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PROFESSOR: The fields of a circular cylindrical solenoid
can be found using the Biot-Savart law.
If these parallel plates were of infinite extent,
and applied voltage would give an electric field that
is uniform, what this plane parallel plate capacitor is
to electric fields, an infinitely long solenoid
is to magnetic fields.
That is, if this solenoid were infinitely long,
a current in the winding would produce a magnetic field
intensity that is essentially uniform inside
and equal to the surface current density.
Outside, the field would be essentially zero.
This solenoid is large enough that we
can poke around in it with a magnetometer probe,
and see how the magnetic field is actually distributed
when the length is finite.
The solenoidal coil consists of n turns, essentially uniformly
wound over the length of D, each turn carrying
the same current i.
The radius is a.
The current i is essentially phi directed.
This axial hole probe shows us the magnitude and direction
of the field.
The probe measures the field along its axis.
The lower scope trace records the probe output.
The upper scope trace shows the coil current.
When the probe is oriented this way,
the axial component of magnetic field is measured.
When the probe is oriented this way,
the radial component of the magnetic field is measured.
Let's measure the axial magnetic field component
along the axis of the cylinder.
On the axis, the radial component
is, of course, 0 because of symmetry.
Well inside the solenoid, the field intensity
in the z direction tends to be uniform everywhere.
The infinitely long solenoid can be
regarded as the analog for MQS systems of the EQS plane
parallel plate capacitor.
Well away from the ends, the axial field
just inside the coil should be equal to the surface
current density.
The surface current density is the number
of turns times the current per turn divided by the length d.
The probe measures the flux density
mu 0 H. To convert from SI units Tesla to Gauss,
we multiply by 10,000.
There are 141 turns.
Our current is 1 amp RMS, and the coil length
is 70 and 1/2 centimeters.
The predicted flux density just inside the coil is 2 and 1/2
Gauss RMS.
This is on the order of the steady flux density produced
by currents inside the Earth.
So that we can distinguish our fields,
we are using a 60 Hertz current.
Let's measure the magnetic field well inside the solenoid.
We measure about 2.3 Gauss just inside the winding,
as compared to the prediction of 2 and 1/2 Gauss.
Let's see what the tangential flux density is just outside,
at the same axial location.
Just outside, the flux density falls by more than a factor
of 10 to 0.15 Gauss.
Just as a capacitor can be constructed
to create a uniform electric field between the plates
with zero field outside the region bounded by the plates,
so, too, the long solenoid gives rise
to a uniform magnetic field throughout the interior region.
Our solenoid length is finite, so the field inside
is only approximately uniform, and the exterior field
is only approximately zero.
The magnetic field along the z-axis
can be found by simple integration
of the Biot-Savart law.
The essentially phi-directed current
along the entire winding generates
a z-directed magnetic field along the axis of the cylinder.
This is on-axis distribution of the axial field intensity
for our solenoid, which has a length to diameter
ratio of 2.58.
We can compare the prediction of this formula to our experiment
here at the end of the solenoid.
Biot-Savart law predicts a flux density of about half
of the 2 and 1/2 Gauss in the interior.
Here's the on-axis magnetic field well inside the solenoid,
about 2.3 Gauss.
Here at the end of the coil, we measure about 1.2 Gauss,
about half of the interior field.
This is where our data points fall on the theoretical curve.
Here's the interior point.
Here's the point at the end of the coil.
The accuracy with which theory and experiment degree
is likely to be limited only by such matters as the uniformity
of the winding, the care taken in mounting
the probe, and the calibration of the Gauss meter.
Another way to demonstrate the discontinuity
in magnetic field between inside and outside the solenoid
is to use this transverse probe.
This probe measures magnetic fields
transverse to its flat surface.
Here, for example, is the on-axis field.
Turning the probe by 90 degrees gives
a 0 signal, which shows the field is purely axial.
We now place this probe just outside the solenoid
in the orientation to measure the small tangential field
component.
As the probe passes through the current-carrying winding,
the magnetic field abruptly rises to the interior field
value.