Lab IX: Electromagnetic Induction
George Wong
Instructor: Geoffrey Ryan
Experiment Date: 17 April 2012
Due Date: 24 April 2012
Figure 1: Magnetic Flux through Loop
1
Objective
The objective of this examination was to examine the way that magnets and changing currents
can induce magnetic fields and to consider the differences in these generated fields as functions
of differences in how they were induced.
2
Theory
The electromotive force (EMF) can be mathematically described as the line integral of the
electric field around a loop or circuit. The EMF is measured with the units V, volts. Just
as electrical current produces magnetic field, so too can it be said that magnetic field can
induce current. Further, the EMF induced in a coil is related to the rate of change of the
magnetic field in the area. That is to say that the time derivative of the magnetic field in an
area is directly related to the strength of the induced EMF in a coil.
R
~ · dA,
~ or the amount of magnetic
, wherein Φ is flux, defined to be B
Specifically, EMF= −dΦ
dt
field perpendicular to some certain area.
where N is the
When a coil is involved, voltage induced follows the formula: V = +N dΦ
dt
number of turns of the coil.e
EMF can be induced in a coil several ways, including: moving a magnetic object through or
near (it must be relative motion) the coil; moving a coil with a current running through it
near a second coil (EMF induced in second coil); changing current in a coil.
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Set Up
In this laboratory examination, we used the prefabricated circuit board with the inductor coil
in conjunction with the another coil and the SWS suite. We also made use of a permanent
metal bar magnet and the iron inductor coil that was a part of the prefab circuit board setup.
We used the power amplifier within the SWS program to generate either AC or DC voltage
1
Figure 2: Circuit Board with Coil
to run through the coils, and then measured various other voltages with the voltage probes,
hooked into SWS.
For one part of the lab, we used one of the resistors (the 10Ω one) in series with the inductor
coil on the prefab board by connecting the second voltage lead to one of the terminals at the
other end of the resistor (putting it in series and forcing it to be a part of the circuit).
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Procedure
1. PART ONE: EMF BY PERMANENT MAGNET
• Using SWS, voltage across the measuring coil was measured by attaching a voltage
probe across the terminals. The voltage reading was ran into a graph display.
• A bar magnet was aligned vertically to fit through the hole in the coil.
• The magnet was thrust into the coil hole, left stationary, then removed.
• The magnet was moved through the coil at different speeds (slower, faster) and in
different orientations (upside down, perpendicular).
• The magnet was held stationary and the coil moved around it to produce the same
relative movements.
• With the coil above the surface of the desk, the magnet was dropped through the
hole (and onto the desk below) to induce a voltage.
2. PART TWO: EMF BY MOVED COIL WITH STEADY CURRENT
• A second coil was hooked up to the power amplifier part of the SWS suite; a 3V
DC current was ran through it.
• The voltage sensor across the primary coil was set to have increased sensitivity so
as to be able to measure the induced current.
• The second coil was moved with respect to the primary coil, the induced voltage
was noted.
2
• The way that the second coil was moved was changed (speed); polarity was reversed in the second coil (by changing the leads).
3. PART THREE: EMF BY CHANGED CURRENT IN COIL
• A voltage sensor was attached across the terminals of the first coil, this measurement was ported to an oscilloscope display.
• The second channel on the scope was directed to take the signal of the output
power amplifier, programmed at 1V DC sinusoid 1000 Hz.
• The second coil (with the switching voltage running through it) was laid on top
of the primary coil (with induced voltage being measured).
• The two leads were switched in the secondary coil; coils were otherwise manipulated as in by being separated.
• The frequency of the sinusoidal wave going to the secondary coil was changed (to
500, 2000, and 4000 Hz in addition to the original 1000 Hz).
• An iron core was inserted part-way into the two-coil contraption, with its effect
upon the induced voltage noted at various times during the process.
4. PART FOUR: SELF INDUCTANCE, COIL INDUCING VOLTAGE IN ITSELF
• Voltage sensors were hooked up to channels A and B in the SWS suite software
system. A went to channel one of the oscilloscope, B went to channel two.
• Voltage probe A was attached across the inductor; probe B was attached across a
10Ω resistor in series with the inductor coil.
• A voltage of 1.5V at 1000 Hz sinusoidal was ran across the coil and resistor. The
voltage across the resistor was measured to calculate the current; phase difference
was also noted.
• The above step was completed for other signal voltages at 500, 2000, and 4000 Hz.
• Coil resistance was measured using the multimeter.
5. Materials were put away.
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Data
8.2mH Inductor Coil
10% uncertainty in reported inductor inductance
1.5 volts going through inductor coil ⇒ 0.25 ± 0.01 amperes current measured
10Ω Resistor
1% uncertainty in reported resistor resistance
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frequency (Hz)
500
1000
2000
4000
resistor voltage (volts) [peak; peak]
0.461; -0.476
0.254; -0.265
0.104; -0.149
0.060; -0.074
Uncertainty in voltage 0.001 volts.
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Calculations
V = IR ⇒ I = V /R ] 0.0461amperes = 0.461volts/10ohms
−0.0265amperes = −0.265volts/10ohms
···
frequency (Hz)
500
1000
2000
4000
resistor voltage (volts) [peak; peak]
0.461; -0.476
0.254; -0.265
0.104; -0.149
0.060; -0.074
current (amperes) [peak; peak]
0.0461; -0.0476
0.0253; -0.0265
0.0104; -0.0149
0.0060; -0.0074
R = V /I
6Ω = 1.5V / 0.25A; therefore, it is reasonable to assume that the inductor has an effective
resistance of about 6 ohms.
Inductive resistance XL is given by the formula Lω = L2πf
L2πf : (0.0082H)2π(1000Hz) = 51.5Ω ∼ 52Ω
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Error Analysis
When we wish to find uncertainties with relation to some formula, we take the partial derivatives of that formula with respect to each variable (with uncertainty,
q that is) and then combine
2
)2 + ( VRδR
them in quadrature. Thus, for I = V /R, we end up with δI = ( δV
2 ) .
R
σI =
q
(0.001/10)2 + ((0.476/102 )0.1)2 amperes
frequency (Hz)
500
1000
2000
4000
resistor voltage (V) [p;p]
0.461; -0.476
0.254; -0.265
0.104; -0.149
0.060; -0.074
current (A) [p;p]
0.0461; -0.0476
0.0253; -0.0265
0.0104; -0.0149
0.0060; -0.0074
q
2
R = V /I ⇒ δR = ( δVI )2 + ( VIδI
2 )
σR = 0.2ohms (of a total calculated 6Ω)
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uncertainty current (A) [p;p]
5×10−4 ; 5×10−4
3×10−4 ; 3×10−4
1×10−4 ; 1×10−4
1×10−4 ; 1×10−4
q
Inductive resistance is given by L2πf which ends up turning into (2πf δL)2 + (2πLδf )2 for
our uncertainty formula.
We can effectively say that δf = 0 so we end up with a ±5Ω uncertainty in our calculated
value for inductive resistance.
The way in which we measured voltage across the various coils necessarily lead to the generation of “error”, or at least what might be termed uncertainty. Especially in the case where
a coil with a steady current was used, the values reported by the SWS suite (as in onto
the scope) might very well have been greatly influenced by electronic ticks in the code and
program. The sensitivity scales at which the voltmeters were set to read at this point were
on the order of tenths or hundredths of volts, which leads for quite a bit of uncertainty.
Further, the way that the magnetic fields were created (and thereafter voltages induced) was
somewhat haphazard in that it was necessary for the experimenter to physically, manually
move the coil or magnet to produce the field. As the human was an important, integral part
of the process, obviously human imprecision became a part of the measurements.
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Questions
1. When dropping the magnet through the coil to induce voltage, which peak is
higher and why?
When dropping the magnet, the second voltage peak was higher than the first. This can be
explained by the fact that the voltage induced is essentially a function of the speed at which
the magnet was falling (which would obviously be greater for the second peak—acceleration
due to gravity and all), which can be seen in the area-differential term in the original equation.
2. With the two coils laying on top of each other, one with an AC current: how
are the two scope traces related? What happens if the leads are reversed? What
happens if the coils are moved apart? What does “mutual inductance” have to
do with this?
With the two coils laying on top of each other, the two scope traces match each other (there is
obviously a difference in voltage scale) but are π/2 out of phase with each other. This makes
sense, as current is directly related to voltage—the derivative of current must be directly
related to the derivative of voltage. If the leads are reversed, then the magnitude of one of
the traces is flipped about the line v = 0. If the coils are moved apart, then the induced
voltage goes down noticeably (that is, the reported measured voltage in the primary coil is
lesser). This is related to the concept of mutual inductance, in that with a greater distance
between the two coils, the amount of “transfer” from one coil to the other is down; it might
be said that the two coils are less mutually inductive. Mutual inductance is somewhat a
property of the two-coil system, suggesting how well a voltage through one will induct a
voltage in another.
3. Again with the two coils, one having AC current, what happens to the induced
voltage in the coil when the metal core is inserted part way; when the core is
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Figure 3: Voltage across Resistor and Inductor Added
inserted fully?
When the metal core is first inserted into the two coils with induced voltage, the overall
induction goes down (that is, the reported, measured induced voltage is less); however, when
the metal core is inserted fully, the reported induction goes up (that is, the voltage is a net
increase).
4. For the part with self inductance: on the scope trace, do the shapes of the
voltage across the coil and the resistor make sense (especially for triangular and
square waves)? What happens when an iron rod is inserted into the coil and why
(at 1000 Hz)?
The traces of the voltage across the coil and across the resistor add up to the voltage being
sent across the entire circuit, following the current by a π/2 phase shift, which makes perfect
sense (after all, the voltage around a loop needs to add up to zero after all is said and done).
Otherwise, the voltage in the resistor and inductor should (and do) generally follow the
patterns outlined in the RL lab from before, wherein the inductor represents the derivative
and the resistor is the difference between total and the derivative. For the triangular wave,
the trace basically follows a triangular wave; however, it is somewhat rounded at the top. For
the square wave, there is a slight pick-up time while the slope trace is compensating. When
an iron rod is inserted into the coil, the inductance of the coil overall goes up (according to
the way inductors work), and summarily bring the overall voltage down (according to the
formula, as well as experimental results).
5. Assuming a voltage v(t) = VP cos (ωt) across an inductor L, integrate the power
over one cycle and show that the net energy into the inductor is zero.
We start with the equation VP cos (ωt) and take the integral of it with respect to t for one
cycle (which is 0 → 2π). Simply integrating with respect to t leads to VωP sin (ωt), evaluated
from t = 0 to t = 2π. We can take an easy shortcut here and note that any modification of
a function sin on the interval of t = 0 to t = 2π (corrected for any other changes in time by
the factor ω) will lead to a net of 0 overall. Thus, it has been shown that net energy into the
inductor = 0.
6. Is there an induced voltage when the magnet (or coil) is perpendicular to the
axis of the measuring coil?
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There is no induced voltage when this is the case. (Experimentally, there is a slight bit of
induced voltage; however, there should be none.) This is because of the fact that flux is a
function of differential area and the component of the magnetic field that is perpendicular to
the area. When the magnet is perpendicular, it generates a similarly perpendicular field, that
ends up being parallel to the differential area. In the equation, therefore, we have a ‘zero’
term: BA cos (θ) ⇒ BA(0) and overall induced voltage becomes (or at least approaches) 0
V.
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Conclusion
Nothing incredibly spectacular was done for this lab, indeed, while it all was rather “cool”,
no new properties of physics were seen, nor were any true fundamental constants (or nonfundamental constants) measured. It seemed as if the examination was more of the qualitative
sort, showing how magnetic fields are produced and how these fields induce voltages via EMF.
This having been said, the properties of induced voltages that were examined were absolutely
comparable to various understood (or postulated) behaviors as given by the laws of Faraday
and Henry.
When comparing values for the inductive resistance and the calculated resistance of the coil,
we find that the inductive resistance is much greater (an order of ten) than the inductor’s
resistance, which should rightly lead us to conclude that the inductor’s resistance is relatively
unimportant for what we are considering here. The inductive resistance was found to be
52 ± 5Ω whereas the inductor’s resistance was found to be 6 ± 0.2Ω. As can be seen, the
inductor’s resistance is almost within the tolerance of uncertainty for the inductive resistance.
These bounds suggest that the theory we used was acceptable; for our theory, we assumed
the inductor coil had no effective resistance of its own.
Several properties of the relationship between induced voltage and magnet movement /
pseudo-generation were noted, though, over the course of the experiment. One, the faster the
movement of the magnet (or magnet-like object such as coil), the greater the magnitude of the
induced voltage in the primary coil. Two, this relationship can be related to the “jerk” with
which one of the magnets was moved in relation to the other; greater change led to greater
induced voltage. Three, a change in the polarity of the magnet (either by physically flipping
the permanent magnet or reversing the polarity of the current) led to opposite magnitude
voltages being produced in the primary observation coil. Four, increasing (or changing, for
that matter) the frequency at which the AC current was oscillating when it was acting as the
magnet did not, overall, have a major impact upon the magnitude of the voltage induced.
(As it could be argued that there was a greater overall change, the relative constancy of the
induced voltage suggests a lesser current was being “used” each time.)
The above findings agree with the
laws and behaviors suggested by Faraday and Henry
R
~
~ emf (EMF= dΦ ), and the relationship between
especially dealing with flux (Φ = B · dA),
dt
that and voltage induced in a coil (as measured).
Finally, and definitely hinted at before, it might be somewhat said that a coil with a DC
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voltage through/across it can be considered the same as a permanent magnet, and a coil
with an AC voltage through it might be the same as a permanent magnet flipping back and
forth (switching polarity, NORTH-SOUTH SOUTH-NORTH) at some oscillatory rate.
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