Chapter 23 Electromagnetic Induction, AC Circuits, and Electrical Technologies
Chapter 23 Electromagnetic Induction, AC Circuits, and Electrical Technologies
23.5 Electric Generators
Summary
- Calculate the emf induced in a generator.
- Calculate the peak emf which can be induced in a particular generator system.
Electric generators induce an emf by rotating a coil in a magnetic field, as briefly discussed in Chapter 23.1 Induced Emf and Magnetic Flux. We will now explore generators in more detail. Consider the following example.
Example 1: Calculating the Emf Induced in a Generator Coil
The generator coil shown in Figure 1 is rotated through one-fourth of a revolution (from [latex]{\theta = 0^{\circ}}[/latex] to [latex]{\theta = 90^{\circ}}[/latex]) in 15.0 ms. The 200-turn circular coil has a 5.00 cm radius and is in a uniform 1.25 T magnetic field. What is the average emf induced?
Strategy
We use Faraday’s law of induction to find the average emf induced over a time [latex]{\Delta t}[/latex]:
We know that [latex]{N = 200}[/latex] and [latex]{\Delta t = 15.0 \;\text{ms}}[/latex], and so we must determine the change in flux [latex]{\Delta \phi}[/latex] to find emf.
Solution
Since the area of the loop and the magnetic field strength are constant, we see that
Now, [latex]{\Delta (\text{cos} \;\theta) = -1.0}[/latex], since it was given that [latex]{\theta}[/latex] goes from [latex]{0^{\circ}}[/latex] to [latex]{90^{\circ}}[/latex]. Thus [latex]{\Delta \phi = -AB}[/latex], and
The area of the loop is [latex]{A = \pi r^2 = (3.14 \cdots )(0.0500 \;\text{m})^2 = 7.85 \times 10^{-3} \;\text{m}^2}[/latex]. Entering this value gives
Discussion
This is a practical average value, similar to the 120 V used in household power.
The emf calculated in Example 1 is the average over one-fourth of a revolution. What is the emf at any given instant? It varies with the angle between the magnetic field and a perpendicular to the coil. We can get an expression for emf as a function of time by considering the motional emf on a rotating rectangular coil of width [latex]{w}[/latex] and height [latex]{ \ell}[/latex] in a uniform magnetic field, as illustrated in Figure 2.
Charges in the wires of the loop experience the magnetic force, because they are moving in a magnetic field. Charges in the vertical wires experience forces parallel to the wire, causing currents. But those in the top and bottom segments feel a force perpendicular to the wire, which does not cause a current. We can thus find the induced emf by considering only the side wires. Motional emf is given to be [latex]{\text{emf} = B \ell v}[/latex], where the velocity [latex]{v}[/latex] is perpendicular to the magnetic field [latex]{B}[/latex]. Here the velocity is at an angle [latex]{\theta}[/latex] with [latex]{B}[/latex], so that its component perpendicular to [latex]{B}[/latex] is [latex]{v \;\text{sin} \;\theta}[/latex] (see Figure 2). Thus in this case the emf induced on each side is [latex]{emf = B \ell v \;\text{sin} \theta}[/latex], and they are in the same direction. The total e