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PROFESSOR: Field of a square pair of coils
can be found by using the Biot-Savart law.
The pair is often called a Helmholtz coil
when the windings are connected so that the currents produce
axial fields that reinforce.
It can then be used to make a relatively uniform field
in the mid-region, which is accessible for experiments.
The Biot-Savart superposition integral
can be used to find the field at the observer coordinate r.
Its evaluation amounts to a summation of the field
contributions from each of the differential-sized current
elements at the source coordinate, or prime.
Our square coils are composed of connected straight line
current segments, or current sticks.
The Biot-Savart law is used to find the field
at an arbitrary observer position r associated
with each current stick.
Because the individual current sticks do not
represent the solenoidal current density at its ends,
the magnetic field derived is a physical significance
only if used in conjunction with other current
sticks that together form a continuous current
distribution.
Each of our square coils is composed of four
connected current sticks.
The magnetic field at point P is found by a summation of fields
from each stick.
Each coil has n turns, carrying a current i
with sides of length 2d.
The current i in each coil is in the same direction
so that the magnetic field induced by each coil
reinforce along the minus z-axis.
For the direction of current shown,
the right-hand rule shows that the axial magnetic field
is in the minus z direction.
Because of the axial symmetry, the fields
along the z-axis induced by each of the sides of the coil at z
equal d are the same.
Evaluation of the Biot-Savart law for both square coils,
therefore, gives this variation of the axial magnetic field
as a function of axial position z.
The first term is due to the coil at z equal d.
The coil at z equal minus d makes a similar contribution,
except that the term 1 minus z over d is replaced by 1 plus z
over d.
In the experiment, the axial field
produced by currents in the two square coils
is probed by means of a whole magnetometer.
The output is connected to the vertical trace
of a high-persistence scope.
The probe is mounted on a motor-driven carriage that
is attached to a potentiometer in such a way
that there is an output voltage proportional
to the horizontal position of the probe.
This is used to control the horizontal scope deflection.
The result is a trace that follows
the predicted z dependence of the magnetic field
along the symmetry axis.
Here are our square coils, each with 50 turns and sides
of length 45 centimeters.
The axial magnetometer probe is mounted on this carriage
to record the z component of magnetic field.
Before we apply current to the coils,
we have compensated for the small part of the Earth's
magnetic field that is in the horizontal direction
by zeroing the magnetometer.
The Earth's magnetic field is essentially vertical
with a strength of about a half a gauss
and could contribute a slight error to the measurement
if not accounted for.
A current of 2 and 1/2 amperes--
[CLICKS]
--is applied to the coils.
We have placed the theoretical variation of the magnetic field
versus position z as a solid line
directly on the oscilloscope screen.
Let's see if the scope trace follows this theoretical curve.
We have adjusted the scope vertical gain
so that the display matches the theoretical waveform
at one position.
We'll see if the scope trace follows the shape
of this theoretical curve.
We'll check the field magnitude later.
The probe is initially here.
We now turn on the probe carriage.
[WHIRRING]
The scope horizontal gain has been
adjusted so that the horizontal deflection of the scope
matches the z position of the probe.
The magnetic field is largest directly under a coil, now
the left one at z equal minus d and is
fairly uniform in the central region between coils,
with the minimum at the center, z equals 0.
The field increases again to a maximum
under the right coil at z equal d
and decreases in the region outside the coils.
Let's compare the theoretical and measured field
magnitude at z equals 0, midway between the coils.
We bring the carriage back to the center, z equals 0.
[WHIRRING]
Full-scale deflection is 10 gauss.
We measure a magnetic field of about 2.4 gauss.
Let's compare to the predicted value.
With 50 turns, a current of 2 and 1/2 amperes
and a half-side dimension of 22 and 1/2 centimeters,
the predicted magnetic flux density is 2.6 gauss.
To convert from teslas to gauss, we have multiplied by 10,000.
We measured 2.4 gauss.
In our Helmholtz coil experiment,
the current in each coil was in the same direction
so that the axial magnetic fields from each coil
reinforced.
With the current reverse in the left coil,
the axial magnetic fields of each coil
are oppositely directed and cancel in the center.
This results in a sign reversal for the field contribution
from the left coil.
Now the magnetic field is an odd function around z equals 0.
It is 0 in the center, as expected.
This reversing switch reverses the direction
of current in the left coil.
[CLICKS]
Let's see how the measured field compares with the prediction.
We have aligned to zero field lines
and adjusted the vertical gain to match
the waveform at one point.
We now turn on the probe carriage.
[WHIRRING]
The field is a negative maximum under the left coil,
is 0 at the center, and reaches a positive maximum
under the right coil.
Using a pair of square coils, we have demonstrated the power
of the Biot-Savart law in predicting the magnetic field
distribution.
Given the distribution of current
and given that there are no effects
of magnetizable materials, this law gives precise predictions.