The Faraday disk and a self-sustained dynamo
A perfectly conducting disk, having radius a and thickness h ≪ a,
is kept in rotation with uniform angular velocity ω (parallel to the
disk axis), in the presence of an uniform and costant magnetic field
B parallel to ω.
a) Find the electric field E in the disk in steady state conditions,
and the corresponding potential drop between the center and the
boundary of the disk (hint: the total force on charge carriers must
be zero).
b) Using brushing contacts, a closed and fixed circuit connects the
center and the border of the disk. Let R be the resistance of the
circuit. Calculate the momentum of the external forces necessary to
keep the disk in rotation at constant angular speed.
c) Assume the external circuit to be connected to an external
solenoid, having n coils per unit length, that generates the B field
(there are no other external sources of the magnetic field). Find a
condition on ω in order for the current to flow in a steady state without additional electric generators (but in the presence of mechanical
motors to keep the rotation constant).
1
ω
R
I
a
B
Solution
a) The v × B force (per unit charge) generates and electromotive force E between the center (r = 0)
and the border (r = a) of the disk:
Z
Z a
E = v × B · dl =
ωrBdr = ωBa2 /2 ,
(1)
0
being v = ω × r. Of course E is also equal to the voltage drop associated to the electrostatic field
E that (in the case without a closed circuit)
must be generated to balance the magnetic force, i.e.
R
E + v × B = 0 corresponding to E = − E · dr. The rotating disk is thus a voltage generator, called
a Faraday disk.
b) The power dissipated by Joule heating is Pd = E 2 /R. In a steady state, Pd must balance
the mechanical power Pd = Pm = Mext · ω where Mext is the torque of external forces. Thus,
Mext = V 2 /(Rω) = ω(Ba2 /2)2 /R.
We may also calculate directly the torque of magnetic forces exerted by B on the currents flowing
in the disk. In a steady state and assuming cylindrical symmetry, the condition ∇ · J = 0 leads
to a radial current Jr = I/(2πrh). The torque on an infinitesimal volume element dV = 2πrhdr is
dMm = r × [J × B]dV = −ẑIBrdr, thus the total torque is
Z a
Ba2
(Ba2 /2)2
Mext = −Mm = ẑIB
rdr = I
ẑ = ω
ẑ .
(2)
2
R
0
c) If the disk acts as the generator for the current in the solenoid and hence for the magnetic field,
we must have
B = µ0 nI = µ0 nV /R = µ0 n(ωBa2 /2R) ,
(3)
from which ω = 2R/(µ0 na2 ) follows independently from the value of B. This is an elementary model
for a dynamo self-sustained by rotation, such as the generation mechanism of the Earth’s magnetic
field.1
1
See e.g. Ronald T. Merrill, Our magnetic Earth: the Science of Geomagnetism (Chicago University Press, 2010)
chapter 3.
2