Understanding Lenz's Law: Solved Example on Homopolar Generator

[andrew410]

The homopolar generator, also called the Faraday disk, is a low-voltage, high-current electric generator. It consists of a rotating conducting disk with one stationary brush (a sliding electrical contact) at its axle and another at a point on its circumference, as shown in the figure below.
Figure: http://east.ilrn.com/books/sepsp06t/pse6e.31.24p.e.jpg
A magnetic field is applied perpendicular to the plane of the disk. Assume the field is 0.935 T, the angular speed is 3180 rev/min, and the radius of the disk is 0.383 m. Find the magnitude of the emf generated between the brushes.

I know I have to use Faraday's law of induction to solve this, but there is no change in the field or the area. Any help on this would be great! Thx in advance! :)

 

[Doc Al]

I know I have to use Faraday's law of induction to solve this, but there is no change in the field or the area.

Ah, but the area does change. Imagine a radius of the disk sweeping out an area as the disk rotates.

Note that this is equivalent to calculating the "motional EMF" as the conducting disk sweeps through the magnetic field.

 

[wais]

Lenz's Law states that the direction of an induced current in a conductor will always oppose the change that produced it. In this case, the rotating disk is creating a changing magnetic flux through the disk as it rotates, which induces an emf (electromotive force) in the disk according to Faraday's law of induction.

To solve this example, we can use the equation e = -N(dΦ/dt), where e is the emf, N is the number of turns in the coil, and dΦ/dt is the change in magnetic flux over time. Since the disk is rotating at a constant speed, there is a constant change in the magnetic flux through the disk, and we can simplify the equation to e = -NΦω, where ω is the angular speed.

In this example, we are given the value of the magnetic field (B = 0.935 T), the angular speed (ω = 3180 rev/min = 333.3 rad/s), and the radius of the disk (r = 0.383 m). We can calculate the magnetic flux through the disk using the equation Φ = BAcosθ, where A is the area of the disk and θ is the angle between the magnetic field and the normal to the disk. Since the magnetic field is perpendicular to the disk, θ = 90° and we can simplify the equation to Φ = BA.

Now, we can substitute the given values into our equation for emf: e = -NΦω = -NBAω. We know the number of turns in the coil is 1 (since there is only one stationary brush), so we can simplify the equation further to e = -BAω.

Finally, we can plug in our values and solve for the emf: e = -(0.935 T)(0.383 m)(333.3 rad/s) = -117.7 V. Since the emf is negative, it means that the induced current will flow in the opposite direction of the rotation of the disk, in accordance with Lenz's Law.

In summary, Lenz's Law helps us understand the direction of the induced current in a homopolar generator. By using Faraday's law of induction and the given values for the magnetic field, angular speed, and radius of the disk, we can calculate the magnitude of the emf generated between the brushes.