Faraday Disk Dynamo model (a) in external magnetic field and (b) attached to a coil. Green arrow shows generated current and yellow vectors show magnetic field.

# Faraday Disk Dynamo Model

The EJS Faraday Disk Dynamo shows a conducting disk that rotates in a magnetic field. This produces a current (homopolar generator) and for certain configurations, it is a self-exciting dynamo. A self-exciting dynamo is the mechanical analog of the proposed mechanism to produce the earth and sun's magnetic fields. Ejs must be installed to explore and change the model.

## Background:

This model explores different configurations of a "Disk Dynamo" beginning with a disk rotating in an external magnetic field. In a 1955 article, Edward Bullard suggested this dynamo might serve as a model for the Earth or sun's magnetic field since it was stable and could account for changes in the magnetic field. To begin, the emf across a rotating disk (between center and edge) in an external field is
emf = ω ∫ B r dr
and for the attached "circuit" (of disk, support rod and connecting wire),
emf = L di/dt + R i
where i is the current in the circuit, L is the inductance and R is the resistance. The equation of motion is given by (equating the torques):
I dω/dt = τ - i ∫ B r dr
where I is the moment of inertia, τ is the external torque and the second term is the drag due to the current flowing in a magnetic field. If the magnetic field is a constant (due to an external field, the Ext B setting for the model), then setting
A = ∫ Br dr (constant)
and doing some algebra results in a differential equation for a damped oscillator:
IL d2i/dt2 + IRdi/dt + iA2 = τ A.
If, however, the magnetic field is not constant (not due to an external field), but instead is generated from the generated current (by traveling through a coil, as in the Coil configuration in the model), the integral A is no longer constant, but proportional to the current:
∫ B r dr = iM
where M is 2π times the mutual inductance. Simplifying gives two coupled equations:
Idω/dt = τ - i2M
L di/dt + Ri = iω M
Note that if ω is fixed (Configuration: Coil, constant ω), the solution is
i = i0 e (Mω - R)t/L
so for ω < R/M, i exponentially drops to 0 while for ω > R/M, i approaches infinity.
However, if the torque, τ, is fixed, Bullard points out that the dynamo is self-exciting and oscillates between between values of the angular velocity and current, allowing for magnetic field flips (as required for a model of the magnetic field of the earth.) This is the configuration Coil, contant torque in the model.
Hide, Skeldon and Acheson (and later Moroz) explored the addition of a capacitor to the configuration resulting in a chaotic system (In the model, try the capacitor, constant torque configuration and see the phase space plot). The addition of a capacitor of capacitance, C and resistivity r (to allow for leakage current-- necessary for a chaotic system), gives an additional coupled differential equation of
dq/dt = i - q/(rC)
and adds a term q/C to the equation for the emf (but leaves the equation for the torque the same):
L di/dt + Ri + q/C = iω M
Idω/dt = τ - i2M
The capacitor configuration models these three differential equations and shows a phase space plot of charge, current and angular velocity.

### References:

- Sir Edward Bullard, "The stability of a homopolar dynamo," Mathematical Proceedings of the Cambridge Philosophical Society, 51 (1955) 744-760.
- Raymond Hide, Anne C. Skeldon and David J. Acheson, "A study of two novel self-exciting single-disk homopolar dynamos: Theory" Proceedings fo the Royal Society A, 452 (1996) 1369-1395.
- Irene Moroz, "The Hide, Skeldon, Acheson dynamo revisited," Proceedings of the Royal Society A, 463 (2007) 113-130.

### Credits:

The Faraday Disk Dynamo Model and Exercises were created by Anne J Cox using the Easy Java Simulations (EJS) authoring and modeling tool.

You can examine and modify a compiled EJS model if you run the program by double clicking on the model's jar file. Right-click within the running program and select "Open EJS Model" from the pop-up menu to copy the model's XML description into EJS. You must, of course, have EJS installed on your computer.

Information about EJS is available at: <http://www.um.es/fem/Ejs/> and in the OSP ComPADRE collection <http://www.compadre.org/OSP/>.