Question

An alternating voltage is represented as $\mathrm{E}=20\sin 300\mathrm{t}$. The average value of voltage over one cycle will be. (a) zero (b) 10 volt (c) $20\sqrt{2}$ volt (d) $(20/\sqrt{2})$ volt

Short answer

The average value of voltage over one cycle is (a) zero.

Step-by-step solution

Find the Time Period of the Alternating Voltage

In the given voltage function, E(t) = 20 sin(300t), the angular frequency ω is 300 rad/s. To find the time period (T) of the alternating voltage, we can use the formula: T = $\frac{2\pi }{\omega }$ Now, plug in the value of ω: T = $\frac{2\pi }{300}$ s

Calculate the Average Value of the Voltage Over One Cycle

To find the average value of the voltage over one cycle, we will integrate the voltage function E(t) over one time period and then divide the result by the time period. Since the voltage is alternating, we need to find the average of the absolute value of the voltage function: Average Voltage = $\frac{1}{T}{\int }_0^T|E(t)|dt$ First, substitute the given voltage function and the calculated time period T: Average Voltage = $\frac{1}{\frac{2\pi }{300}}{\int }_0^{\frac{2\pi }{300}}|20\sin (300t)|dt$ Now, the absolute value of sin function varies from 0 to 1, so the absolute value of the given voltage function is simply the voltage function itself: Average Voltage = $\frac{1}{\frac{2\pi }{300}}{\int }_0^{\frac{2\pi }{300}}20\sin (300t)dt$ Integrate the function: Average Voltage = $\frac{1}{\frac{2\pi }{300}}[-\frac{1}{15}\cos (300t){]}_0^{\frac{2\pi }{300}}$ Evaluate the definite integral: Average Voltage = $\frac{1}{\frac{2\pi }{300}}[-\frac{1}{15}\cos (300(\frac{2\pi }{300}))+\frac{1}{15}\cos (0)]$ Average Voltage = $\frac{300}{2\pi }[\frac{1}{15}(1-1)]$ Average Voltage = 0 So the correct answer is (a) zero.

Key concepts

Angular Frequency

Angular frequency is a fundamental concept in the study of alternating voltages and waves. It describes how quickly a wave oscillates over time. Imagine a wave as it cycles up and down; angular frequency tells us how fast these cycles occur.
Angular frequency is denoted by the Greek letter $\omega$ (omega) and is expressed in radians per second (rad/s). It can be visualized as the rate of change of the phase of the sinusoidal waveform.
In the equation $E=20\sin (300t)$, $300t$ represents the phase angle, meaning 300 is the angular frequency. This high value indicates rapid oscillations. Understanding angular frequency allows you to derive other useful wave properties such as the time period and wavelength. To calculate the angular frequency, you may use the relation:

• Angular Frequency $\omega =2\pi f$
  'f' is the frequency of the wave.

Recognizing the angular frequency is a crucial step in interpreting wave behaviors such as resonance, phase shifts, and power transfer in AC circuits.

Time Period Calculation

The time period of a wave is a measure of the time taken for one complete cycle of the wave to pass a given point. It is inversely related to the frequency and provides insight into the periodicity of the waveform.
For a sinusoidal wave like $E(t)=20\sin (300t)$, knowing the angular frequency helps in determining the time period. The formula to calculate the time period $T$ is:

• $T=\frac{2\pi }{\omega }$

In this formula, $\omega$ is the angular frequency. As solved, the time period $T$ for our given alternating voltage is $\frac{2\pi }{300}$ seconds. This reflects how long it takes for the alternating voltage to complete one full cycle.
The understanding and calculation of the time period are essential for predicting the behavior of alternating currents in circuits, such as timing for electrical components and signal processing.

Average Voltage Calculation

Calculating the average voltage, particularly for an alternating voltage, gives insight into the net effect over a cycle. However, due to the symmetrical nature of sine waves, the average over a full cycle is zero, as positives and negatives cancel out.
To calculate the average voltage, you integrate the absolute value of the voltage function over one complete cycle and divide by the cycle's duration. The formula used for average voltage is:

• Average Voltage = $\frac{1}{T}{\int }_0^T|E(t)|dt$

In our task, we considered absolute values because the sine function swings above and below zero, causing straightforward averages to cancel out.
Steps in solving this problem included:
- Identifying $E(t)=20\sin (300t)$ as the function
- Performing integration over $0$ to $\frac{2\pi }{300}$ (one time period)
The result showed an average voltage of zero, emphasizing the cancellation effect of alternating phases.

Integration in Physics

Integration is a mathematical process critical in physics for finding areas, volumes, or any quantity that changes continuously. In the context of alternating voltages or currents, it helps compute averages over time.
The typical integration operation involves summing infinitely small parts to find a total effect. In our example with alternating voltage, integration enabled us to derive the average voltage over a cycle.
For sine-based voltage functions, as seen with $E(t)=20\sin (300t)$, integrating serves to account for the complete range through which the waveform moves:

• Calculate the integral of the function within limits specific to one cycle.
• Apply definite integration if bounds are known, providing a precise value rather than a general expression.

This step involves the fundamental theorem of calculus which connects differentiation and integration. Integration, particularly of trigonometric functions, is foundational for physics problems involving periodic functions like oscillations in circuits or mechanical vibrations.