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PROFESSOR: Faraday's integral law
shows how a voltmeter reading induced by magnetic induction
at the terminals of a coil provides a measurement
of the magnetic flux density.
Faraday's integral law states that the circulation
of electric field E around the closed contour C
is determined by the time rate of magnetic flux
linking the surface and closed by the contour.
The circulation of electric field E
around the closed contour C is determined
by the time rate of magnetic flux
linking the surface enclosed by the contour.
In this experiment, the contour C
is traced out by the wire in this coil
so that the surface S is the area having the entire length
of the coil as its edge.
In this coil, the surface area S is
8 centimeters by 21 centimeters multiplied by 20 turns.
We will use this coil with Faraday's law
to find the magnetic flux associated
with current in this wire.
Here is the wire, carrying a current I,
producing this magnetic field, h sub phi.
Here is our coil.
In accordance with Ampere's integral law,
the magnetic field intensity circulates around the current
carried by the wire.
The magnetic flux passing through the coil
contributes to the surface integral
on the right in Faraday's law.
The current, and hence the field, are varying at 60 Hertz
so this integral has a time rate of change.
Applied to the coil, the line integral on the left
becomes the negative of the terminal electromotive
force or voltage.
Here is the induced voltage in the coil.
The total flux linked by the coil is lambda sub f.
The upper trace of this oscilloscope
records the electromotive force of voltage at the coil
terminals.
The lower trace is the current in the wire, measured
by the voltage drop across a 1 ohm series resistor.
With the coil rotated so that it links no magnetic flux,
no voltage is seen.
With the coil now placed so that the magnetic flux passes
perpendicularly through the coil,
the maximum signal is recorded.
With the coil straddling the wire,
the flux passing through the coil in the upper section
is oppositely directed to the flux through the lower section.
With it symmetrically positioned,
the net flux is zero so the measured voltage is zero.
The scope traces are triggered on the wire current, the lower
trace, so we can see the change in phase
as the coil is moved from below the wire to above the wire.
The lower trace current is 90 degrees out of phase
with the upper trace induced voltage.
This is because the coil voltage is the time rate of change
of the magnetic flux and hence of the time
rate of change of the current.
The coil voltage is 90 degrees out of phase with the current.
The coil height is h.
The length is l.
And the number of turns is N. In terms of these parameters,
this is the flux linkage.
The flux linkage for a single turn is the surface integral.
The total flux length is obtained by multiplying by N. R
is the distance from the wire to the top of the coil.
In the experiment, this distance is R. Faraday's law
gives this as the coil voltage.
To put data on a plot of normalized voltage
as a function of normalized spacing, we compute V sub 0.
The coil has 20 turns, a length of 21 centimeters.
And the frequency is 60 Hertz.
The peak wire current is 6 amperes.
Then V sub 0 is 1.9 millivolts peak.
With the spacing R equal to 6 and 1/2 centimeters,
the peak core voltage measured at the terminals
is about 2 millivolts peak.
Here is the normalized voltage predicted
by Faraday's integral law as a function of normalized spacing.
Our data point goes here.
These are normalized voltages measured at other spacings.
The discrepancy is due to the large sensing coil size
as compared to the length of the current carrying wire
and the proximity of the nearby return current wire.
Faraday's integral law predicts the voltage
induced at the terminals of a coil that links
a time varying magnetic flux.