Faraday’s Homopolar Generator

Consider a perfectly conducting disk of radius r₀ in a constant magnetic field B perpendicular to the plane of the disk. Sliding contacts are provided

Figure 1.1

at the edge of the disk (C₁) and at its axle (C₂) (see Figure 1.1). This system is Faraday’s “homopolar generator.” When turned at constant angular velocity, it provides a large direct current with no ripple. A torque is produced by a mass M hung on a long string wrapped around the perimeter of the disk.

a) Explain how and why a current flows. Give a quantitative expression for the current as a function of angular velocity.

b) Given a long enough string, this system will reach a constant angular velocity ω_f . Find this ω_f and the associated current.

SOLUTION

a) Consider an electron at a distance r from the axle (see Figure 1.2). It experiences a Lorentz force

(1)

with v = ω × r, so we have a radial force F_r acting on the electron:

(2)

where is the electron charge. Therefore, the equivalent electric field E = -(1/c)ωRr, and the voltage between C₂ and C₁ is

The current i through the resistor R is given by

(3)

Figure 1.2

b) The power P dissipated in the resistance can be found from (3)

The kinetic energy of the disk

(4)

where I is the moment of inertia of the disk. From energy conservation, we may write

(5)

For a constant angular velocity  we have

(6)

So

(7)

and

(8)