D.E. Shaw
©Villanova University
January 3, 2007
(Edited April 3, 2016, M. DeGeorge)
Lab #11 - Electromagnetic Induction I - 850
Fall 2016
Voltage probe to Ch. A
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.
Signal generator
n
A
I(t)
inner
solenoid
Equipment: 12 cm. Pasco
concentric solenoids, one Pasco voltage probe, two long
wires with banana plugs at both ends, Pasco RCL
experiment board, bar magnet, 2 meter vertical rod, one
table clamp and two bar clamps, sand bag and 1 meter
plastic tube (see fig 4a at end).
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:
dΦ
E =−
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
outer
"a" is zero then the time at "c" is the
B
N solenoid
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
Fig. 1
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:
B(t ) = µ 0 nI (t ) ...(2)
where "n" is the number of turns
per unit length of the inner
solenoid.
Neglecting the field outside the
inner solenoid, the total flux
through the outer solenoid is:
+I0
c
a
T
0
time
2
0
-I0
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
1
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 waveform 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 red and black OUTPUTS terminals of the
Pasco interface box to the two terminals of the inner
solenoid.
a
b
Emf
• Connect a Pasco voltage probe
to the Analog Input A. Connect
the other ends of the probe to the 0
terminals of the outer solenoid to
measure the induced Emf. The
I
currents and potentials in this
0
experiment are quite small so it
time
is important that the plugs make
good electrical contact. Turn the
Fig. 3
plugs back and forth in their
sockets a few times to clean the
surfaces.
• To reduce noise on the signal make sure the input and
output wires do not run parallel to each other and are well
separated.
• Load the Capstone program. Under Hardware Setup, click
on the icon for Outputs and select Output 1 (upper
connectors) then select Voltage-Current Sensors. Next click
on the Analog Input A and select the Voltage Sensor. Click
the Signal Generator and set up the parameters for 850
Output 1 as follows:
• In the dropdown window select the following:
• Triangle
• Sweep Type off
• Frequency 40.00 Hz
•
•
•
•
•
Amplitude 0.30 V
Voltage Offset 0.00V
Voltage limit 1.00 V
Current limit 1.50 A
Turn on Auto start
NOTE: In electrical engineering, the term for the return
side of a circuit is often called ground or common.
For the rest of this part of the experiment do not change
the generator amplitude.
• Set the Common sample rate to 10.00 kHz.
• Using the Recording Conditions button, set a Timed Stop
of 0.1 seconds. Choose a Measurement Based Start
Condition of Output Current, Ch 01 Rises Above 0.000A.
• Drag a graph icon to a fresh worksheet and add a second Y
axis graph. Choose Current in the inner solenoid, Output
Current, Ch 01 versus time to display on the lower graph.
On the upper graph plot the Emf induced in the outer coil
(Voltage, Channel A) versus time.
• Collect a data set. If the data do not have the same phase
and polarity as Figure 3 then reverse the wires on the input
or output of the solenoid.
• 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 maximum amplitude, I0, of the triangular current which
should not change significantly 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 stop
120
.02
condition times according to the
140
.02
values given in the table.
160
.02
Analysis:
180
.02
• Are the shapes of your induced
200
.02
Emf versus time curves consistent
500
.01
with Faraday's Law? Explain in
detail.
• Test Faraday's Law in a quantitative fashion by plotting, in
Excel, 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,
2
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 known 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:
ε = (4 µ0 nΝΑ)x
… (4 a)
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?
Part B: Faraday's Law with Square and Sine Wave
Currents
We use exactly the same experimental conditions as used in
Part A except we try different types of applied currents to
the inner solenoid.
• Select a square wave of the same amplitude that you used
in Part A and a frequency of 40 hz. Use a common data
collection rate of 10,000 hz. Collect a
data set and plot the current applied
to the inner solenoid versus the time.
magnet
Note: you may need to change the
Plastic tube not
stop time to get a few cycles of data
show for clarity
on the plot,( maybe 0.1 sec). On the
same graph plot the Emf measured in
the outer coil versus the time.
Pasco
Interface box
According to Faraday's Law the
I
induced Emf should depend on the
B I
time derivative of the current applied
A
to the inner solenoid since the flux
through the outer solenoid is directly
proportional to this current. Is your
Fig. 4
data consistent with Faraday's Law?
Explain clearly.
• Repeat the above step for a 40 Hz sine wave current of the
same amplitude used in the previous step and answer the
same question. Now, using the sine wave data, construct a
plot of the Emf in the outer solenoid versus the current
applied to the inner solenoid. Use points only and set the
symbol size to 2. This plot is often referred to as a phase
plot. What do you conclude from this phase plot?
Part C: The Laws of Faraday and Lenz Applied to the
Motion of a Magnet through a Coil
Objectives:
In this part of the experiment a permanent magnet is moved
through a fixed coil to produce a changing flux in the coil.
The dependence of the induced Emf on magnet speed is
observed and Lenz's Law is used to predict the direction of
the induced current flow in the coil.
Theory:
A magnet moving through a coil produces a changing flux
and therefore an induced Emf. Lenz's Law provides a simple
way of determining the direction of the induced current in
the coil as the magnet is moved through it. Lenz's Law states
the induced Emf will create an induced current whose
magnetic field will oppose the change in the magnetic flux
which is occurring.
Experimental Details:
Magnet Warning: The bar magnet must be handled very
carefully since it is very brittle and easily broken. You must
keep the magnet far from CRT monitors, and magnetic
storage devices to prevent damage.
• Mount the Pasco RLC circuit board at the side of the table
using the stand and clamps and slide the long plastic
cylinder into the top side of the coil (see fig 4a in the
appendix). Be sure to leave space between the bottom of the
RLC board and sand bag so that the magnet can exit the coil
completely.
• When inserting or removing the plastic tube hold the coil
directly. Do not twist the tube in or out of the coil by
holding the printed circuit board. Twisting the board will
cause the coil to break off of the board!
• Use a rubber band to attach a 1 meter
length of string to the north side of the
N
magnet. The North Pole of the magnet
is marked by a groove that may be
difficult to see.
S
• Connect the Pasco voltage probe to
B
Analog Input A and the other end to
coil
coil on the circuit board with the red
and black plugs connected to the
ds
terminals marked "R" and "B"
R
respectively in Fig. 4. Disconnect the
wires from the generator to the
RLC board
solenoid. Set the voltage sensor data
collection rate of 2.00 kHz and a data
stop time of a few seconds. The
potential difference measured by the
voltage probes (which is also the induced Emf in the coil)
will be positive when current flows in the direction from
terminal "R" to the interface box (A) and back out to
terminal "B" in the direction shown by the solid red arrow.
When the measured Emf is positive, the current inside the
coil is in the direction from "B" towards "R" as shown by the
dotted green arrow. When the measured Emf is negative the
3
current flows in the opposite direction. This information is
important when we apply Lenz's Law.
• Construct a graph to display the induced Emf measured by
channel A versus the time. Turn off the data points for
greater clarity by using the gear icon on the graph and under
Active Data Appearance unselect show data points and
Show Run Symbols. By means of an attached string, hold
the magnet in the plastic tube above the coil with the South
Pole at the bottom as shown. Measure the height of the
bottom of the magnet above the coil. Click on Record and
let the magnet fall through the coil from the top and land on
the sand pad free of the coil. Prevent the magnet from
hitting the floor and breaking by catching it immediately
after it hits the sandbag! Do not pull the sting sideways;
this will not raise the magnet and will only cause the
apparatus to rotate from vertical. If the measured Emf's
show a positive peak and then a negative peak either the
connections of the voltage probe are incorrect or the
poles of the magnet are reversed from Fig. 4. In order to
understand Fig. 5 it is important to know that the magnetic
field of a permanent magnet decreases rapidly with the
distance from the pole. Initially the magnet is at rest and no
flux change occurs in the coil no Emf is induced. When the
magnet begins to move the magnetic field is small and the
flux change at the location of the coil produces an induced
Emf that is too small to be seen in the figure. At the time t1
the magnetic field in the coil begins to increase more rapidly
with time and the induced Emf becomes significant. Once
the South Pole is inside the coil the rate of change of flux
begins to decrease and at time t2 the rate of change of flux is
zero and so is the Emf. The induced Emf produced from t3 to
t4 is associated with the changing flux as the North Pole
leaves the coil.
• The change in flux, ∆Φ12 , that occurs from t1 to t2 can be
found using Faraday's Law:
E=−
dΦ
dt
⇒ −dΦ = Edt
t2
− ∆Φ 12 = − ∫ dΦ = ∫ Edt ...(5)
t1
Similarly,
t4
− ∆Φ 34 = ∫ Edt
t3
The flux changes ∆Φ12 and ∆Φ34 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.
• According to theory, the Emf will be zero when the flux is
zero or not changing with time. On the plots this would
occur before t1, after t4 and between t2 and t3. To achieve
the best measurements of the area under the Emf curves it is
beneficial to have these parts of the curve align with zero
Emf.
• Drag a measurement box around the data in an area where
the Emf is expected to be zero. Use the mean value found
and subtract this value from each data run using the
calculator function. You may label the new, calculated
value ZEMF, corresponding to the zeroed EMF values on
the plots.
• For the ZEMF data 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 ∆Φ12 is the negative of this area. In the same
way obtain ∆Φ34. You may increase the precision of the
area measurement using the Data Summary icon
and
selecting the Calculator tab, use the gear icon
associated with Equations, ZEMF=[Voltage(v)…] input and
select Numerical Format. Change the Number of Decimal
Places to something higher, say 4 or 5 significant figures.
Close all open tabs. Re-plot the data and display the area
measurement. The ZEMF area should now display with
more precision.
• Record the values h, ∆Φ12 and ∆Φ34 for each trial.
• Move the magnet to a higher distances h (in the plastic
tube) and let it fall through the coil at least six times. Repeat
all measurements.
• Find the mean value of ∆Φ12 for your trials. Why is this
value almost constant for all trials with different speeds?
Also find the mean of ∆Φ34 for all trials.
• Find the percent difference between the absolute values of
the mean values of ∆Φ12 and ∆Φ34. Why are these flux
changes expected to be almost the same?
• Plot max Emf vs. velocity above the center of the coil.
Comment on how the maximum induced Emf varies with the
height, and consequently, the speed of the magnet through
the coil.
• Explain why the maximum induced Emf increases as (t2 t1) or (t4 - t3 ) decreases.
• Are you quantitative and qualitative observations
consistent with Farady's Law of induction? Explain.
Lenz's Law:
In order to answer the following questions you need to
examine Fig. 4 and 5 and the italicized text in the second
bullet point of the "Experimental Details" section on page 3.
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.
4
Rod
E
0
t1
t2
a
t3
b
t4
table
Plastic tube
Clamps
time
Fig. 5
Appendix: Coil Constants:
The value of "n" (940 m-1) was found 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 there are approximately 315
loops of wire in one layer of loops. By measuring the
thickness of all the layers the number of layers of wire was
found to be 16 +/- 2. The best estimate of the total number of
loops is (315 x 16) or 5040 with an error range of +/- 630.
Coil
floor
Pasco RLC
board
Sand Bag
Figure 4a
5