Problem 54 - Fundamentals of Physics

Question

What is the maximum value of an ac voltage whose rms value is \(100 \mathrm{~V} ?\)

Step-by-Step Solution

Answer

The maximum value is 141.4 V.

Step 1: Understanding RMS Voltage

RMS (Root Mean Square) voltage is a method of expressing AC (Alternating Current) voltage in a way that represents an equivalent DC (Direct Current) voltage's ability to do work. For a sinusoidal AC waveform, the RMS voltage is defined as the maximum (peak) voltage divided by the square root of 2.

Step 2: Formula to Find Peak Voltage

To find the peak voltage from the RMS voltage, we use the formula: \[V_{ ext{peak}} = V_{ ext{rms}} imes \sqrt{2}\] where \(V_{ ext{rms}}\) is the RMS voltage.

Step 3: Inserting RMS Value

Insert the given RMS value into the formula: \[V_{ ext{peak}} = 100 imes \sqrt{2}\]

Step 4: Calculating Peak Voltage

Calculate the peak voltage by computing the result: \[V_{ ext{peak}} = 100 imes 1.414 \] \[V_{ ext{peak}} = 141.4 ext{ V}\]

Step 5: Conclusion

The maximum value or peak voltage of an AC voltage with an RMS value of 100 V is 141.4 V.

Key Concepts

AC Voltage

AC voltage, or Alternating Current voltage, is an electrical voltage that changes direction and magnitude periodically. Unlike DC (Direct Current) voltage, where the flow of electric charge is constant in one direction, AC voltage shifts back and forth. This alternating nature of AC voltage makes it ideal for power transmission over long distances, which is why it is the most common form of electricity used globally.

Understanding its behavior is crucial for many applications. AC voltage is often described using terms like amplitude, frequency, and phase. Its value changes continuously over time, typically following a sinusoidal pattern. This consistent change in direction helps AC systems to efficiently transmit electricity on power grids and into homes.

AC voltage is often measured in terms of its RMS (Root Mean Square) value. This approach provides a meaningful way to express the AC voltage's effectiveness in terms of its capacity to do work, similar to how DC voltage is represented.

Peak Voltage

Peak voltage is the maximum voltage level that an AC waveform can reach. In the context of a sinusoidal waveform, peak voltage is an important parameter as it indicates the highest point of voltage reached during one cycle.

To determine the peak voltage from an RMS value, a specific formula is used:

• For sinusoidal AC waveforms, the peak voltage (\( V_{\text{peak}} \) ) can be computed by multiplying the RMS voltage (\( V_{\text{rms}} \) ) by the square root of 2.

This relationship stems from the mathematical properties of a sinusoidal waveform. For instance, if the RMS value of an AC voltage is 100 V, the peak voltage is calculated as:

• \[V_{\text{peak}} = 100 \times \sqrt{2} = 141.4 \text{ V}\]

This means the voltage can peak at 141.4 V at the top of its wave cycle.

Knowing the peak voltage is essential in designing electrical systems to ensure they can handle max voltages without damage.

Sinusoidal Waveform

Sinusoidal waveforms are a fundamental aspect of AC voltage. A sinusoidal waveform is a smooth, repetitive oscillation that is shaped like a sine wave. It is characterized by its natural, flowing shape and is mathematically described by the sine function. This waveform is significant because it can describe the pattern of an AC signal over time.

The sinusoidal waveform is ideal for AC current because it can be easily transformed with devices like transformers and has minimal electromagnetic interference. Key features of a sinusoidal waveform include:

• Amplitude: The height of the wave, representing the peak voltage.
• Frequency: How often the waveform repeats in one second, measured in hertz (Hz).
• Phase: The position of the waveform relative to time zero.

Recognizing these features helps in understanding how AC systems function and are managed. Not only does the sinusoidal nature allow for efficient energy transmission, but it also facilitates downscaling and distribution across various electrical devices easily.