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PROFESSOR: Currents induced in a conducting shell
oppose the penetration of a suddenly applied magnetic field
and cause a force that tends to crush the shell.
A thin conducting cylindrical shell
is placed on top of a coil.
This is the same apparatus used with Edgerton's Boomer
in demonstration 10.2.1.
We generate a large current in the coil
by discharging an initially charged capacitor
C through the coil.
This current generates a large magnetic field
which tries to penetrate the cylindrical shell conductor.
Axial currents induced in the conducting cylindrical shell
create a self-magnetic field which
opposes the penetration of the suddenly applied magnetic
field.
These currents cross with the applied field cause
a radial force inwards that tends to crush the cylinder.
Here, again, is Edgerton's Boomer apparatus,
composed of the driver coil and large capacitor
bank with total capacitance of 48 microfarads.
The capacitor bank is charged to 4 kilovolts.
Discharging the capacitor into the coil with this trigger
generates a large transient current
in the coil which, in turn, generates
a large magnetic field.
Currents induced in the thin foil cylinder
to oppose the penetration of magnetic field--
[POP]
--cause a radial inwards force on the cylinder
that tends to crush it.
Let's see that again.
[POP]
Here is what the magnetic field would look like immediately
after the field is turned on if the applied field were uniform.
At first, the shell acts like a perfect conductor.
The magnetic field within the shell at t equals 0 is 0.
This happens because the axially directed surface currents are
induced to buck out the penetration
of the applied magnetic field through the shell.
At the shell's surface, the radial component
of the magnetic field is 0, while the tangential component
is terminated by the surface current.
The cross product between the axial current
with the tangential component of magnetic field
then gives the J cross B magnetic force
as radially inwards.
This tends to crush the aluminum foil cylinder.
As time goes on, the currents decay to 0
and the imposed field penetrates through the shell
as if it were not there.
For a step magnetic field, the surface current exponentially
decays from its initial value when
t equals 0 to 0, with magnetic diffusion time
constant tau sub m.
The time constant tau sub m is the conductivity
of the shell, sigma, times the shell thickness, delta,
times the shell radius, a, times the shell magnetic
permeability, here, M u naught, the permeability of free space.
Tau m is the continuum version of the circuit l
over r time constant.
Let's evaluate tau mm our experiment.
The aluminum foil cylinder radius
is about 4 and 1/2 centimeters.
The foil, wrapped as two layers, has a total thickness
of about 60 microns.
We use the magnetic permeability of free space
and a conductivity of aluminum of 3 and 1/2 times 10
to the 7th siemens per meter.
The calculated value of tau m is about 60 microseconds.
With this time constant in mind, let's see
what the actual fields are.
Because this is a single shot experiment,
we use this storage oscilloscope to record and store
measured waveforms.
We use this current probe to record the current
in the coil in the upper oscilloscope trace.
Its gain has been adjusted to 500 amps per centimeter.
This small 200 turn coil is placed just above the driver
coil to record the time derivative of magnetic flux
on the lower oscilloscope trace.
We charge the capacitor to 4 kilovolts.
As we discharge the capacitor bank,
we record the current oscillations
in the driver coil on the upper scope trace
and the time derivative of the magnetic flux on the lower
scope trace.
We see decaying oscillations characteristic of a ROC
circuit.
For our circuit parameters, the frequency of oscillation
is about 4 kilohertz.
We stored these waveforms and reset the scope trigger.
We now place the foil cylinder around the magnetic flux
sensing coil.
Remember, the upper trace records
of the driver coil current which imposes the magnetic field,
while the lower trace records the sensing coil
signal now inside the cylinder.
[POPS]
We see that the driver coil current waveform
has diminished somewhat, but that the sensing coil
waveform within the foil cylinder
has decreased even more.
The magnetic field has been shielded out of the cylinder.
This modest shielding is expected,
because the product of angular frequency, at 4
kilohertz with tau m of 60 microseconds is of order unity.
With a thicker cylinder, tau m is increased
and the shielding should be more complete.
Let's measure the field inside another aluminum cylinder
that is about 100 times thicker.
The bottom scope trace shows the field without the cylinder.
Let's reset the scope and put the thick cylinder
around the sensing coil.
[POP]
The new lower trace is essentially 0.
For a mega tau m of order 100, the magnetic field
inside the cylinder is greatly reduced.