Electromagnetic Induction, AC Circuits, and Electrical Technologies
Electromagnetic Induction, AC Circuits, and Electrical Technologies
48 Motional Emf
Learning Objectives
- Calculate emf, force, magnetic field, and work due to the motion of an object in a magnetic field.
As we have seen, any change in magnetic flux induces an emf opposing that change—a process known as induction. Motion is one of the major causes of induction. For example, a magnet moved toward a coil induces an emf, and a coil moved toward a magnet produces a similar emf. In this section, we concentrate on motion in a magnetic field that is stationary relative to the Earth, producing what is loosely called motional emf.
One situation where motional emf occurs is known as the Hall effect and has already been examined. Charges moving in a magnetic field experience the magnetic force [latex]F=\text{qvB}\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}\theta[/latex], which moves opposite charges in opposite directions and produces an [latex]\text{emf}=\mathrm{B\ell v}[/latex]. We saw that the Hall effect has applications, including measurements of [latex]B[/latex] and [latex]v[/latex]. We will now see that the Hall effect is one aspect of the broader phenomenon of induction, and we will find that motional emf can be used as a power source.
Consider the situation shown in Figure 48.1. A rod is moved at a speed [latex]v[/latex] along a pair of conducting rails separated by a distance [latex]\ell[/latex] in a uniform magnetic field [latex]B[/latex]. The rails are stationary relative to [latex]B[/latex] and are connected to a stationary resistor [latex]R[/latex]. The resistor could be anything from a light bulb to a voltmeter. Consider the area enclosed by the moving rod, rails, and resistor. [latex]B[/latex] is perpendicular to this area, and the area is increasing as the rod moves. Thus the magnetic flux enclosed by the rails, rod, and resistor is increasing. When flux changes, an emf is induced according to Faraday’s law of induction.
To find the magnitude of emf induced along the moving rod, we use Faraday’s law of induction without the sign:
Here and below, “emf” implies the magnitude of the emf. In this equation, [latex]N=1[/latex] and the flux [latex]\Phi =\text{BA}\phantom{\rule{0.25em}{0ex}}\text{cos}\phantom{\rule{0.25em}{0ex}}\theta[/latex]. We have [latex]\theta =0º[/latex]
and [latex]\text{cos}\phantom{\rule{0.25em}{0ex}}\theta =1[/latex], since [latex]B[/latex] is perpendicular to [latex]A[/latex].
Now [latex]\Delta \Phi =\Delta \left(\text{BA}\right)=B\Delta A[/latex], since [latex]B[/latex] is uniform. Note that the area swept out by the rod is [latex]\Delta A=\ell \Delta x[/latex]. Entering these quantities into the expression for emf yields
Finally, note that [latex]\Delta x/\Delta t=v[/latex], the velocity of the rod. Entering this into the last expression shows that
Making Connections: Unification of Forces
There are many connections between the electric force and the magnetic force. The fact that a moving electric