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 MIT OpenCourseWare
at ocw.mit.edu.
PROFESSOR: The non-uniqueness of voltage in an MQS system
can seem paradoxical, yet it is essential to the behavior
of a common transformer.
When the electric field intensity
is essentially irrotational, we can
count on its having no circulation around a closed
path.
This means that its integral between two points
is the same regardless of path.
If the magnetic flux density in Faraday's law is important,
the line integral of the electric field intensity
is path dependent.
This experiment emphasizes this by showing
how the voltages measured by these leads
and the oscilloscope are path dependent.
Here's the ground lead of the oscilloscope.
Here's the top trace lead, and connected to the same point
is this bottom trace lead.
A magnetic flux is created in a toroidal magnetizable core
by driving this winding with a sinusoidal current.
Because it is a highly permeable ferrite,
the core guides a magnetic flux density
B that is much greater than that in the surrounding air.
Looped in series around the core are two resistors
of unequal values.
The terminals of these resistors are connected together
to form a pair of nodes.
One of these nodes is grounded.
The other is connected to high impedance oscilloscope inputs
through the two leads that follow different paths.
Here's the ground lead, and we'll
use a dual trace oscilloscope to display the voltages.
Here's the lead connected to the top trace,
and then this lead is connected to the bottom trace.
The voltages observed with the leads connected
to the same node not only differ in magnitude
but are 180 degrees out of phase.
The ratio of the voltage magnitudes is 3.
This is the ratio of the resistances.
The probes are connected to the same node,
yet the voltages differ by a factor of 3
and are 180 degrees out of phase.
We checked to see that we don't have
the traces on different scales by reversing
the trace connections.
Reversal of the leads results in a reversal of the traces.
Faraday's integral law accounts for this seeming failure
of Kirchhoff's voltage law, for how
it is that different voltages are
measured by probes connected between the same nodes.
This is a cross section of the core.
The pair of resisters are connected in series.
The node that is connected to the ground
terminal of each probe is here.
One scope terminal is connected to the other node like this
while the other is connected like this.
The scope resistances are very large compared to either R1
or to R2, so the current carried by the oscilloscope
leads is negligible.
This means that, if there is a current I through one
of the series resistors, it must be the same as that
through the other.
Because it encloses the time varying magnetic flux carried
by the core, there is a current circulating
through the resistors.
The induction of this current is predicted by Ohm's law
and Faraday's integral law.
Here, the key contour is the path of circulation
through the resistors, and the surface cuts through the core.
The voltage measured by trace 1 is
determined by applying Faraday's integral law to this contour.
Because it links a negligible magnetic flux, what is measured
is just what is expected, the negative of the current times
the resistance R sub 1.
The voltage measured by trace 2 follows from this contour
and is the same current but multiplied by plus R sub 2.
The current out of the positive terminal of R sub 1
is that into the positive terminal of R sub 2.
When voltage 1 is positive, voltage 2 must be negative.
The ratio of the voltages is the negative ratio
of the resistances.
In our case, the ratio of the resistances
is 30 ohms to 10 ohms, 3, just as we measured.
If the probe leads follow the same paths,
the signals should be the same.
If we remove one of the leads and then connect the other one
to its own ground, after having looped
the lead through the core, what we see is this.
Just as the series resisters form the secondary
of a transformer, so now the lead
itself is a one turn secondary.
The measured voltage is simply the time rate
of change of the flux through the core.