15.2.1 Motional EMF

It is observed that when a wire is moved across a constant magnetic field, an emf is induced between the two ends of the wire. Can this phenomenon (called motional emf) be explained by Faraday’s Law?

Firstly, unlike a coil, a wire does not have any area. So there is no \displaystyle \phi =BA to talk about. Secondly, the magnetic field is a constant one. So there is no \displaystyle \frac{d}{{dt}} to talk about. At first glance, \displaystyle \varepsilon =-\frac{{d\Phi }}{{dt}} has nothing to do with this phenomenon. Right?

Not so fast. A wire may not have any area, but it does sweep out an area when it is moving. So we can associate a magnetic flux Φ to the area swept by the wire. Hence \displaystyle \displaystyle \frac{{d\phi }}{{dt}} can be understood as the rate at which the magnetic flux is being “cut” by the wire. Faraday’s Law can now be used to explain motional emf as well!

For a straight wire of length L moving at speed v (perpendicularly) across a uniform magnetic field B, the wire would be sweeping out a rectangular area of Lx, which has an associated flux of BLx. So the rate of flux cut by the wire is

\displaystyle \begin{aligned}\left| \varepsilon  \right|&=\frac{{d\phi }}{{dt}}\\&=\frac{{d(BLx)}}{{dt}}\\&=BL\frac{{dx}}{{dt}}\\&=BLv\end{aligned}

As for the direction of the induced emf, we can use the Fleming’s Right Hand Rule (FRHR). As always, the index and the middle finger points in the direction of the magnetic field B and conventional current I (inside the moving wire) respectively. The thumb points in the direction of the wire’s motion (i.e. velocity v).

Which is the positive/negative terminal?

Notice that the two ends of the moving wire must be closed by an external circuit before the induced current can flow. The wire should be regarded as an emf source just like an alkaline battery, except that it derives its emf from electromagnetic induction rather than chemical reactions. The positive terminal is the end from which the induced current leaves the wire. The negative terminal is the end from which the induced current returns to the wire.

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