The voltage across the terminals of an ac power supply varies with time according to Eq. (31.1) v = Vcosωt. The voltage amplitude is V = 45.0 V. What are (a) the root-mean-square potential difference V_rms and (b) the average potential difference V_av between the two terminals of the power supply?

Verified step by step guidance

1. Understand the given equation: The voltage across the terminals of the AC power supply is given as \( v = V \cos(\omega t) \), where \( V \) is the voltage amplitude (45.0 V in this case), \( \omega \) is the angular frequency, and \( t \) is time. The problem asks for the root-mean-square (RMS) potential difference \( V_{\text{rms}} \) and the average potential difference \( V_{\text{av}} \).
2. To calculate \( V_{\text{rms}} \): The RMS value of an AC voltage is defined as \( V_{\text{rms}} = \frac{V}{\sqrt{2}} \), where \( V \) is the voltage amplitude. Substitute the given value of \( V = 45.0 \ \text{V} \) into this formula to find \( V_{\text{rms}} \).
3. To calculate \( V_{\text{av}} \): The average potential difference over a full cycle of a sinusoidal AC voltage is zero because the positive and negative halves of the cycle cancel each other out. However, if the problem refers to the average of the absolute value of the voltage (the rectified average), the formula is \( V_{\text{av}} = \frac{2V}{\pi} \). Substitute \( V = 45.0 \ \text{V} \) into this formula to find \( V_{\text{av}} \).
4. Perform the substitution for \( V_{\text{rms}} \): Use the formula \( V_{\text{rms}} = \frac{V}{\sqrt{2}} \) and substitute \( V = 45.0 \ \text{V} \). Simplify the expression to find the numerical value of \( V_{\text{rms}} \).
5. Perform the substitution for \( V_{\text{av}} \): Use the formula \( V_{\text{av}} = \frac{2V}{\pi} \) and substitute \( V = 45.0 \ \text{V} \). Simplify the expression to find the numerical value of \( V_{\text{av}} \).

Key Concepts

Root Mean Square (RMS) Voltage

The root mean square (RMS) voltage is a statistical measure of the magnitude of a varying voltage. For an AC voltage described by the equation v = Vcos(ωt), the RMS voltage is calculated as V_rms = V/√2. This value represents the equivalent DC voltage that would deliver the same power to a load, making it crucial for understanding AC circuits.

Average Voltage

The average voltage over one complete cycle of an AC waveform is the mean value of the instantaneous voltage. For a cosine waveform, the average potential difference V_av is zero over a full cycle, but for practical applications, it is often calculated over a half-cycle, yielding V_av = (2V/π). This concept is important for understanding the effective voltage delivered to devices.

AC Voltage Waveform

An AC voltage waveform is a periodic function that varies with time, typically represented as a sine or cosine function. The equation v = Vcos(ωt) describes how the voltage oscillates between positive and negative values, affecting how electrical devices operate. Understanding the characteristics of AC waveforms is essential for analyzing power supply behavior and its impact on electrical systems.