# 13.3 Motional emf

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By the end of this section, you will be able to:
• Determine the magnitude of an induced emf in a wire moving at a constant speed through a magnetic field
• Discuss examples that use motional emf, such as a rail gun and a tethered satellite

Magnetic flux depends on three factors: the strength of the magnetic field, the area through which the field lines pass, and the orientation of the field with the surface area. If any of these quantities varies, a corresponding variation in magnetic flux occurs. So far, we’ve only considered flux changes due to a changing field. Now we look at another possibility: a changing area through which the field lines pass including a change in the orientation of the area.

Two examples of this type of flux change are represented in [link] . In part (a), the flux through the rectangular loop increases as it moves into the magnetic field, and in part (b), the flux through the rotating coil varies with the angle $\theta$ .

It’s interesting to note that what we perceive as the cause of a particular flux change actually depends on the frame of reference we choose. For example, if you are at rest relative to the moving coils of [link] , you would see the flux vary because of a changing magnetic field—in part (a), the field moves from left to right in your reference frame, and in part (b), the field is rotating. It is often possible to describe a flux change through a coil that is moving in one particular reference frame in terms of a changing magnetic field in a second frame, where the coil is stationary. However, reference-frame questions related to magnetic flux are beyond the level of this textbook. We’ll avoid such complexities by always working in a frame at rest relative to the laboratory and explain flux variations as due to either a changing field or a changing area.

Now let’s look at a conducting rod pulled in a circuit, changing magnetic flux. The area enclosed by the circuit ‘MNOP’ of [link] is lx and is perpendicular to the magnetic field, so we can simplify the integration of [link] into a multiplication of magnetic field and area. The magnetic flux through the open surface is therefore

${\text{Φ}}_{\text{m}}=Blx.$

Since B and l are constant and the velocity of the rod is $v=dx\text{/}dt,$ we can now restate Faraday’s law, [link] , for the magnitude of the emf in terms of the moving conducting rod as

$\epsilon =\frac{d{\text{Φ}}_{\text{m}}}{dt}=Bl\frac{dx}{dt}=Blv.$

The current induced in the circuit is the emf divided by the resistance or

$I=\frac{Blv}{R}.$

Furthermore, the direction of the induced emf satisfies Lenz’s law, as you can verify by inspection of the figure.

This calculation of motionally induced emf is not restricted to a rod moving on conducting rails. With $\stackrel{\to }{F}=q\stackrel{\to }{v}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\stackrel{\to }{B}$ as the starting point, it can be shown that $\epsilon =\text{−}d{\text{Φ}}_{\text{m}}\text{/}dt$ holds for any change in flux caused by the motion of a conductor. We saw in Faraday’s Law that the emf induced by a time-varying magnetic field obeys this same relationship, which is Faraday’s law. Thus Faraday’s law holds for all flux changes , whether they are produced by a changing magnetic field, by motion, or by a combination of the two.

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