Problem 39 - Physics for Scientists and Engineers with Modern Physics

Question

(II) Show that the rms output of an ac generator is \(V_{\mathrm{rms}}=N A B \omega / \sqrt{2}\) where \(\omega=2 \pi f\).

Step-by-Step Solution

Answer

The RMS voltage is \( V_{\mathrm{rms}} = \frac{N A B \omega}{\sqrt{2}} \) where \( \omega = 2 \pi f \).

Step 1: Understand the problem

We need to show that the RMS (Root Mean Square) output voltage of an AC generator is given by the formula \( V_{\mathrm{rms}} = \frac{N A B \omega}{\sqrt{2}} \) where \( \omega = 2 \pi f \). The variables \( N \), \( A \), \( B \), and \( \omega \) represent the number of turns, the area of the coil, the magnetic field strength, and the angular frequency respectively.

Step 2: Recall the induced EMF formula

The instantaneous induced EMF (Electromotive Force) for a coil rotating in a magnetic field is given by \( V = NAB\omega \sin(\omega t) \), where \( \omega t \) is the angular position of the coil at time \( t \).

Step 3: Calculate RMS Voltage

The RMS value of a sinusoidal voltage \( V = V_0 \sin(\omega t) \), where \( V_0 \) is the peak voltage, is given by \( V_{\mathrm{rms}} = \frac{V_0}{\sqrt{2}} \). Here, \( V_0 = NAB\omega \). Substitute \( V_0 \) to get \( V_{\mathrm{rms}} = \frac{NAB\omega}{\sqrt{2}} \).

Step 4: Verify the expression for \(\omega\)

From the problem, \( \omega \) is given as \( 2\pi f \), where \( f \) is the frequency. This is consistent with the standard formula for angular frequency. Therefore, substituting \( \omega = 2 \pi f \) into the expression does not change its form, proving it represents the RMS voltage correctly.

Key Concepts

RMS Voltage

RMS Voltage, or Root Mean Square Voltage, is an important concept when dealing with alternating current (AC) systems. It helps you determine the effective voltage that performs equivalent work as a direct current (DC) circuit. Instead of dealing with varying values throughout a cycle, RMS provides a single steady value.
In a sinusoidal waveform, the RMS voltage comes from the peak voltage, understood with the formula:

• For a sine wave: \( V_{\mathrm{rms}} = \frac{V_0}{\sqrt{2}} \).
• The \( V_0 \) or peak voltage is the maximum voltage attained.

This conversion is vital because it allows you to relate AC circuits directly to DC circuits in terms of power delivery and heating effect.

Electromotive Force (EMF)

Electromotive Force (EMF) is a term used to denote the energy provided per charge in a circuit. In the context of an AC generator, it is the voltage induced when the coil rotates through a magnetic field. This concept is central to generating electrical energy.
The induced EMF in a rotating coil within a magnetic field is expressed as:

• \( V = N A B \omega \sin(\omega t) \)
• Where \( N \) is the number of turns, \( A \) is the area of the coil, \( B \) is the magnetic field strength, and \( \omega t \) is the angular position over time \( t \).

The sine function indicates that the EMF changes cyclically, showing maximum values when the coil is perpendicular to the magnetic field lines.

Angular Frequency

Angular Frequency, symbolized as \( \omega \), gives insight into how fast the coil in an AC generator rotates. It is vital for understanding the time-based behavior of the system.
Angular frequency is defined by the formula:

• \( \omega = 2\pi f \)
• Where \( f \) is the frequency of rotations in hertz (cycles per second).

This formula connects the linear frequency to the angular aspect of rotation, underlining the cyclical nature of alternating currents. It defines how many radians the coil rotates per second, crucial for calculating induced voltages in the system.

Magnetic Field

The Magnetic Field, represented by \( B \), is an essential element in the generation of EMF within an AC generator. It provides the necessary force to induce voltage as the coil moves through it.
Characteristics of the magnetic field in this scenario include:

• The magnetic field is uniform and allows for consistent EMF generation.
• Its strength is directly proportional to the induced EMF, which implies stronger fields generate higher voltages.

The interaction between the magnetic field and coil is what generates electricity, making it an indispensable part of the AC generation process.