# Charge Trajectory in the Faraday disk

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)?