most commonly used for average power in AC circuits is:

• \( P = \frac{V_{rms}^2}{R} \)

This formula implies that average power (\( P \)) is dependent on the square of the RMS voltage (\( V_{rms} \)) and inversely on resistance (\( R \)). Multi-phase AC uses different forms of the power formula, but the concept of average power remains central to the analysis of how much effective power a device uses.

Understanding RMS Voltage

RMS (Root Mean Square) voltage is a key concept in AC circuits. It’s a kind of "average" value for voltage that allows us to use simple DC formulas to evaluate power in an AC context. The RMS value helps us grasp how an AC voltage or current affects a circuit compared to a constant DC.

Unlike peak voltage, which only quantifies the maximum at a certain instant, RMS provides a more practical measure for power calculations.

• For instance, to find RMS voltage: \( V_{rms} = \sqrt{P \times R} \).
• It literally comes down to finding the square root of the product of power and resistance.

Through such simplification, engineers make informed decisions on the required power rating and capacity for AC devices.

Resistance in Circuits

Resistance in circuits plays a pivotal role in determining how much power is dissipated and how current flows through a circuit. Measured in ohms (\( \Omega \)), resistance is that part of a circuit that resists the flow of electricity, transforming electrical energy into heat or work.

In our situation with the stereo speaker represented as a resistance:

• We view it as a resistor with a value of \( 4.0 \Omega \).
• The relation \( P = \frac{V_{rms}^2}{R} \) also shows how critical knowing your resistance is for accurate power calculations.

A change in this resistance can readily affect the RMS voltage and average power that the circuit handles.

Peak Voltage

Peak voltage (\( V_{peak} \)) refers to the maximum value of voltage in an AC cycle. It represents the high point of the voltage wave wherein it reaches its highest value.

To link RMS voltage to peak voltage, the following relation is used:

• \( V_{peak} = V_{rms} \times \sqrt{2} \)

This conversion is essential because it tells us how high the voltage gets at its peak during each cycle, which maximizes its effect on a device.

Knowing the peak voltage is vital for protecting electronic components, as it impacts their durability and performance. Without accounting for peak voltage, devices may improperly handle excess voltage, leading to malfunctions or damage.