Calculating Maximum Value and Angular Frequency for AC Voltage

The problem asks us to find the maximum value and "wavelength" for a sinusoidal alternating voltage with a given RMS value and frequency. In the context of alternating current (AC) circuits, the term "wavelength" is typically associated with travelling waves or transmission lines and represents a spatial dimension. However, the provided options include values in "rad/sec", which is the unit for angular frequency ($\omega$). It is highly probable that "wavelength" in the question is a typo and it is intended to mean "angular frequency". We will proceed by calculating both the maximum value and the angular frequency of the given sinusoidal alternating voltage.

Understanding the Given Values

We are given the following information:

• Type of voltage: Sinusoidal alternating voltage
• Frequency (f): 50 Hz
• RMS value ($V_{rms}$): 200 V

Calculating the Maximum Value (Peak Value)

For a sinusoidal alternating voltage, the relationship between the RMS value ($V_{rms}$) and the maximum value (or peak value, $V_m$) is given by the formula:

$$V_m = V_{rms} \times \sqrt{2}$$

Let's plug in the given RMS value:

$$V_m = 200 \text{ V} \times \sqrt{2}$$

Using the approximate value $\sqrt{2} \approx 1.4142$:

$$V_m \approx 200 \times 1.4142 \text{ V}$$

$$V_m \approx 282.84 \text{ V}$$

This is the calculated maximum value based on the given RMS value of 200 V.

Calculating the Angular Frequency

The angular frequency ($\omega$) of an alternating voltage is related to its linear frequency ($f$) by the formula:

$$\omega = 2 \pi f$$

where $\pi$ is the mathematical constant pi (approximately 3.14159).

Let's plug in the given frequency:

$$\omega = 2 \times \pi \times 50 \text{ Hz}$$

$$\omega = 100 \pi \text{ rad/sec}$$

Using the approximate value $\pi \approx 3.14$:

$$\omega \approx 100 \times 3.14 \text{ rad/sec}$$

$$\omega \approx 314 \text{ rad/sec}$$

This is the calculated angular frequency based on the given frequency of 50 Hz.

Comparing Calculations with Options

Our calculations yielded:

• Maximum Value ($V_m$): Approximately 282.84 V
• Angular Frequency ($\omega$): Approximately 314 rad/sec

Let's look at the options provided:

1. 272.2 V, 300 rad/sec
2. 280.2 V, 308 rad/sec
3. 282.2 V, 314 rad/sec
4. 277.4 V, 302 rad/sec

Option 3 provides 282.2 V for the maximum value and 314 rad/sec for the angular frequency. Our calculated maximum value (282.84 V) is very close to 282.2 V, and our calculated angular frequency (314 rad/sec) matches exactly with 314 rad/sec when using $\pi \approx 3.14$. Therefore, option 3 is the closest match based on standard calculations.

Summary of Results

Based on our analysis and assuming "wavelength" refers to angular frequency:

• The maximum value of the voltage is approximately 282.8 V (closest option is 282.2 V).
• The angular frequency is exactly 314 rad/sec (using $\pi \approx 3.14$).

Revision Table: AC Voltage Parameters

Additional Information: AC Circuit Concepts

Understanding the different parameters of a sinusoidal AC voltage is crucial in electrical engineering. Here's a bit more detail:

• Sinusoidal AC Voltage: This is a voltage that varies over time according to a sine or cosine function. It is the most common type of AC voltage found in power systems.
• Frequency (f): Measured in Hertz (Hz), it represents the number of complete cycles the voltage waveform completes in one second. A frequency of 50 Hz means 50 cycles per second.
• RMS Value ($V_{rms}$): The Root Mean Square value is a way to represent the effective value of an AC voltage or current. It is the equivalent DC voltage or current that would dissipate the same amount of power in a resistive load. For a sine wave, $V_{rms} = V_m / \sqrt{2}$.
• Maximum Value ($V_m$): Also known as the peak value, this is the highest instantaneous voltage value reached during each cycle of the waveform.
• Angular Frequency ($\omega$): Measured in radians per second (rad/sec), it represents the rate of change of the angle of the sinusoidal function. It is related to the linear frequency by $\omega = 2\pi f$. It's often used in formulas involving capacitors and inductors.
• Relationship between parameters: The formulas $V_m = V_{rms} \sqrt{2}$ and $\omega = 2\pi f$ are fundamental for converting between these common AC voltage parameters.