Experiments in Electromagnetism
27 September 2012
Experiments in Electromagnetism
These experiments reproduce some of the classic discoveries in electromagnetism,
specifically those performed by Ampère and Faraday.
These fundamental
experiments were later generalised and famously summarised by Maxwell yielding
the four equations that are commonly known by his name.
In his derivation,
Maxwell discovered that one additional term was necessary to add to Ampère’s law
in order to ensure full consistency between these equations.
Later, when
establishing the theory of Special Relativity, Einstein showed that the equivalence of
apparently different electromagnetic phenomena were actually manifestations of the
same physical process. Therefore the experiments you will perform have not only
established electromagnetic theory, but also led to some surprising advances in
modern physics.
You may wish to review the relevant chapters of your textbook before attempting the
experiments.
Experiment 1: Parallel Plate Capacitor
A capacitor is simply a pair of conductors that stores
charge. The simplest configuration for a capacitor
consists of two metal plates separated by a thin
insulating layer. Figure 1 shows a capacitor with its
plates connected by a wire, effectively short-circuited
ensuring that the potential of both plates is the same. In
this configuration free (electrons) and fixed (static
positively charged metal atoms) charges are uniformly
distributed throughout the circuit. Now if we add a
battery supplying a voltage V, electrons are pulled from
the positive terminal and pushed to the negative
terminal. The charge distribution in the circuit now looks
like the lower part of the diagram with an excess
positive charge on one plate and excess negative
charge on the other. This separation of charge across
the insulating layer sets up an electric field in a direction
opposite to that imposed by the battery trying to drive
current around the circuit. When the two are equal there
is no further flow of electrons. However the separation
of charge represents stored electrical energy. If we
removed the battery and replaced it with a short circuit
electrons would flow around the circuit again (albeit in
the opposite direction). This method of storing electrical
energy is used in many applications, for example in a
camera flash.
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Figure 1: Diagram showing charge
distribution on an uncharged and
charged capacitor.
Experiments in Electromagnetism
27 September 2012
The quantity of charge delivered to the capacitor plates is given by Q = CV where C is the
value of the capacitor measured in Farads. The farad is a very large unit, so typically a
capacitor would have values of pF or nF.
The ability of capacitors to hold charge was discovered in the
18th century when early pioneers in electromagnetism
inadvertently gave themselves electric shocks by touching a
conductor that had been previously charged. In this lab
experiment we will develop a quantitative and painless method
for measuring capacitance and then investigate the properties
of a capacitor.
If a charged capacitor is suddenly connected to a resistor,
current will flow from one plate to another through resistor R.
Apparatus for performing this is shown in figure 2.
Figure 2: Diagram showing
apparatus for charging and
discharging a capacitor.
The current that flows through resistor R, will depend upon the voltage across the capacitor
V. Recalling Ohm’s law (V=IR) and the definition of capacitance above, we obtain:
Recalling that current is the flow of charge in time we obtain the differential equation
which if the capacitor has an initial charge Q0, has the solution
(1)
Therefore, we can expect an exponential discharge from the capacitor with a time constant
of 1/RC. Since Q=CV and we can measure voltage against time accurately with the
oscilloscope, this will form the basis for measuring capacitance.
Ultimately, the aim in this experiment is to determine the capacitance of the large parallel
plate capacitor sitting on your desk, at different plate separations. By plotting a graph of
capacitance against separation, it will be possible to estimate the universal constant ε0.
Experiment 1a: Measuring the capacitance of a known capacitor:
To develop your method for measuring capacitance, you will first measure the capacitance of
a known ceramic capacitor, commonly used in electronic circuits. Rather than switching
between a battery and resistor, as suggested in figure 2, you will use the function generator
to perform this task. Use a 10V peak-peak square wave, and set the Offset dial so that the
waveform starts at 0V and rises to +10V (rather than the default +5V to -5V). Set the
frequency to approximately 5kHz and check the waveform using the oscilloscope. Your
waveform should resemble that shown in figure 3a.
Now construct the circuit shown in figure 3b using the 470pF capacitor provided, and a
resistor of approximately 100kΩ. Take care to ensure that the oscilloscope is connected with
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Experiments in Electromagnetism
27 September 2012
the correct polarity. The negative terminals of the signal generator and oscilloscope are
grounded, so must be connected with a common ground, as shown. On the oscilloscope you
should observe an exponential rise and decay of the
voltage across the capacitor. You may wish to use the
second channel on the oscilloscope to superimpose the
function generator signal over that measured across the
capacitor 1. What effect does the frequency have on the
trace that you observe?
Try switching the resistor and capacitor, so that the
oscilloscope now measures the voltage across the
resistor. Why is this trace different?
There are two methods that you can attempt for
measuring capacitance. The first involves measuring the
voltage across the capacitor and then using equation 1,
which re-arranged and with appropriate substitution
yields:
✓ ◆
ln
V
V0
=
t
RC
Taking measurements of V at various times t and
measuring the peak voltage V0, will enable you to plot a
straight line graph of ln(V/V0) vs t, which should have
gradient -1/RC. Since you know the value of R, you can
determine the value of C.
Figure 3: (a) Pulse shape for
charging and discharging the
capacitor. (b) Circuit diagram
showing the measurement of
voltage across the capacitor.
Alternatively, you can measure the voltage across the resistor and use this to determine the
total charge that flowed onto the capacitor. Since the peak charge Q0 will correspond to the
peak voltage V0 across the capacitor, the capacitance can be determined from:
Q0 = CV0
where to determine Q0 it is necessary to take a series of current readings at various time
intervals and integrate to give the maximum charge.
Q0 =
Z
T
Idt
0
You will need to take a number of datapoints to obtain an accurate integral and you can
choose the integration method, either trapezium rule or the simple graphical method of
counting squares on standard graph paper. The accuracy of your value for Q0 and hence C
will depend upon the number of datapoints you take and the integration method you use.
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27 September 2012
Experiment 1b: Properties of the Parallel Plate Capacitor:
Having established a method for measuring capacitance, you can now replace the ceramic
capacitor with the parallel plate capacitor on the bench. From theory, we expect the
capacitance to scale inversely with plate separation, it being related by the expression
below.
C=
k✏0 A
d
where k is the dielectric constant, d is the plate separation, A is the plate area and ε0 is the
permittivity of free space.
Make a couple of measurements of capacitance at different plate separations and make a
graph of capacitance against 1/d to determine the extent to which this expression holds.
The dielectric constant of air is unity, so by making an estimate for the plate area, obtain a
value for ε0 and compare it with the accepted value of 8.854x10-12 F.m-1
Experiment 1c (Extension) - Properties of dielectrics:
On the lab bench, you will find some sheets of material. These have been chosen since
they have different dielectric constants. Referring back to figure 1, we learned that charge
will accumulate on the plates of a capacitor when an external voltage is applied. Placing a
material between these plates can polarise the charge on this material which in turn attracts
further charge to accumulate on the capacitor plates. The degree to which this occurs
depends upon the charge density of the material and leads to a value for the dielectric
constant > 1.
Determine the dielectric constant of some of the materials on the bench and see if they
correlate with known values.
You should be aware that the dielectric behaviour of most materials only remains constant
over specific frequency ranges. When driven by a continuously varying AC field, the charge
will be continuously fluctuating and there will be certain frequencies where this frequency
resonates with electronic transitions in the material. At these points the dielectric “constant”
will vary abruptly, but away from these resonant frequencies, the dielectric constant will be
largely frequency independent. You should bear in mind however that dielectric constants
may be quoted at optical frequencies (1014Hz) and potentially quite different to those that
you measure in your low frequency experiment.
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Experiment 2: Ampère’s Law
The magnetic field associated with a flow of charge is described very generally by Ampère’s
law. In your electromagnetic lectures, you will consider the general form of Ampère’s law
and apply it to specific geometries. In this experiment we are only concerned with the
magnetic field associated with electrical current travelling along a straight wire. In this case
Ampère’s law can be stated as:
B=
µ0 I
2⇡r
where B is the magnetic field strength, I is the current, r is the radial distance from the wire
and µ0=1.257x10-6 H.m-1 is the permeability of free space . The field pattern and geometry is
illustrated above.
Experiment 2a: Measuring the magnetic field associated with current flow.
You will start by repeating work performed in 1820 by Ørsted who was among the first to
notice that a compass needle was deflected by the passage of current along an adjacent
wire.
You will use the 2A power supply as a current source. Since the power supply can control
either voltage or current, you should set the voltage at open-circuit to a large value, say 10V.
This will ensure that the power supply limits the current flow, without restriction on the
voltage necessary to achieve the specified current.
Next, suspend a length of wire between the two retort stands provided. Place the compass
on a stage below the wire enabling the B field produced by the wire to be determined from
the deflection of the compass needle. You now want to position the compass so that it sits
just below the wire (a couple of millimetres at most). This can be done using the plastic
spacers to raise the compass above the stage. To achieve maximum deflection, you will
want to arrange the experiment so that the magnetic field due to the wire (BF) is
perpendicular to the Earth’s magnetic field (BE.); the deflection angle can then be related to
the relative magnetic field strength of the wire to the Earth’s magnetic field. Once you’ve
decided on the direction, pull the wire taut, then wrap it round the clamp stand and tape it
down. Ensure the wire is as straight and with as few kinks as possible. Having excess wire
at each end is not a problem, so long as it is kept away from the point of measurement.
Connect the wire through the multimeter via the 10A socket to the power supply. Turn on the
power supply and set the current to maximum. You should observe a deflection of the
compass needle. It may be necessary to gently tap the compass to encourage the needle to
move. Experiment with your technique to find a method that yields consistent results. Turn
the power supply off when you are satisfied that you can observe the effect.
You now want to make some careful measurements of the deflection of the compass needle.
Make a note of the initial angle of the compass needle and estimate the separation between
the needle and the wire. Vary the current passing through the wire, ensuring to use the
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maximum range available and record the deflected compass needle angle. For each
measurement, tap the compass first to force the needle to move, as sometimes it gets stuck,
and then measure the new needle position. Plot a graph of Tan(displacement angle) against
current. If Ampère’s law is obeyed you should obtain a straight line, the gradient of which
will enable you to determine the Earth’s magnetic field BE (with associated error).
Experiment 2b: Measuring the magnetic pattern from the wire
Ampère’s law predicts that the magnetic field strength will scale with the inverse of the radial
distance from the wire. You can measure this by using the same apparatus as above, but
setting the current to maximum (2A) and varying the separation between the compass and
the wire by removing the plastic spacers. If Ampère’s law holds, a graph of magnetic field
strength vs 1/r will yield a straight line. Consider carefully the measurement intervals for r
since you will want roughly equal spacing of datapoints on a graph of field strength against
1/r. You can again estimate the value of BE and compare with your earlier value.
Experiment 2c (Extension) - Magnetic Field of Coils:
The magnetic field associated with a single straight wire is relatively weak, but can be
enhanced by winding the wire into a coil; each coil effectively adding to the field strength of
the last. Make a simple coil by winding wire around a pen or pencil and investigate the field
strength and pattern by placing the compass at different locations around the coil. From
your observations, try to establish a relationship between the number of coils and the
magnetic field strength.
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Experiments in Electromagnetism
27 September 2012
Experiment 3: Faraday’s Law
In 1831 Faraday performed what looks like a very simple experiment
with a coil and bar magnet shown in figure 5. Faraday observed that
a changing magnetic field induces an electric field, or in the case of
this experiment, the motion of a permanent bar magnet results in a
voltage appearing across the ends of a coil.
We can express
Faraday’s findings in the simple expression:
"=
d
dt
where ε is the induced electromotive force (voltage) and dΦ/dt is the
rate of change of magnetic flux through a single loop.
Figure 5. Schematic
diagram showing
Faraday’s classic
experiment
The consequences of Faraday’s experiments are profound, in
particular the equivalence of moving either the magnet and coil which ultimately led Einstein
to the foundation of special relativity some 74 years later2. Einstein’s work goes beyond
what we can reasonably cover in a lab experiment but the experiment you are about to
perform puzzled some of the finest minds of the 19th century and ultimately led to one of the
landmark achievements of modern physics.
Experiment 3a: Repeating Faraday’s experiment
You will first reproduce Faraday’s experiment shown in figure 5 using a coil, your
oscilloscope and a permanent magnet. On the bench you will find two coils, a large 100 turn
coil and a smaller 50 turn coil. Connect the smaller coil to the oscilloscope and try moving a
permanent magnet through the coil. The time base setting is unimportant since you are
simply observing the effect of the non-periodic motion of the magnet on the current in the
coil. Observe the effect of moving the magnet faster or slower through the coil and does the
orientation of the magnet matter?
Experiment 3b: Electromagnet and Coil:
To make a more controlled experiment it is convenient to apply
your findings of Ampère’s law to this experiment. We will replace
the permanent magnet with an electromagnet and check that the
relative motion of these two coils also obeys Faraday’s law.
Connect the larger coil to the power supply as shown in figure 6
and pass 2A of current though it. Try moving the two coils
relative to each other and check that Faraday’s law still applies.
Figure 6: concentric coil
a rra n g e me n t wit h t h e
primary coil connected to
the PSU, the secondary
coil connected to the
oscilloscope.
Next, with the coils arranged as shown in figure 6, observe what
happens when you turn the power supply off. The current in one
of the coils will drop abruptly and induce a large voltage momentarily in the second. You
may be able to record a peak voltage on the oscilloscope but to obtain quantitative results, a
more controlled means of controlling dΦ/dt is required.
2
Introduction to Electrodynamics, D.J.Griffiths, 3rd Ed, Addison Wesley (1999).
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The function generation provides a convenient means for
controlling dΦ/dt. Since we need to pass an appreciable
current through the large, primary coil, the output of the
function generator is fed into an amplifier and then passed
into the coil via a 10Ω resistor3. Set up the circuit shown in
figure 7 and configure the function generator to ~5kHz,
square wave, zero offset and about 1v pk-pk amplitude. You
can use the oscilloscope to measure the output from the
Figure 7: concentric coil
amplifier, the magnetic field produced by the primary coil and
arrangement with the primary
the induced voltage in the secondary coil.
coil driven using a periodic
With the square wave, you should observe something similar signal supplied by the function
generator.
to switching the power supply on and off, since dΦ/dt is zero
except when there are abrupt changes in primary coil
current. The principle advantage of this arrangement is that the signal is now periodic and
more easily measured on the oscilloscope screen.
Switch to the triangle waveform and observe the waveform appearing across the secondary
coil. Does the induced voltage that you measure correspond to what you would expect from
Faraday’s law? What is the effect of using a sine wave, does the shape of the waveform
change?
Experiment 3c: (Extension) - Investigating different magnetic cores
When measuring capacitance with the parallel plate capacitor, you observed that different
dielectric materials can increase the ability of the capacitor to store charge. Similarly
inserting a core into our coils, we can concentrate the magnetic field in the core. This arises
since the magnetic field produced by the primary coil, serves to align the internal magnetic
moment of electrons in the core material with the applied field. When the electrons align
parallel to the field, the effect is called paramagnetism, when in opposition to the applied
field the effect is called diamagnetism. Some materials exhibit persistent magnetisation
parallel to the applied field and are called ferromagnetic.
Inserting a core into the coil with a different magnetic material has the result that the
magnetic field is concentrated in the core of the material. Inside the core (and therefore in
its vicinity) the permeability in Ampère’s law is no longer that of free-space but modified by
the material to µ =Km µ0 The effect is small for paramagnetic and diamagnetic materials4,
but extremely large for some ferromagnetic materials where Km can exceed 3000. It is
convenient to characterise materials using the magnetic susceptibility Χm=1-Km, some typical
values are given in the table below.
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On the bench you will find several different cores that you can insert into the smaller coil.
Investigate the effect of inserting these different cores into your two coils. Do they behave
as you would expect? If not, can you find an explanation for their behaviour?
Returning to the method used to investigate Ampère’s law, you can use the deflection of the
compass needle as a means of assessing the magnetic field strength due to a coil wound
around an iron core. A large nail is provided for this purpose. Try to estimate the magnetic
field strength of your electromagnet with and without the core.
Material
Type
Susceptibility
Uranium
Paramagnetic
4.0 x10-4
Aluminium
Paramagnetic
2.1x10-5
Carbon
Diamagnetic
-2.1 x10-5
Copper
Diamagnetic
-9.7 x10-6
Iron
Ferromagnetic
3000
Nickel-Zinc Ferrite
Ferromagnetic
20-15,000
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