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PROFESSOR: A classic application of Ampere's integral law
is to the prediction of the magnetic field of a line
current.
Ampere's integral law states that the integral
of the magnetic field intensity around a closed contour C
is equal to the net current conduction
plus displacement that passes through the surface
S that is enclosed.
The integral of the magnetic field intensity
around a closed contour C is equal to the net current
that passes through the surface S that is enclosed.
In this demonstration, the current density
and associated fields are at 60 hertz.
Because this frequency is low, the time rate of change
is relatively small.
As a result, the displacement current contribution
to Ampere's law is negligible.
In this demonstration, a line current
essentially extends to minus and plus infinity.
This line current passes through a surface S enclosed
by this contour C. Ampere's law says that there
is an integral of H around the contour C. Symmetry requires
that only the azimuthal component of H be non-zero
and that this component only depend on the radial position
r.
The line integral is simply the product
of H phi and the circumference 2 pi r.
The surface integral is the total current i.
H phi is inversely proportional to r.
H phi is proportional to i.
In our experiment, the wire is horizontal.
Here is the wire.
Ampere's law predicts the magnetic field intensity
as it depends on r.
H sub phi is inversely proportional to r.
This Hall effect probe measures the magnetic field intensity.
The top trace on the oscilloscope
is a measure of the probe output.
The lower trace measures the wire current.
The wire current is measured by measuring the voltage drop
across a series 1 ohm resistor.
We measure 6 volts peak, so the current is 6 amperes peak.
This probe measures magnetic field intensities perpendicular
to its flat surface.
When aligned in this direction, it
measures the z component of magnetic field, about 0.
When aligned in this position, it
measures the radial component of magnetic field, again about 0.
When aligned in this position, it
measures the phi component of the magnetic field,
which seems to be the only non-zero component
of magnetic field.
For our current of 6 amps, Ampere's integral law
predicts this one overall radial dependence for H sub phi.
Our measurements are limited to distances,
r, small compared to the distances
to the end of the wires or to the return wires below.
Because the Hall effect probe fundamentally
exploits the Lorentz force law, it
measures the flux density mu naught
H. A common unit for flux density is the gauss.
We have to take this into account
in evaluating Ampere's law.
For conversion of units, 10,000 gauss
equal 1 tesla where the tesla is the SI unit.
With the probe at 1 centimeter, the measured flux density
is 1.2 gauss peak.
i is 6 amps.
r is 1 centimeter.
So we calculate mu 0 H phi to be 1.2 gauss.
Here is the data point on our plot.
Here are some other measured values.
We have demonstrated the relationship
between current through a surface S
and magnetic field intensity integrated around the contour C
given by Ampere's integral law.