Faraday's first dynamo: An alternate analysis

Abstract

The steady-state charge densities, electric potential, and current densities are determined analytically in the case of the first dynamo created by Michael Faraday, which consists of a conducting disk rotating between the poles of an off-axis permanent magnet. The results obtained are compared with another work that considered the same problem using a different approach. We also obtain analytical expressions for the total current on the disk and for the dynamo's electromotive force.

The induction motor

“…We use basically the same notation for the dimensions and coordinates used in [8,9], which discuss Faradayʼs dynamo. The conducting disc has radius A, thickness h, electrical conductivity σ, permeability 0 μ , permitivity ϵ, and rotates with angular speed along the z-axis: zω ω ⃗ =…”

“…. Here we use a procedure analogous to that used in [8], to find a second field E ⃗ ′, which must be added to 3) valid for region I), but with a change of sign and a constant c α in place of the constant c, where α is a dimensionless parameter to be determined: ⃗ , and its induced electric field inside the conducting disc is identical with that of the induced charges on the surface A ρ = . Thus, being E E E i ⃗ = ⃗ + ⃗ ′, and using equations (1), we obtain: in region II, and…”

“…We also calculate the induced currents due to the motional electromotive force, which are responsible for the damping torque in the disc. These eddy currents are the currents induced on Faradayʼs first dynamo [8,9]. Their analytic expressions were obtained in [8] from the analysis of space charges induced in a conductor moving in a static magnetic field.…”

“…These eddy currents are the currents induced on Faradayʼs first dynamo [8,9]. Their analytic expressions were obtained in [8] from the analysis of space charges induced in a conductor moving in a static magnetic field. The torques due to all these currents are calculated and an analytical expression for the no-load angular speed is obtained.…”

Analytical results for rotating and linear magnetic brakes

“…These space charges and the fields associated with them have been studied in a number of conducting systems rotating in static and uniform magnetic fields (see, for example, [6,7,8]). Depending on the geometries of the conductor and the magnetic field, these space charges may be accompanied by an electric current density J, as in the case of the Faraday's first dynamo [9,10] and the induction motor [11]. References to these space charges on a magnetic brake made up of a disk rotating under the magnetic field of a single pole can be found in literature, [12,13,14,15,16,17] in the context of analytical calculations of forces and torques.…”