PROFESSOR: The inductive attenuator
demonstrates the Laplacian nature
of magnetic fields and free space--
the way the currents distribute themselves
on the surfaces of perfect conductors
and the use of the vector potential
in describing magnetic fields in two-dimensional systems.
In this model, the floor is the xy plane.
The altitude of the surface is the vector potential.
Here's the direction of periodicity.
Here's the direction of decay.
Because it represents a solution to Laplace's equation,
every point on the surface having a positive curvature
in the y direction must have a negative curvature
in the x direction.
This is why periodicity in one direction
implies decay in the other.
For the solution to Laplace's equation,
three of the rectangular boundaries
have a constant vector potential.
The fourth is a sinusoid.
In our physical situation, this fourth boundary
is elevated to a uniform vector potential.
This can be represented by a superposition
of these solutions.
In our model, lines of constant altitude
are lines of constant vector potential.
In two dimensions, lines of constant a sub z
are lines of magnetic flux density.
So the wires are the lines of magnetic flux density.
This is the actual apparatus.
Here are the sidewalls insulated from each other.
Here's the top plate.
Current from this source passes forward through the upper plate
and is returned by the shorts through the sidewalls.
Here are the connections at the back end.
The source is connected by the center
conductor of a coaxial cable to the upper electrode here.
The driving current is returned to the source
by these wires and the coaxial conductor.
These leads are connected to the top trays of our scope.
The top trace is the driving voltage.
Here at the back, the bottom plate
is shorted to the sidewalls.
It is also shorted at the front through these wires.
We measure the current induced in this electrode
by means of this current probe.
The lower trace on the scope represents the output current
from the probe.
It represents the net current induced in the lower electrode.
Here's how the current induced in the lower electrode
varies as the driving electrode is lowered.
As the spacing is increased, again,
with the driving voltage essentially held fixed,
the induced current decreases.
As the driving electrode approaches the lower electrode,
it increases again.
With the electrodes this close, almost all
of the current in the upper electrode
is imaged in the lower electrode.
As the distance between electrodes increases,
the fraction of the current, the returns in the lower electrode,
is reduced.
The inductive attenuator has illustrated
the Laplacian nature of free space magnetoquasistatic
fields.
The magnetic field, induced by applying a voltage
to the upper electrode, decays, essentially exponentially,
into the rectangular region, bounded
by the essentially perfectly-conducting sidewalls
and the bottom electrode.