Chapter 23 Electromagnetic Induction, AC Circuits, and Electrical Technologies
Chapter 23 Electromagnetic Induction, AC Circuits, and Electrical Technologies
23.3 Motional Emf
- 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 F=qvBsinθF=qvBsinθ size 12{F= ital “qvB””sin”θ} {}, which moves opposite charges in opposite directions and produces an emf=Bℓvemf=Bℓv size 12{“emf”=Bℓv} {}. We saw that the Hall effect has applications, including measurements of BB size 12{B} {} and vv size 12{v} {}. 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 [link]. A rod is moved at a speed
vv
along a pair of conducting rails separated by a distance
ℓℓ
in a uniform magnetic field BB size 12{B} {}. The rails are stationary relative to BB size 12{B} {} and are connected to a stationary resistor RR size 12{R} {}. The resistor could be anything from a light bulb to a voltmeter. Consider the area enclosed by the moving rod, rails, and resistor. BB size 12{B} {} 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, N=1N=1 size 12{N=1} {} and the flux Φ=BAcosθΦ=BAcosθ size 12{Φ= ital “BA””cos”θ} {}. We have θ=0ºθ=0º
and cosθ=1cosθ=1, since BB is perpendicular to AA.
Now ΔΦ=Δ(BA)=BΔAΔΦ=Δ(BA)=BΔA size 12{ΔΦ=Δ ( ital “BA” ) =BΔA} {}, since BB size 12{B} {} is uniform. Note that the area swept out by the rod is ΔA=ℓΔxΔA=ℓΔx size 12{ΔA=ℓΔx} {}. Entering these quantities into the expression for emf yields
Finally, note that Δx/Δt=vΔx/Δt=v size 12{Δx/Δt=v} {}, the velocity of the rod. Entering this into the last expression shows that
is the motional emf. This is the same expression given for the Hall effect previously.
There are many connections between the electric force and the magnetic force. The fact that a moving electric field produces a magnetic field and, conversely, a moving magnetic field produces an electric field is part of why electric and magnetic forces are now considered to be different manifestations of the same force.