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Physics 241 Lab: Solenoids
http://bohr.physics.arizona.edu/~leone/ua_spring_2009/phys241lab.html
Name:____________________________
“When I Heard the Learn’d Astronomer”
When I heard the learn’d astronomer,
When the proofs, the figures, were ranged in columns before me,
When I was shown the charts and diagrams, to add, divide, and measure them,
When I sitting heard the astronomer where he lectured with much applause in
the lecture-room,
How soon unaccountable I became tired and sick,
Till rising and gliding out I wander’d off by myself,
In the mystical moist night-air, and from time to time,
Look’d up in perfect silence at the stars.
-Walt Whitman
Important:
 In this course, every student has an equal opportunity to learn and succeed.
 How smart you are at physics depends on how hard you work. Work problems daily.
 Form study groups and meet as often as possible..
 Join professional organizations.
 Physicists help people: science => technology => jobs.
Section 1:
1.1.
A current carrying wire creates a magnetic field around the wire. This magnetic field may be
 I  ds  rˆ
found using the Biot-Savart Law dB  o
. However, this law can be difficult to use. If there
4
r2
is a high degree of symmetry, you may choose to find the magnetic field produced by the current
carrying wire using Ampere’s Law
 B  ds  oItotal enclosed by. Ampere’s Law simplifies finding a

whole
Amperian
loop
Amperian loop
magnetic field in the same manner that Gauss’s Law simplifies finding an electric field.
Examine the ideal solenoid (also called “inductor”) where a very long coil of wire carries a
current I is found to have a nearly uniform magnetic field inside the coils and a nearly zero magnetic
 sectional picture is provided:
field outside the coils. A cross
In the laboratory, an experimenter may easily control the current through a solenoid, and so it is
desirable to determine the strength of the magnetic field inside the solenoid as a function of the current.
Because of the high degree of symmetry in the ideal solenoid, Ampere’s Law may be used to find the
magnetic field inside the solenoid. Go over the following derivation with your lab partner since you
will need to understand it for your lab report.
When the current in a wire changes, the strength of the magnetic field created by the current also
changes. Therefore, if you know the time dependent behavior of the current, you can find the time
dependent behavior of the magnetic field inside the solenoid. If the solenoid is
carrying a sinusoidally oscillating current, then it will produce a sinusoidally
varying magnetic field inside the coils.

1.2.
A solenoid with 38,000 loops per meter is placed in series with a 10  resistor and driven by a
sinusoidally varying source voltage. Using an oscilloscope, the voltage across the resistor is found to
be Vres (t)  4.2sin 325t . Write a time dependent equation with numerical values to describe the time
dependent strength of the magnetic field inside the solenoid. Then provide two specific times when
the current and magnetic field have both simultaneously reached their respective amplitudes. Note that
o=4x10-7 [Tm/A].
Your answers:
Section 2:
2.1.
Two very important concepts in the study of electricity and magnetism are Ampere’s Law and
Faraday’s Law. Ampere’s Law states that a current in a wire creates a magnetic field. Faraday’s Law
states that if the magnetic field changes (oscillates) the changing magnetic field will create a voltage in
the wire. This voltage that is created by the changing magnetic field is call the “back EMF” or the
“self-induced EMF”. Thus the logical flow of ideas is the changing current inside the solenoid wire
causes a changing magnetic field between the solenoid coils that then causes a changing back EMF
inside the solenoid wire. This is shown graphically for a sinusoidally oscillating circuit with a resistor
and inductor in series:
2.2.
Experimentally observe the 90o phase shift that shows that the oscillating voltage across an
inductor leads the oscillating voltage across a resistor by /2. Use a sinusoidal source voltage to drive
an inductor in series with a resistor. Use your oscilloscope to measure the voltage across the resistor
and inductor simultaneously so that you can view the 90o phase shift. Make a quick sketch of your
observations below. Record your choices for resistance values, driving frequencies, etc. that enabled
you to see the inductor lead the resistor.
Warning: at very low frequencies, the self-induced back EMF goes to zero because dB/dt is
nearly zero so you would think that you would measure zero volts across the inductor. However, these
inductors have ~1 km of wire and therefore some small resistance so at low frequencies you may
register the voltage drop of the solenoid acting as a pure resistor. In this case, there is no phase shift.
Check for yourself that the solenoid acts as a pure resistor at very low frequencies ~10 Hz.
Your parameter choices and quick sketch:
2.3.
If a magnetic field is changing through the area of a loop of wire, then Faraday’s law says that
this changing magnetic field will induce a negative voltage (called the back EMF) in the circuit:
d
 Faraday’s law:
L (t)   B ,
dt
where the magnetic flux B is simply the magnetic field integrated over the cross sectional area of the
wire loop:
 Magnetic flux:
B  Aarea  B.

Since a solenoid has N loops each with a cross sectional area that is constant in time, this means that:
dB
 Faraday’s law for fixed area and N loops: L (t)  NA .
dt

Due to the way we think about electronics,
we need to remove the negative sign from this
equation. This is similar to our use of V=IR for resistors in a circuit when technically the resistor
causes a voltage drop (therefore a negative voltage change). Another way to think about this is that we
 adding all the voltages in a circuit together should yield
can say that due to conservation of energy,
zero: Vsource Vresistor  0. But we can always rearrange to have Vsource  Vresistor. Thus you should write
dB
when measuring inductance in a circuit.
L (t)  NA
dt

 2.4. Find the time dependent back EMF created bythe RL circuit given in section 1.2. if the cross
sectional area of the inductor is 0.006 m2 and the inductor is 9 cm long. Your answer:
Restating the ideas of the previous section, if you take a single solenoid and apply an
alternating source voltage, then it will have an alternating current. The alternating current of this
single solenoid will produce an alternating magnetic field within the area of its own coils. Now, there
is a changing magnetic field within the area of the coils. The changing magnetic field induces a
voltage in this solenoid (the back EMF). This process is called self-inductance.
The parameter L (SI units of Henry) is used to describe the solenoid’s ability to self-induce a
back EMF L. This parameter is defined as:
 (t)
Lsolenoid  L .
dI L 
 
 dt 
dB
Substituting the previous results L  NA
and Binside  onIL gives:
dt
solenoid



dB 
NA 
dB 
dt 
Lsolenoid 
 NA  A  o  N  n .
dI L 
dI L 
 
 dt 
2.3.
Find the self-inductance parameter L for the inductor described in sections 1.2. and 2.3. Your
answer:

2.4.
If you allow a magnet to swing at the end of a pendulum just barely above some nonmagnetic
sheet of copper, the pendulum-magnet will lose energy and come to a stop. Explain where the energy
goes using the concepts of eddy currents &Lenz’s Law. Your explanation:
Section 3:
3.1.
Now you will experimentally examine the magnetic field produced by a solenoid with current
flowing through it. Note that a real solenoid produces a magnetic field qualitatively similar to a bar
magnet of the same dimensions.
First check that your compass has not been “flipped”. The compass arrow should align itself
with the local magnetic field produced by the Earth. Remember that the Earth’s north magnetic pole is
at the geographic south pole. This causes the local magnetic field to point toward the north geographic
pole. The compass aligns with this magnetic field and thus points toward the north geographic pole. If
your compass has been flipped, fix it with a strong magnet or tell your TA so they can fix it.
Examine the direction of the windings of your solenoid. You can tell which way the current
circulates by how the wire enters the solenoid. Then you can use the right-handed screw rule to
determine the direction of the magnetic field inside the solenoid (which side is north and which is
south).
Apply constant current using the constant voltage supply (a few volts should be good) to your
solenoid. Use the right-handed screw rule to predict the positive direction (north pole) of the magnetic
field produced within the solenoid and the magnetic field surrounding the solenoid in general. Test
your prediction using the compass and sketch the magnetic field surrounding the solenoid below. Be
sure to draw your solenoid in such a way as to show the reader how the current flows through the
solenoid (see previous pictures in this manual for hints).
Your sketch of the observed magnetic field within and surrounding a real solenoid with constant
current:
Section 4:
4.1.
In this section you will experimentally observe the process of self-inductance. Place a solenoid
in series with a 100  resistor and drive this simple RL circuit with a sinusoidal source voltage with 10
Hz (very slow) and 10 to 20 volts source amplitude. Set your oscilloscope to measure the voltage
across the resistor. The resistor voltage tells you how much power the circuit is removing from the
function generator. This also tells you about how much current is flowing through the resistor and
therefore the solenoid.
Qualitatively examine what happens to the current in the solenoid (via resistor voltage) as you
increase the driving frequency from 100 Hz to higher frequencies. Your observations:
The impedance L of an inductor is found to be L  driveL where L is the self-inductance
parameter of the solenoid and drive is the angular driving frequency. The total circuit impedance
V
without a capacitor is given by Z  R2   L2 . Since Iresistor  source, amplitude , explain how IR would
Z
amplitude

change if drive was increased. (This is an alternate explanation of the previous question.) Your
explanation:



Section 5:
5.1.
Restating previous ideas, Faraday’s Law states that a changing magnetic field will induce a
d
back EMF voltage: L   B . Technically this equation shows that the back EMF produced is
dt
proportional to the rate of change of the magnetic flux, but the magnetic flux is just the magnetic field
integrated over the surface area of the circuit.
In simple situations this just becomes
d B dA  B
dB

. Since the presence of a back EMF is effectively like adding a battery to a
  A
dt
dt
dt
circuit, a current can be induced in a circuit without any other voltage source present!
An example of this interesting situation is when a magnet approaches a solenoid in series with a
resistor:
Here the magnet’s north pole is approaching the solenoid from the left. The magnetic field
dB
strength is growing inside the inductor coils as the magnet approaches
 0 . Lenz’s Law is a very
dt
simple rule that allows you to figure out the direction of the current flow caused by the induced back
EMF. Lenz’s Law states that the back EMF will cause a current that generates it’s own reactionary
magnetic field that opposes that change of the magnetic field inside the solenoid. In this case the
 states that there will be a reactionary
magnetic field is growing to the right. Therefore, Lenz’s Law
magnetic field growing to the left:
Now you can simply use the right-hand screw rule to find the direction of the current traveling
around the inductor coils that will produce the correct reactionary magnetic field and trace this current
through the wires to find the direction of the current through the resistor.
5.2.
Now solve some “simple” applications of Lenz’s Law and the right-hand screw rule for finding
the current induced in a circuit with no source voltage present.
Your answer:
Your answer:
Your answer:
Your answer:
Section 6:
6.1.
Now you will experimentally observe how a changing magnetic field induces a voltage in an
unpowered solenoid. Check your bar magnet with your compass to see that it is labeled correctly.
Remember that the compass’s needle labeled “north” should point to south magnetic poles (like the
north geographic pole of the earth).
Hook up your solenoid to the oscilloscope so you can measure the electric potential difference
between both ends. Don’t bother placing a resistor in series; there won’t be large enough currents to
measure with the oscilloscope. Instead you will just examine the direction of the back EMF by
checking the sign of the voltage across the solenoid caused by the back EMF.
6.2.
Move the north pole slowly toward one side of your solenoid. Write what you see on your
oscilloscope (positive or negative induced voltage). You should see some induced voltage or you are
moving too slowly. Try using 500 ms/div and 50 mV/div to clearly see the response. Your
observation:
6.3.
Move the north pole quickly toward same side of your solenoid as in 6.2. Write what you see
on your oscilloscope and compare the amplitude of the induced voltage with part 6.2. Your
observation and comparison:
6.4.
Use the equation for Faraday’s Law L  
d B
to explain why the amplitude 6.3 was larger
dt
than 6.2. Your answer:

6.5.
Now move the south pole quickly toward the same side of your solenoid as in 6.2. Write what
you see on your oscilloscope and compare your result with that of 6.3. Your observation and
comparison:
6.5.
Use Lenz’s Law to explain the difference between your observations in 6.3 and 6.5. Your
explanation:
Section 7:
7.1.
If you take a solenoid and drive it with an alternating current, it will produce an alternating
magnetic field inside its coils. If you then take another solenoid that is unpowered and place it nearby
so that the changing magnetic field of the first solenoid reaches inside the coils of the second solenoid,
then a voltage will be induced in the second solenoid despite the fact that the wires of each solenoid in
no way touch each other. This is the process of mutual inductance and is the basis of transformers in
transmission lines.
The definition of the mutual-inductance is:
M transmitter 

receiver
.


dI
to receiver
 transmitter 
 dt 
The mutual inductance parameter M can be used to encapsulate all the geometric information
of the combined solenoids. It is used in engineering so that you can calculate the transmitted voltage
from an applied current without having to calculate the magnetic field and geometric factors.
The equation for M istrue for all times so it is easiest to determine it experimentally using the
time when the right hand side components have simultaneously reached their maximums:
M transmitter 
receiver, amplitude
dI transmitter 


 dt amplitude
Some digital oscilloscopes can measure derivatives in which case you could determine the

dI

1 dV
  resistor, transmitter 
denominator using  transmitter 
. If not, you would have to take the
 dt amplitude R 
dt
amplitude

dI


 Vresistor .
derivative yourself to find that  transmitter 
 dt amplitude R transmitter
to receiver
amplitude

Not a question.

Section 8:
8.1.
In this section you will experimentally study the process of mutual inductance. Use a function
generator to drive a solenoid without any resistance. Use a sinusoidally oscillating source voltage of
10-20 volts amplitude. This solenoid is the transmitter.
Hook up the other undriven solenoid directly to the oscilloscope. This solenoid is essentially
the receiver. Place the solenoids together as close as possible so that you can measure the maximum
induced voltage amplitude in the receiver. Placing soft iron inside the coils can increase the magnetic
field inside the solenoids so that transmission is increased.
8.2.
Now create a graph showing the
maximum induced voltage amplitude in the
receiver for several different driving
frequencies. Collect about 10 data points (it is
wisest to find fmax transmission first and collect data
on either side). Then use your data to create a
graph. A poor example of what you should see
is provided. Find the frequency that maximizes
transmission. Collect your data in the space
below, make your graph on graph paper, and
report
your
maximum
transmission
frequency below:
8.3.
Use the mutual inductance equation and the self-inductance equation to explain which part of
your graph is dominated by mutual inductance and which part of your graph is dominated by selfinductance. (Be sure to check your answer with your TA.) Your thorough explanation:
Section 9:
9.1.
Your answer to 2.1.b should state that L = R when you measure VR, amplitude = L, amplitude. If
you find the frequency where this happens, you can use L  driveL to find L. Use this single
measurement method to find L for your solenoid.

9.2.
There is a second multi-measurement method to find L. The voltage amplitudes of the
sinusoidally driven RL circuit are very similar to those for the sinusoidally driven RC circuit you
encountered a few weeks ago:

R

 
V (t) resistor  Vsource sin  drivet  and V (t) inductor  L Vsource sin  drivet  .
Z amplitude
Z amplitude 
2 
This gives the following relationships for the amplitudes:
R

Vresistor  Vsource and Vinductor  L Vsource .
Z amplitude
Z amplitude
amplitude
amplitude


 L  Vsource 
amplitude 



Vinductor
Vinductor
Z

  L
amplitude
amplitude


Dividing these two equations gives
. Therefore,  L  R
.




R  Vsource
Vresistor
Vresistor
R
amplitude 
amplitude

amplitude


Z


In order to test the relationship L  driveL and experimentally determine L for your solenoid, simply

combine the last two equations:
Vinductor

 driveL  R amplitude .
Vresistor

amplitude
Vinductor
Therefore if you graph R
amplitude
Vresistor
vs.  drive, you should obtain a linear graph with a slope equal to L.
amplitude

Find L by collecting data for multiple driving frequencies, making a graph and finding the slope.
Make your observations and graph now. Then write your work and result for L:

Section 10:
Authentic Assessment: 2 points. Use your solenoid to search for stray alternating magnetic fields in
the lab. There are many strange high-frequency electromagnetic oscillations (radio waves) permeating
the laboratory. Use an unpowered solenoid to detect the oscillating magnetic fields of these radio
waves in order to measure the frequencies of these signals.
This is the perfect opportunity to learn the use of your oscilloscopes FFT (fast Fourier transfer)
math function. The FFT math operation changes the time-axis to a frequency-axis. You may then
adjust the frequency scale using the “seconds/div knob”, zoom in using the “math > FFT zoom
function”, horizontally scan using the “horizontal shift knob”, and Record your observations below for
your TA to examine. Your observations:
Show your TA once you are successful and have them initial here:______________
Section 11: (Open-ended question / creative lab design)
Imagine you work for an engineering firm that constructs transformers for power lines. Your
customers most likely don’t understand the intricate relation between Faraday’s Law, Ampere’s Law,
etc. They just want to know what the output voltage is of the transformers your company sells if they
put in a given current. With this in mind, determine the mutual inductance of your two-solenoid
transformer with a soft iron core.
At the following prompts, design an experiment to find the mutual inductance of your twosolenoid transformer. Then implement your experiment and record your observations. You may
“cheat” by talking to other groups for ideas, but not “cheat” by already knowing the answer or looking
it up.
Your planned experiment, sketch of actual implementation and any theoretical calculations:
Your observations:
Your explanations & conclusion:
I, the physics 241 laboratory TA, have examined this worksheet and found it to be thoroughly
completed excepting any sections that I have marked herein. TA signature: ____________________
Report Guidelines: Write a separate section using the labels and instructions provided below. You
may add diagrams and equations by hand to your final printout. However, images, text or equations
plagiarized from the internet are not allowed!
 Title – A catchy title worth zero points so make it fun.

Goals – Write a 3-4 sentence paragraph stating the experimental goals of the lab (the big
picture). Do NOT state the learning goals (keep it scientific). [~1-point]

Concepts & Equations – [8-points] Be sure to write a separate paragraph to explain each of
the following concepts.
Explain the following items in order:
 Explain Faraday’s Law and Ampere’s Law.
 Explain what causes the back EMF in a solenoid (self-inductance).
 Explain why L(t) (voltage across the inductor) is 90o phase-shifted before VR(t).
 Explain why IR decreases when the driving frequency drive is increased.
 Explain how to induce current in an unpowered circuit and the use of Lenz’s Law.
 Explain how to transmit electrical power in wires that don’t touch.
 Derive Binside  onIL .
solenoid
dB
.
dt
 Derive Lsolenoid  A o  N  n .

 Compare the frequency dependence of the inductive reactance to the capacitive
reactance.


Procedure
– Do not write a procedure [0-points]


Derive L  NA

Results – Summarize the quantitative or qualitative results of each section of the lab with a
separate, short paragraph. Then write a lengthier paragraph(s) about your observed results in
your open-ended experiment. Be sure to reference any graphs as results in the appropriate part
this section (using the title of your graph). This must be written in sentence-paragraph form
without bullet points. [1-points]

Conclusion – Write at least three paragraphs where you analyze and interpret the results you
observed or measured based upon your previous discussion of concepts and equations. It is all
right to sound repetitive since it is important to get your scientific points across to your reader.
Write a separate paragraph analyzing and interpreting your results from your open-ended
experiment. Do NOT write personal statements or feeling about the learning process (keep it
scientific). [5-points]

Graphs – All graphs must be neatly hand-drawn during class, fill an entire sheet of graph
paper, include a title, labeled axes, units on the axes, and the calculated line of best fit if
applicable. [5-points]
o The two graphs from sections 8 and 9.

Worksheet – thoroughly completed in class and signed by your TA. [5-points & mandatory
for report credit.]
Introductory Electricity & Magnetism Laboratory Learning Outcomes:
Introductory electricity & magnetism laboratory students should be prepared for leadership roles in
an increasingly diverse, technological and highly competitive world. To this end, these students
should…
 Understand the role of science in our society – How a solenoid affects an AC circuit.
 Have a firm grasp of the theories that form the basis of electricity and magnetism – Use of
Faraday’s Law, Ampere’s Law and Lenz’s Law.
 Be able to apply the principles of physics to solve real world problems – Design an experiment
to measure mutual inductance. Also to detect ambient electromagnetic radiation and
determine its frequency.
 Be familiar with basic laboratory equipment, and should be able to design and carry out
experiments to answer questions or to demonstrate principles. – Skills developed throughout
lab especially the open-ended component. Use of modern digital oscilloscope.
 Be able to communicate their results through written reports – See the general report writing
guidelines in the syllabus and the specific report writing guidelines at the end of the handout.
Graduates of the introductory electricity & magnetism laboratory program should…
 Have a broad education that will allow them to succeed in diverse fields such as business, law,
medicine, science writing, etc. – Writing skills, critical thinking skills, creative problem-solving
skills, communicating ideas & information, teamwork skills, leadership skills, working with
time constraints.
 Have mastered the introductory theoretical techniques and electricity and magnetism
experimental techniques that are commonly expected for students at this level. – Across the
world, all introductory electricity and magnetism students must be able to
o Use a digital oscilloscope to measure rapidly changing voltages.
o Build circuits and measure their properties.
 Be familiar with the principles and practice of engineering and should be able to apply their
knowledge to solve state of the art problems, both individually and as part of a team. –
Individuals must be able to build and test simple circuits. Teams must devise an experiment to
measure mutual inductance.
Teaching Tips:
o Review the theory for finding a magnetic field: Ampere’s Law (compare structure to Gauss’s Law)
& Biot-Savart Law.
o Review right-hand wrap rule for finding direction of B inside a solenoid.
o Review the theory for using a magnetic field: Faraday’s Law and Lenz’s Law.
Demonstrate together:
 Quick demo of Faraday’s law with solenoid hooked directly to o-scope and bar magnet.
 Quick demo of mutual inductance.
Instructional Equipment Needed:
