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I've learnt that an Electromotive force (EMF) is induced in a loop when the magnetic flux through it changes with time. The induced EMF $\mathcal{E}$ is given by the Faraday's law of electromagnetic induction: $$\mathcal{E}=-\frac{d\phi}{dt}$$ where $\phi=\int\vec B.d\vec s$ is the magnetic flux through the loop. The flux can be varied by either changing the magnetic field, or the area of the loop, or the angle between the magnetic field and the area vector of the loop.

Let us consider a metal rod of length $l$ rotating about an end with a uniform angular velocity $\omega$. A uniform magnetic field $\vec B$ exists in a direction perpendicular to the plane of the screen and is going into it. The free end of the rod slides along an annular metallic disk as shown in the following diagram:

enter image description here

As the rod is rotating about one of its ends, all points on the rod will move in a circle of radius $r$ with a speed given by $v=r\omega$ where $r$ is the distance of the point from the axis of rotation and $\omega$ is the angular speed. Let us consider the following diagram:

enter image description here

The element $dr$ is at a distance $r$ from the axis of rotation and moves with a speed $r\omega$. Due to its speed in the magnetic field, it can be replaced by a cell of EMF $d\mathcal{E}=vBdr=r\omega B dr$. Doing the same thing for each and every element, we can replace the entire rod with a series combination of such cells. The overall EMF of the rod between the points $A$ and $B$ is:

$$\mathcal{E}=\int_0^l r\omega B dr=\frac 1 2 B\omega l^2$$

On this basis, I can clearly understand why an EMF must be induced between the point at the centre and the annular disk. However, when we analyse the system using Faraday's law, it seems there must be no EMF induced as the flux remains constant (both magnetic field and the area of the loop is constant). Why is there a contradiction when we analyse the system in this way? Both methods are well established, and I don't see where I'm going wrong while analysing this using Faraday's law.

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  • $\begingroup$ You need to define the area which is used to evaluate the magnetic flux. $\endgroup$
    – Farcher
    Mar 6, 2020 at 8:57
  • $\begingroup$ @Farcher: Isn't the area simply the area of the circular loop if we neglect the dimensions of the rotating rod? $\endgroup$
    – Vishnu
    Mar 6, 2020 at 8:58
  • $\begingroup$ If that is the case then there is no change in the magnetic flux and no emf induced in the loop ie the annular metallic disc. $\endgroup$
    – Farcher
    Mar 6, 2020 at 9:15
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    $\begingroup$ An article which you might find informative Is Faraday’s Disk Dynamo a Flux-Rule Exception? $\endgroup$
    – Farcher
    Mar 6, 2020 at 9:18

2 Answers 2

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The time rate of change of flux is $B.\frac{ dA}{dt}$. The area swept out $\frac { dA}{dt}$ is $\frac{l^2 \omega }{2}$, so that matches your other result.

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  • $\begingroup$ +1 Thank you for your answer. Now, I understand that if we consider the area as the area swept by the moving part, then we arrive at the correct result. However, could you explain why we must not consider the area of the loop to be that of the annular disk in the question? Clearly it gives a contradictory result and I'm interested in the reason for its failure. $\endgroup$
    – Vishnu
    Mar 6, 2020 at 9:56
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I guess there is no contradiction. EMF is induced along the metal rod, as you analyzed using $dE=vBdr=r\omega Bdr$. There is no EMF along the annular ring because the flux it bounds remains constant.

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