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PROFESSOR: An artificial dielectric
made from an array of conducting spheres demonstrates
how a polarization of media gives rise
to the properties of a linear dielectric.
When the region between electrodes
is filled with a dielectric material,
the capacitance is increased.
This increase is due to polarization
of molecules of the material.
We form an appreciation for how this happens
by replacing the molecules with microscopic spheres suspended
in air.
This artificial dielectric is composed
of what we can think of as molecules
that are actually, essentially, perfectly conducting spheres.
Each has radius R, and they're in a cubic array
with spacing S.
Application of voltage to the electrodes
results in the spheres acquiring positive and negative surface
charges on their poles.
In addition to the uniform field,
due to the applied voltage, the field outside the spheres
also includes the dipole fields from the array of spheres.
The spheres, like the molecules of a material,
are polarized by an applied field.
The artificial dielectric is composed
of an array of ping pong balls with conducting coatings.
We're going to measure the effective permittivity
by measuring the change in capacitance
as the array is placed between the plates of this plane
parallel plate capacitor.
The cage is grounded for shielding
and to keep us from touching the bottom electrode.
It's driven to 400 volts RMS at 250 hertz
by this transformer, which is itself driven
by an oscillator and amplifier.
Here's the top electrode.
To avoid fringing effects, only the charge
induced on this central section will
be measured when we measure the capacitance.
Here's how the dielectric is inserted between the plates.
This is a picture of the experiment.
Here's the transformer driving the lower electrode.
Here's the top plate connected to one side
of a differentiable amplifier.
Because the spacing between spheres
is more than 2 diameters, the increase
in capacitance caused by the spheres is small.
To measure the small change in capacitance
that results from inserting the array,
the parallel plate capacitor is in one leg of a bridge.
These capacitors are in the other legs of the bridge.
It helps to draw the circuit like this.
Here are the ping pong balls.
These capacitors are those of the balanced amplifiers
on the scope.
So the oscilloscope displays the output voltage, V sub-0.
With the array removed, this capacitor, C sub-2,
is adjusted to know the output voltage, V sub-0.
The output voltage resulting from the insertion of the array
is a measure of the change in capacitance.
To simplify the interpretation of this voltage,
the resistances, R sub-S, are made small,
compared to the impedance of the parallel plate capacitor.
They're 100 kiloohms.
As a result, almost all of the applied voltage
V from the transformer appears across the lower legs
of the bridge.
With the introduction of the array,
there's a change in current through the parallel plate
capacitor.
This results in a change in current
in the resistance in the right leg,
and hence, a change of voltage across the resistance.
Because the current through the left leg
has remained essentially the same, this change in voltage
is the measured output voltage.
With many widely spaced and therefore noninteracting
spheres, the effective susceptibility
and the change in capacitance caused
by inserting the array between the plates
are given by these expressions.
Remember R is the radius of a ping pong ball,
and s is the spacing between balls.
A is the active area of an electrode,
and d is the spacing.
So this is the capacitance without the array.
The radius of the ping pong balls 1.87 centimeters.
The spacing, s, is 8 centimeters.
The electric susceptibility is then 0.16.
This means that the relative permeability is 1.16.
To compute the resulting change in capacitance,
we multiply the susceptibility we've
computed by the electrode area, 0.4 meter
squared, the permittivity of free space,
and divide by the electrode spacing, D--
15 centimeters.
So the predicted change in capacitance is 1.5 picofarads.
What change in voltage should we measure
when the array is inserted?
Here's what we predict.
For our experiment, our sub-s is 100 kiloohms.
The frequency F is 250 hertz, delta C is 1.5 picofarads,
and V is 400 volts RMS or 566 volts peak.
For this applied voltage, this change in capacitance
should result in a change in voltage of 133 millivolts peak.
Before balancing the bridge, we remove
the array of ping pong balls.
We've already set the driving voltage to about 400 volts RMS.
Here are the resisters in the two upper legs of the bridge.
They shunt the scope differentiable amplifier.
So the scope signal is the voltage V sub-0.
The balancing capacitance, C2, is here.
This is the capacitor C1, which is there for fine tuning.
We can balance the signal with this capacitor.
So we tuned C2 to null the bridge voltage and C1
to fine tune the null.
We're ready to insert the artificial dielectric.
As the array is inserted, the bridge
becomes unbalanced, indicating that the capacitance has
changed.
We measure a voltage of $135 millivolts peak.
Remember what we predicted was a change of 133 millivolts peak.
The agreement is perhaps better than we should expect.
Has the capacitance actually been
increased by the insertion of the spheres?
Let's try the experiment again.
We can see that it has, by looking at the balancing
capacitor.
Here the array is out, and we balance the capacitor.
Now we insert the array.
Note the position of the tuning capacitor plates.
If the capacitance has been increased
to balance the bridge, the knob must
be turned so that they have greater area of overlap.
The balancing capacitance is increased.
So the artificial dielectric does
increase the capacitance of the plane parallel plate capacitor.
Our model for how the polarization of the spheres
increases the capacitance appears to be a good one.
It is because the individual spheres have a polarizability
that we can calculate, and because they're well
removed from each other that we can
predict the equivalent primitivity
of the artificial dielectric.
Insertion of the artificial dielectric
increases the capacitance.