Download Lab 8: Faraday Effect and Lenz` law Phy208 Spring 2008

Lab 8: Faraday Effect and Lenz’ law
Phy208 Spring 2008
Name_____________________________
Section___________
This sheet is the lab document your TA will use to score your lab. It is to be turned in
at the end of lab. To receive full credit you must use complete sentences and explain
your reasoning clearly.
What’s this lab about?
In this lab you investigate effects arising from magnetic fields that vary in time.
There are three parts to the lab:
PART A
Move a bar magnet through a coil of wire to investigate induced EMF and current.
PART B
Drop a strong magnet through various copper tubes to investigate the forces caused
by induced current.
PART C
Quantitatively investigate Faraday’s law by using a time-dependent current through a
coil of wire to generate a time-dependent magnetic field.
Why are we doing this?
Time-varying magnetic fields are all around us, most commonly in electromagnetic
waves. But we most often see effects generated by physically moving permanent- or
electro-magnets, or by changing the current through an electromagnet. The EMF,
electric currents, and forces generated by these can be quite impressive, even enough to
help brake a subway car.
What should I be thinking about before I start this lab?
Last week in lab you looked at the properties of static (time-independent) magnetic
fields, produced by permanent magnets and by loops of current. These static fields
varied throughout space in direction and magnitude, but were the same at all times.
This week you discover some very unusual properties of time-varying magnetic fields. In
particular, a time-varying magnetic field produces an electric field. This means that there
is more than one way to make an electric field.
You can make an electric field with electric charges, as around a point charge or
between charged capacitor plates, but also an electric field accompanies a time-varying
magnetic field. Wherever there is a time-varying magnetic field, there is also an electric
field. For instance, waving a permanent magnet in the air produces an electric field.
In this case the electric field is said to be induced by the time-varying magnetic field. This
induced electric field exerts a force on charged particles, and so work is required to
move a charged particle against this field.
Pretty much like the electric fields we’ve worked with before, except that there aren’t
any electric charges around producing this electric field.
A. Induced fields in a loop of wire
A simple way to measure this induced electric field is to put a piece of wire where you
want to measure the electric field. At any point in ra conductor
where
there is an
r
r
electric field, there is also a current, according to j = "E , where j is the current
r
density, E is the electric field, and " is the conductivity.
This means that there can be an electric field in the wire, and a current in the wire,
!
!
without a battery anywhere.
!
!
Faraday’s law: Faraday discovered a quantitative relation between the induced electric
field and the time-varying magnetic field.
He found that the EMF around a closed loop is equal to the negative of the time rate of
change of the magnetic flux through a surface bounded by the loop:
.
" = # d$
dt
The EMF around a closed loop represents the work/Coulomb required for you to move
a positive charge around that closed loop.
!
One other difference between EMF and electric potential difference is that the potential
difference between two points depends only on the beginning point and the end point,
and not the path between them. The electric potential works great for electric fields
and forces generated from charges because those forces are conservative. The work
done to move a charged particle from point A to point B against these fields depends
only on the location of points A and B.
This means that an electric field line like this
cannot be generated by fixed electric charges.
A1. Give one reason why the electric field line above cannot be generated by fixed
electric charges.
1. Field lines can start only at positive charges, and end only on negative charges.
2. The electric potential is not well-defined, since going around the loop once leads to a potential
difference between a point and itself.
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But these are exactly the kind of field lines that are generated by Faraday’s mechanism!
Your book calls these non-Coulomb electric fields, and the ones that can be generated
from static charges it calls Coulomb electric fields.
A2. Now you measure the EMF induced in coil of wire by a changing magnetic flux. You
will start by using the 800 turn coil on your lab table. Remember that this induced
EMF will cause a current to flow in the wire. So connect the 800 turn coil to the
Digital Multimeter (DMM), making sure that the DMM is set to measure current.
Take the long bar magnet and push it into the hole through the coil, watching the
current on the DMM. (this is easiest with the coil laying on its side).
Turn the bar magnet around, and do it again.
Summarize your results in the table below.
The long arrow on the coil indicates that the coil is wound
clockwise from the bottom terminal to the top terminal
Connect the top terminal to the red terminal of the DMM for this
measurement.
Motion
Sign of flux
N pole moving toward coil
N pole moving away from coil
S pole moving toward coil
S pole moving away from coil
Magnet centered in coil,
move N pole away
Magnet centered in coil,
move S pole away
A3. Explain whether this is consistent with Lenz’ law.
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Sign of
change in
flux
Red
Black
Sign of
induced
current
A4. Now connect the current sensor (small silver box) to
channel A of the Pasco system. (There are only four of
these, so you will need to share). Open the
Lab8Settings1 file from the course web site and record
some data: try pulling the magnet slowly at constant
speed, and then a little more quickly at constant
speed. The current sensor only works up to about
28 mA, so keep your currents below that.
Qualitatively explain the time-dependence of the
current using ideas of magnetic flux and Faraday’s law.
The current sensor:
This box provides the
interface with a voltage
proportional to the current
flowing into the red
banana plug connection
and out the black. There is
very
close
to
zero
resistance between the red
and black terminals, so it
as if they are connected by
a wire. The conversion is
1V per 0.01 amp.
The current is proportional to the emf, which is proportional to
the time rate of change of the flux. Even though you move the
magnet at constant speed, the flux through the loop does not
change at a constant rate. This is because the field is stronger near the magnet than farther
away. So the current increases as you move the magnet toward the coil at a constant rate.
When you stop moving the magnet, the current goes to zero.
A5. Look at the maximum current you obtained in A4 when moving the magnet. What
EMF around the loop does this correspond to?
(Hint: remember that your DMM can also measure resistance).
The current is produced by an emf according to ohm’s law. So the current is equal to
emf/resistance.
The resistance of the 800 Ω coil is about 9.7 Ω.
Some students wonder why the resistance ‘changes’ when they move the magnet in the coil while
the voltmeter measures resistance. This is because the voltmeter measures resistance by putting
a fixed current through the coil, and measuring the voltage drop. When they move the magnet
into the coil, they generate an emf which contributes to the voltmeter reading. The voltmeter
doesn’t know that they moved a magnet through the coil, and so reports a varying “resistance”.
A6. You can also make a measurement that gives you the EMF directly. Unplug the coil
from the current amplifier, and the current amplifier from the Pasco interface. Plug
the coil directly into channel A of the Pasco interface. Take some data running the
bar magnet into and out of the coil. How does this qualitatively compare to your
measurement of the current in A4?
The reason why this works is a little subtle. A voltmeter has very large impedance, so connecting
the coil to the pasco voltmeter in this way essentially breaks the loop, meaning that no current
will flow. Which means that something is canceling the induced electric field. Charge flows
initially to the broken ends of the loop, where it builds up until the electric field from this
built-up charge cancels the nonconservative electric field that the emf represents. The
voltmeter measures this. The measured current and the measured voltage should match up
according to ohms law.
A7. Do A6 again using the 400 turn coil. How do your measurements compare to the
800 turn coil for the same magnet motion?
Voltages should be about half the previous voltages, since the number of turns is half. Think of
the coil as a helix, so that the area of the surface enclosed by the loop is proportional to the
number of turns.
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B. Inducing currents in other conducting objects.
As discussed in class, a time-dependent flux will produce currents in any conducting
object. These are usually called “eddy currents”. They generate a magnetic field that
adds to the original time-dependent flux. According to Lenz’ law, the direction of the
induced current is such that the generated flux opposes the change in the original flux.
B1. In the diagram below, draw the direction of the induced current in the ring.
Induced current generates a flux that opposes the change in the flux from the magnet. So it must
produce a field to the right. This makes the current counterclockwise as shown
Induced
current
N
S
B2. In the same diagram, draw the direction of the force on the bar magnet exerted by
the induced current.
Two ways you can think about this.
1. think about the Lorentz force of the B-field of the magnet on the currents in the loop. By the
right hand-rule (if you can really hold your hand this way!) the net force on the loop is to the
right. There is an equal and opposite force by the currents on the magnet to the right.
2. Replace the current loop with an equivalent bar magnet (both are approximately dipoles).
This bar magnet will have its N pole to the right, and S pole to the left, in order to generate
the same fields as the ring current.
B3. Write a few words below about how you determined the current direction and the
force direction.
See above.
B4. How do the current and force direction change if the North and South pole of the
magnet are switched?
When the magnet is reversed, the flux has the opposite sign, call this negative. As the magnet is
pushed in, the flux becomes large in absolute value, but still has the opposite sign. So the
induced current is opposite. But because the field is also opposite sign, the force is in the
same direction — the force always resists the motion of the magnet.
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B5. You have a 6” length of 1/8” wall copper tube, and a strong NdFeB disc magnet.
Hold the tube vertically above the lab table, and drop the disc magnet down the
tube. Start the magnet so that the disc surface is parallel to table. Describe below
what happened.
The magnet quickly reaches a terminal velocity, after which it falls at
constant speed.
B6. The forces on the magnet are the force of gravity, and the force from the induced
currents, as you investigated in B2 above. Explain why the magnet falls in the tube
at a constant speed.
The force on the magnet from the induced currents is proportional to the time rate of change of
the flux, which in turn is proportional to the speed of the magnet. As long as the force from the
induced currents is less than the force of gravity, the net force is down, and the magnet will
increase its speed downward. Acceleration will stop when the force from the induced currents
exactly cancels the force from gravity. Since at this point there is no acceleration, the speed is
constant.
B7. Now drop the magnet down the tube so that the flat disc surface is perpendicular
to the lab table. Describe below the motion of the magnet, and explain why it does
this. Hint: sketch in the field lines from the magnet.
Think of the flux through a particular path along the circumference of the
tube. By symmetry, whatever field lines go through the surface bounded by
this path also return through that same surface. So if the magnet were to
be dropped with perfect symmetry, there would be no induced currents
and no forces from these on the magnet. So it would accelerate just as if
there were no tube. But there are always some inaccuracies. Suppose the
disc is slightly tiled to start. Then the symmetry is broken, and apparently
perfect alignment is an unstable equilibrium. The result is a torque that
rotates to the (apparently) stable dynamic equilibrium of disc surface
perpendicular to the lab table. I’ll see if I can figure this out in more
detail.
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B8. Now you will make a quantitative measurement of how long it takes the magnet to
drift down various tubes. Use your stopwatch to time the drift. Do the
measurement several times and average, then calculate the terminal velocity.
You will use:
Two 6” tubes ( 1/8” wall, 1” ID) doubled up lengthwise (you may need to borrow)
One 6” length ( ¼” wall, 1” ID) (there is only one of these)
One 6” tube (1/8” wall, 7/8” ID )
Tube
Length
1/8 W, 1 ID
30.5cm
¼ W, 1 ID
15.25cm
1/8 W, 7/8ID
15.25cm
Time1
Time2
Time3
AveTime
Term. Vel.
(cm/s)
B9. Qualitatively explain the ordering of the terminal velocities (e.g. why the slowest is
the slowest and why the fastest is the fastest). Use the following concepts:
Magnetic field distribution of a dipole.
Flux
Faraday’s law
Induced currents
Lorentz force on current in a magnetic field.
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C. Quantitative determination of induced fields.
Dropping magnets through tubes is a lot of fun, but it is difficult to make quantitative
measurements of induced currents. The magnetic field from a permanent magnet is
fairly complicated, and it is difficult to make the flux from it change at a known rate.
In this section you produce a magnetic field by passing a current through a large coil of
wire. It takes a lot of current to make any reasonably sized field, so you use a power
amplifier to supply the necessary current to the coil. The large coil of wire here plays
the role of the moving bar magnet of part A. You don’t physically move the large coil,
but have the computer make the current through the coil change in time. This
produces a magnetic field that varies in time, just as if you were moving a bar magnet.
Just as in part A, you will use this time-dependent field to induce an EMF in a small coil
of wire. This time it is a 2000 turn “sense coil” on a plastic ‘wand’. You will move it
around to various locations to make measurements
Before hooking things up, complete the preliminary questions below.
C1. The current in the large coil produces a magnetic field at its center as described by
the Biot-Savart law dB =
axis of the coil?
µo Idl # rˆ
. How is this field oriented with respect to the
4" r 2
The field is along !
the axis.
C2. Using the Biot-Savart law, write an expression for the magnitude of the
contribution to the total field from a small current element of length dl at the
center of the loop.
Each infinitesimal current contributes
µo Idl
to magnitude of the total field
4" R 2
C3. Add up the contributions of each current element along the entire length of the
! to get an expression for the magnetic field at the center of
coiled wire of 200 turns
the loop. This is the relation between current through the loop and the magnetic
field at the center of the loop.
The total length of the wire is
(N )(2"R) .
The field is then
rdrive = 0.105m as the average radius of the drive coil.
!
!
!
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µo NI
= (0.0012T / A) I , using
2 rdrive
Now you are ready to hook things up.
Connect the output of the power amplifier to the inputs of one of the large coils (red to
red and black to black). This puts current through that coil. The other large coil is
unused.
Use a voltage probe cable to connect Pasco input A to same large coil (red to red and
black to black) as the power amplifier. Input A then measures the voltage drop
across the coil through which the power amplifier drives current.
Plug one end of the 8-pin gray cable into the back of the power amplifier and the other
end into Pasco input C. Plug in and turn on the power amplifier (switch on back).
Use another voltage probe cable to connect the 2000 turn sense coil to Pasco input B.
Position the 2000 turn sense coil at the center of the large coil by sliding it onto the
metal bar.
Click on the Lab8Settings2 file to start up the data acquisition system. Click start to
begin acquiring data.
You should have a real-time display of the voltage sent to the power amplifier, the
voltage drop across the driven coil (input A), and the induced voltage in the 2000turn sense coil (input B).
C4. Use a 10 Hz triangle-wave for the current in the large coil. Describe the timedependence of the induced emf in the small coil, and explain the shape of its
waveform using Faraday’s law (e.g. why is it not a triangle wave like the drive?)
The field at the center of the loop is proportional to the current, which is proportional to the
voltage across the loop. The flux through the sense coil is proportional to the field. The emf is
proportional to the derivative of the flux, which is proportional to the derivative of the voltage
across the drive coil. This derivative alternates positive and negative, giving a square wave.
C5. From your result in C3, calculate the EMF induced in the sense coil when it is at
the center of drive coil. (You should use the average radius of the sense coil). How
does this compare to your measurement?
The field through the small loop is 0.0012I T, where I is in amps. The small sense coil has an
2
2
average radius of Rsense, so flux is " = N sense (#Rsense
)(0.0012I) = N sense (#Rsense
)(0.0012V /R)
where V is the drive voltage, R is the drive coil resistance R=7Ω, and Nsense is the number
N µ $ 1 dVdrive ( t ) '
2
of turns in the sense coil. The emf is "d# /dt = " drive o &
)N sense*R sense .
2R
R
dt
(
drive % drive
!
amp
amp
For the triangle wave, the voltage goes from +Vdrive
to "Vdrive
in half a period, so the
amp
1 dVdrive ( t ) 2Vdrive
1
amp
derivative is
. For the parameters here, this is 17.14A/s.
=
= 4 f Vdrive
!(1/ f ) /2 R
R
dt
N drive µo
2
2!
= 0.0012T / A , and N sense"R sense
= 2000"!(0.0143m) = 1.28m 2 . The emf then is
2Rdrive
0.026V=26mV.
(0.0143 is the average radius of the sense coil).
!
!
!
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C6. Take your bar magnet and wave it around near the drive coil while the data
acquisition is running. Explain what is happening.
This is also putting a time-dependent flux through the sense coil, generating an additional
contribution to the emf. You can see this on the screen.
C7. Change the frequency of the triangle wave, and describe quantitatively the change
in emf of the small coil.
When the frequency doubles, so does the derivative, and hence the emf is proportional to the
frequency.
C8. Change the drive voltage to a 10 Hz square wave, and describe the results. Explain
these in terms of Faraday’s law.
Now the flux is now constant except for brief times when it switches from positive to negative.
This switching point is the only time there will be an induced emf, because that is the only
place there is a d" /dt . This leads to spikes in the time-dependent signal, alternating
positive and negative as the switching is from –V to +V, or +V to –V.
!
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