D.E. Shaw
Villanova University
.
Electromagnetic Induction I
Spring 2008
Voltage probe to Ch. B
Introduction: The primary
objective of this experiment is to
study Faraday’s Law of Induction.
A secondary goal is to verify
Lenz’s Law for the direction of an
induced current. You may be
doing this experiment as a
“discovery experiment” just
before you study the material in
the lecture course.
to power amplifier
n
B
N
A
I(t)
inner
solenoid
Fig. 1
Equipment: 12 cm. Pasco
concentric solenoids, Pasco power amplifier, one Pasco
voltage probe, two long wires with banana plugs at both ends,
Pasco RCL experiment board, bar magnet, 3-leg vertical
support bar, one horizontal support bar, two bar clamps and a
few vernier calipers.
Part A: A Study of Faraday's Law of Electromagnetic
Induction Using Concentric Solenoids
Objective:
The objective of this part of the experiment is to verify
Faraday's Law under conditions where the time rate of change
of flux can be determined directly and accurately.
Theory:
Faraday's Law of induction states that the induced Emf in a
loop is:
E =−
outer
solenoid
dΦ
dt
where Φ is the magnetic flux through the loop.
A current that varies with time in a triangular fashion is
created in the loops of the inner solenoid shown in Figure
1(for clarity only a few loops are shown). This current creates
a magnetic field that also changes with time in the same
manner. This field creates a flux through an outer concentric
solenoid. While the current and resulting magnetic field
increase linearly with time in the inner solenoid the total
magnetic flux through the outer solenoid will also increase
linearly with time. According to Faraday's Law the induced
Emf in the outer solenoid depends on the rate of change of this
flux. Since the flux depends on the field and therefore on the
current, the induced Emf should be constant as long as the
current is increasing at a constant rate that is directly
proportional to the frequency of the triangular applied current.
By changing the frequency of the applied triangular current we
can adjust the rate at which the current changes and determine
if the induced Emf follows the prediction of Faraday's Law.
The triangular shaped current is shown in Fig. 2. The current
between "a" and "c" represents one
complete cycle of the triangular wave
form and if the time at "a" is zero then
the time at "c" is the period, T. The
number of cycles per second is the
frequency "f" and is 1/T. The maximum
current is I0 and is also the current
amplitude. At "b" the current has its
maximum negative value of -I0 at the
time T/2. The linear variation of current
with time between "a" and "b" is:
I (t ) = −4 I 0 ft + I 0
...(1)
The magnetic field inside the inner solenoid is obtained most
easily from Ampere's Law:
c
a
+I
B (t ) = μ 0 nI (t ) ...(2 )
0
T
0
where "n" is the number of
turns per unit length of the
-I0
inner solenoid.
Neglecting the field outside
the inner solenoid, the total
flux through the outer solenoid is:
time
2
0
T
b
Fig. 2
Φ (t ) = B(t )NA = μ 0 nNAI (t )
Φ (t ) = μ 0 nNA(− 4 I 0 ft + I 0 ) ...(3)
where N is the total number of turns of the outer solenoid and
A is the cross sectional area of the inner solenoid since we
neglect any field outside the inner solenoid. We have also
neglected all end effects.
The quantity (μ0 nNA) appearing in Eq. 3 is often replaced by
a single constant "M" which is known as the "mutual
inductance" of the two coils. According to Faraday's Law the
induced Emf in the outer solenoid during the time interval
from "a" to "b" is:
dΦ
⎛ d (− 4 I 0 ft + I 0 ) ⎞
= − μ 0 nNA⎜
⎟
dt
dt
⎝
⎠
E = (4 μ 0 nNAI 0 ) f ...(4 )
E =−
Faraday's Law predicts that the induced Emf should be
constant while the current is changing linearly between "a"
and "b" as shown in Fig. 2. The induced Emf in the outer coil
and the current in the inner coil are shown in Fig. 3. Since the
induced Emf in the outer solenoid depends on the derivative of
the applied current in the inner solenoid the theoretical Emf is
a square wave as shown. The value of this constant Emf
should also be directly proportional to the frequency of the
triangular current wave form as shown by Eq. (4). However,
due to the neglected self induction effects in the inner solenoid
the current is not exactly triangular and the Emf is not exactly
square. Up to this point we have assumed that a triangular
voltage applied to the inner solenoid produces a triangular
current. This would be the case if the inner solenoid acted only
as a resistance and Ohm's Law would predict that the current
and voltage are directly proportional. However, the changing
flux in the inner solenoid will also produce a self induced Emf
that will be in series with the applied voltage from the signal
generator. As predicted by Eq. (4) this complication becomes
more significant as the frequency increases. The self induced
Emf can lead to two significant problems: (i) the current
begins to depart from the triangular wave form leading to an
imperfect square wave for the induced Emf in the outer coil
and (ii) for a constant applied voltage the current amplitude in
the inner solenoid will decrease at higher frequencies. The
frequencies used in this experiment are small in order to
minimize these problems.
Experimental Details:
The equipment consists of two concentric solenoids that are
shown in the figure with the inner solenoid partially removed
for clarity.
• Connect the Pasco power amplifier to channel A of the
Pasco interface box and be sure its power switch is ON.
Connect the output terminals of the power amplifier to the two
terminals of the inner solenoid.
• Connect a Pasco voltage probe to channel B. Connect the
other ends of the probe to the
b Emf
a
terminals of the outer solenoid to
measure the induced Emf. The
currents and potentials in this
0
experiment are quite small so it is
important that the plugs make
I
good electrical contact. Turn the
0
plugs back and forth in their
time
sockets a few times to clean the
surfaces.
Fig. 3
• Load the Data Studio program
and then click on the icon for
channel A and select the Power Amplifier. The Signal
Generator Window will automatically open. Select the
triangular wave function, an amplitude of 0.30 volts and a
frequency of 40 Hz. Click the Measurements and Sample rate
button and select only the output current to measure and
finally select a data collection rate of 5000 hz.
• Click on the icon for channel B and select the Voltage
Sensor to measure the Emf induced in the outer coil. Using the
Sampling Options button select an automatic stop time of 0.1
seconds.
• Create a graph of the current in the inner solenoid “Current,
Ch A” (not the Output Current) versus time. On the same
graph plot the Emf induced in the outer coil “Voltage, Ch B”
versus time. Collect a data set then click on the Settings icon
at the right end of the graph's tool bar to turn off both the Data
Points and Data Symbols options. If everything is set up
correctly you should obtain plots resembling Fig. 3 with the
curve for the induced Emf showing a significant amount of
“noise” coming from electromagnetic interference. The signal
generator may be unable to produce sufficient current which
will cause the triangular current peaks to be chopped off. If
this occurs, reduce the voltage amplitude a little and try again.
For the rest of this part of the experiment do not change this
amplitude. When you have found the best amplitude, use the
"smart tool" to measure the amplitude (shown with an arrow
in Fig. 3) of the flat part of the square wave Emf in the region
between "a" and "b" shown in Fig. 3. Also record the
amplitude, I0 (shown in Fig. 2) of the triangular current which
changes slightly during this experiment.
•Repeat the measurement of the Emf
Freq.
Stop
(hz)
Time (s)
and current amplitude that you made
in the previous step for the different
40
.1
frequencies of the triangular current
60
.1
given in the table but do not change
80
.05
the amplitude of the triangular
100
.05
voltage. Also adjust the automatic
120
.02
stop times according to the values
140
.02
given in the table.
160
.02
Analysis:
180
.02
• Are the shapes of your induced Emf
200
.02
versus time curves consistent with
Faraday's Law? Explain carefully.
• Test Faraday's Law in a quantitative fashion by plotting the
measured induced Emfs versus the frequency “f” of the
current in the inner solenoid. Faraday’s Law as expressed by
Eq. (4) predicts that the induced Emf should increase linearly
with the frequency of the triangular current. Is the shape of
your plot consistent with Faraday's Law of induction?
Do a linear fit (through the origin) of this plot and obtain the
slope.
• The mean cross sectional area “A” of the turns of the inner
solenoid is approximately 2.22x10-4 (m2) .
• Using the average current amplitude I0 of all trials and the
values of n (940 m-1 ), N (5040) and A (2.22x10-4 m2) ,find an
experimental value for μ0, the permeability of a vacuum, from
your measured slope (4μ0nNAI0). Compare this value with the
accepted value which is 4πx10-7 (T.m/A). It should be noted
that this is not an extremely accurate way of finding the
permeability since it is based on some quantities that are not
know accurately and approximations are used in finding the
field of the inner solenoid and the flux through the outer
solenoid. The values of ‘n”, “N” and “A” are discussed in the
Appendix.
• You probably found that the current amplitude decreased
slightly as the frequency increased. The product of the
frequency "f" and current amplitude "I0" can be considered as
a new variable that we will call "x" for convenience. Using
this new variable we can take into account the changing
current amplitude. Equation (4) can then be written in terms
of this new variable as:
E = (4 μ 0 nNA
)x
... (4 a )
keep the magnet far from the monitor, hard drive and floppy
disks to prevent damage.
• Mount the Pasco RLC circuit board above the table using
the stand and clamps.
• Connect the Pasco voltage probe to channel B and the other
end to the circuit board with the red and black plugs connected
to the terminals marked "R" and "B" respectively in Fig. 4.
Delete the power amplifier and the signal generator icons
Part B: Faraday's Law with Square and Sine Wave
using the delete key. Select a data collection rate of 2000 Hz
Currents
and a data collection time of 3 seconds. The potential
We use exactly the same experimental conditions as used in
difference measured by the voltage probes (which is also the
Part A except we try different types of applied currents to the
induced Emf in the coil) will be positive when current flows in
inner solenoid.
the direction from terminal "R" to the interface box and back
• Select a square wave of the same amplitude that you used in
out to terminal "B" in the direction shown by the solid red
Part A and a frequency of 40 hz. Use a
arrow. When the measured Emf is
data collection rate of 10,000 hz.
N
positive, the current inside the coil is
Collect a data set and plot the current
magnet
in the direction from "B" towards "R"
applied to the inner solenoid versus the
as shown by the dotted green arrow.
time. On the same graph plot the Emf
S
When the measured Emf is negative
measured in the outer coil versus the
B
the current flows in the opposite
time. According to Faraday's Law the
coil
direction. This information is
induced Emf should depend on the time
Pasco
important when we apply Lenz's
derivative of the current applied to the
ds
Interface box
Law.
I
inner solenoid since the flux through
the outer solenoid is directly
• Construct a graph to display the
B I
R
proportional to this current. Is your
induced Emf measured by channel B
B
data consistent with Faraday's Law?
versus the time. Turn off the data
RLC board
Explain clearly.
point and symbols for greater clarity.
Fig. 4
The North pole of the magnet is
• Repeat the above step for a 40 Hz
marked by a groove that may be
sine wave current of the same
difficult to see. Hold the magnet above the coil with the South
amplitude used in the previous step and answer the same
pole at the bottom as shown. Click on Start and then carefully
question. Also construct a plot of the Emf in the outer solenoid
pass the magnet through the coil from the top and pull it out
versus the current applied to the inner solenoid. This plot is
the bottom. For the first trial pass the magnet through the coil
often referred to as a phase plot. What do you conclude from
quickly. Your data should resemble Fig. 5. If your measured
this phase plot?
Emf's show a positive peak and then a negative peak either the
• Using the equipment of the previous step, devise a simple
connections of the voltage probe are incorrect or the poles of
experiment to show that the field outside a long solenoid is
the magnet are reversed from Fig. 4. In order to understand
very small.
Fig. 5 it is important to know that the magnetic field of a
permanent magnet decreases rapidly with the distance from
Part C: The Motion of a Magnet through a Coil
the pole. Initially the magnet is at rest and no flux change
Objectives:
occurs in the coil and no Emf is induced. When the magnet
In this part of the experiment a permanent magnet is moved
begins to move the
through a fixed coil to produce a changing flux in the coil. The
E
magnetic field is small
dependence of the induced Emf on magnet speed is observed
and the flux change at
and Lenz's Law is used to predict the direction of the induced
the location of the coil
current flow in the coil.
t2
produces an induced Emf 0 t1
a
b
Theory:
t3
t4
that is too small to be
A magnet moving through a coil produces a changing flux and
seen in the figure. At the
therefore an induced Emf. Lenz's Law provides a simple way
time
time t1 the magnetic field
of determining the direction of the induced current in the coil
in the coil begins to
Fig. 5
as the magnet is moved through it. Lenz's Law states the
increase more rapidly
induced Emf will create an induced current whose magnetic
with time and the induced Emf becomes significant. Once the
field will oppose the change in the magnetic flux which is
South pole is inside the coil the rate of change of flux begins
occurring.
to decrease and at time t2 the rate of change of flux is zero and
Experimental Details:
so is the Emf. The induced Emf produced from t3 to t4 is
Magnet Warning: The bar magnet must be handled very
carefully since it is very brittle and easily broken. You must
In your worksheet create a new column for the variable "x"
and then construct a plot of the Emf versus "x". Do a linear fit
through the origin and from the slope (4μ0nNA) of this graph
find μ0 and compare your value with the accepted value. How
does this result compare with your value from the previous
step?
associated with the changing flux as the North pole leaves the
coil.
• The change in flux, ΔΦ12 , that occurs from t1 to t2 can be
found using Faraday's Law:
E=−
dΦ
dt
⇒ −dΦ = Edt
t2
− ΔΦ 12 = − ∫ dΦ = ∫ Edt ...(5)
t1
Similarly,
t4
− ΔΦ 34 = ∫ Edt
t3
The flux changes ΔΦ12 and ΔΦ34 are the time integrals of the
Emf's. The values of these integrals are also the area under the
Emf versus time curves shown in Fig. 5.
• For the data you just obtained, draw a measurement box
around the data from t1 to t2 and then have the program
calculate the area under the curve. According to Eq. 5 the flux
change ΔΦ12 is the negative of this area. In the same way
obtain ΔΦ34.
• Move the magnet through the coil at least five times. For
each trial vary the speed with which the magnet enters and
leaves the coil. Record the values of ΔΦ12 and ΔΦ34 for each
trial. Be sure to observe, and comment on, how the maximum
induced Emf varies with the speed of the magnet.
• Find the mean value of ΔΦ12 for your five trials. Why is this
value almost constant for all trials with different speeds? Also
find the mean of ΔΦ34 for all trials.
• Find the percent difference between the absolute values of
the mean values of ΔΦ12 and ΔΦ34. Why are these flux
changes expected to be almost the same?
• Explain why the maximum induced Emf increases as (t2 - t1)
or (t4 - t3 ) decreases.
Lenz's Law:
In order to answer the following questions you need to
examine Fig. 4 and the italicized text near the beginning of the
"Experimental Details" section. As the South Pole of the
magnet is approaching and then entering the coil determine:
(a) the direction of the induced magnetic field due the current
that is induced in the coil loops.
(b) the direction of the induced current in the coil loops.
(c) The sign of the Emf induced in the coil.
(d) Does your predicted sign for the Emf agree with the sign
obtained in your plot of the Emf versus time between
t1 and t2 ?
• As the North pole of the magnet leaves the coil answer all of
the above questions for the time interval from t3 to t4.
Appendix: Coil Constants:
I found the value of "n" (940 m-1) by counting the number of
loops of the inner solenoid. The cross sectional area of the
loops of the inner coil was determined in the following
manner. The thickness of the wire in the inner solenoid
(1.063x10-3 m) was calculated by taking the inverse of “n”.
The mean diameter (1.68x10-3 m) of the inner loops was
obtained by measuring the outer diameter (1.79x10-2) and
subtracting the diameter of the wire. The cross sectional area
of the loops of the inner coil is found to be 2.22x10-4 m2. For
the outer solenoid I found that there are approximately 315
loops of wire in one layer of loops. By measuring the
thickness of all the layers I was able to estimate that the
number of layers of wire is 16 +/- 2. The best estimate of the
total number of loops is (315 x 16) or 5040 with an error range
of +/- 630.
Check List: Minimal Requirements for Lab Notebook Report
The significance of each graph must be discussed and the fitted values
(such as the intercept and slope) must be compared with model values
when possible.
Part A: (Triangular Current)
9 A single Data Studio plot the applied current and induced Emf versus
time for one frequency only. Explanation of why the observed
induced Emf is or is not consistent with Faraday’s Law.
9 An Excel worksheet with the frequency, current amplitude and
induced Emf.
9 Plot of the measured induced Emf versus the frequency. Discussion
of the shape of this plot and the qualitative connection with
Faraday’s Law. Linear fit through the origin.
9
9
Calculation of the fundamental constant “μo” from the above slope
and the % difference between your value and the accepted value.
Plot of the induced Emf versus the parameter “x” with a linear fit
through the origin. The calculation of “uo” from this slope and the
comparison of the result with the accepted value.
Part B: (Square and Sine Wave Currents)
9 Data Studio plots of the induced Emf and applied current versus the
time for both square and sine wave currents.
9 Discussion of the shape of each Emf curve and the qualitative
connection with Faraday’s Law.
Part C: Motion of Magnet through a Coil
9 A Data Studio plot of one trial only.
9 Qualitative discussion of how the maximum Emf is related to the
speed of the magnet.
9 An Excel worksheet with the area under each curve and the flux
changes ΔΦ12 and ΔΦ34. Comparison of these flux changes.
Lenz’s Law:
9 Detailed discussion of the direction of the induced current and the
sign of the induced Emf as the magnet enters the coil. The predicted
sign of the induced Emf must be compared with your measured sign.
9 Same discussion as the magnet leaves the coil.