The following content is provided under a Creative
Commons license.
Your support will help MIT OpenCourseWare
continue to offer high-quality educational resources for free.
To make a donation or view additional materials
from hundreds of MIT courses, visit
mitopencourseware@ocw.mit.edu.
PROFESSOR: The field and inductance of a spherical coil
illustrate the model of a distributed surface
current density and the associated Laplacian fields.
This is a coil wound on the surface of a sphere.
Here are the terminals.
A current through the wire results
in a magnetic field inside the sphere that is uniform.
Outside, the field is that of a dipole.
Because these fields are so simple,
this flux-ball example is a valuable resource
for dealing with other situations
and for getting basic notions straight,
such as how to compute the inductance of a coil,
every turn of which links a different amount
of magnetic flux.
Here's the coil in cross-section.
With the neglect of the slight pitch in the turns,
the winding is uniformly distributed from north
to south.
The diameter is 2R.
With each turn carrying the current i,
the surface current density is sinusoidally distributed
with respect to theta.
On the surface of the sphere, the surface current density
varies with a sine of theta, and is proportional to i.
N is the number of turns.
The magnetic field that results from this surface
current density looks like this.
Inside the sphere, it should be essentially uniform.
Outside, it should be the same as would
be produced by a magnetic dipole located here
at the center of the sphere.
At the north pole, the field should be radially outward.
At the equator, it should be tangential.
At the south pole, the field should be radially inward.
A Hall magnetometer with a transverse probe
makes it possible to probe the field.
We use the oscilloscope to show the output of the Hall probe.
The upper trace is the coil current.
The lower is from the probe.
Here at the north pole, the magnetic flux density
should be perpendicular to the sphere surface.
The probe detects the field perpendicular to its flat face.
So we can see that the field is perpendicular to the surface
of the north pole.
At the equator, the field is tangential
and in the opposite direction.
By watching the phase of the lower trace,
we can see that the field at the south pole
is in the same direction as at the north pole.
The phase is the same at the two poles,
so the field is upward at both poles.
Watch the vertical component of the field
as we go from the north to the south.
We have a hole from the south to the north pole.
We can insert another probe.
By contrast with our transference probe,
this axial probe measures the magnetic field along its axis.
The magnetic flux density inside the sphere
is essentially uniform.
Outside, it has the decay of a dipole.
Inside, we measure 5.4 Gauss RMS.
Theoretically, this is the flux density inside the sphere.
We have a total of 61 turns, a current of 1 ampere RMS,
and a radius of 5 centimeters.
So the predicted flux density is 5.1 Gauss RMS.
Remember, what we measured was 5.4 Gauss RMS.
Let's measure the inductance of the flux ball.
Its inductance is determined by measuring the voltage
and current of the coil.
The lower trace of the oscilloscope gives the voltage.
The current is the upper trace.
Is the frequency high enough that the resistance of the coil
plays a negligible role in the terminal impedance?
We can see that it is.
The voltage and current are 90 degrees out of phase,
as they should be for an inductor.
The frequency is high enough that the ratio
of the peak voltage, the lower trace, to the peak current
is the inductive reactance.
At a frequency of 40 kilohertz, the voltage of 13 volts peak
gives 0.3 amps peak.
So the measured inductance is 172 microhenries.
Theoretically, this is the inductance.
For our 61 turns and 5 centimeter radius,
we predict L equal 163 microhenries.
What we measured was 172 microhenries.
The flux ball is a classic example
where the fields are simple to represent--
a uniform field inside and a dipole field outside--
and yet interesting enough to give insight
into more complex situations.
For example, the vertical field at the north pole
should be twice what it is at the equator.