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In the left figure, a Faraday disk rotates in a uniform magnetic field. A circuit is connected to the disk via a sliding contact, as in the figure. The EMF of the circuit is found by integrating the magnetic force on each charged particle over the radius of the disk:

$$ \mathcal E = \int_0^a f_{mag}\text{d}s = \omega B\int_0^as\text{d}s = \frac{\omega B a^2}{2}, $$

where $a$ is the disk radius and $\omega$ is the angular velocity of the disk.

In the figure to the right, the blue curve, a spiral moving outward, is the trajectory of a particular (positive) charge in the disk, under the influence of the combination of the magnetic force and its tangential velocity, denoted by $\mathbf V$

If I stick to the definition of the EMF, which is the work per unit charge done by the driving force over the circuit, then I don't have a problem understanding the calculation of the EMF of the circuit, as shown above.

However, I am having an issue understanding why there can be a current in the loop even though the trajectories of the charges are spirals. Intuitively, I would expect an electron in the disk to move in a straight line from the center to the edge.

Could you explain to me why the trajectories of the charges in the disk do not contradict the fact that we have a current from left to right (with reference to the right figure)?

enter image description here

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Just think of this as a normal circuit with a battery in series with a resistor, where our battery is our spinning disk in the magnetic field.

The spinning disk provides the EMF (work) to move the charges around the circuit. The actual path of the charges in the disk does not change this. Any charge that arrives at the contact on the edge of the disk will then move through the resistor around the circuit.

The same idea of this question could be asked for anything that provides an EMF for a circuit (even your typical battery).

Your charges cannot move in a straight line since the disk is rotating, and the charges are still part of the disk. If the charges were not rotating then there would be no induced EMF to begin with. Or another way to look at it: if the charges did start to move in a straight line then the magnetic field would change their direction anyway.

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  • $\begingroup$ I understand that the charges cannot move in a straight line. What I am looking for is how the spiral trajectories of the charge could still manage to create a current flowing in a straight line. $\endgroup$ – A Slow Learner Feb 27 '18 at 4:42
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    $\begingroup$ I'm confused as to where you want the current to be in a straight line. We have specified that the charges in the disk move in a spiral. The charges in the wire just move along the wire due to the field inside of the wire produced by the EMF. Where are you wanting a straight current? $\endgroup$ – Aaron Stevens Feb 27 '18 at 12:31

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