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Faraday's Law
Electromagnetic Induction: Faraday's Law
About this lab
The studies of Oersted, Ampere, Biot, Savart and others
established the existence and description of a connection between static magnetism and
moving electric charge. Faraday's discovery of a dynamic connection, showing that
changing magnetic fields can generate electric fields, tied together inextricably the two
fields. Maxwell followed by postulating the complementary effect of changing electric
fields generating magnetic fields and, in so doing, elucidated the nature of light.
These discoveries are the bases of our most flexible and important technology.
References: Physics, Cutnell and Johnson, 6th Ed., Chapter 22 (Wiley, 2001)
Physics: Serway and Beichner, 5th Ed.,v 2, Chapter 31 (Saunders, 2000)
Apparatus: Helmholtz and pickup coils apparatus with sound card interface board, PC
w/ FFTScope 1 program and line-in Play Control, magnet, cables
Note: Drive output to speakers is muted. Therefore, lacking audible
feedback, it is possible inadvertently to damage the speakers if the volume
control is left turned up high for long periods. Turn down the speaker
volume control when not in use.
Introduction
The mathematical law that relates the changing magnetic field to the induced current is
(or, more accurately, the induced voltage or emf) called Faraday's Law, named after
Michael Faraday 2 who first observed it in the laboratory around 1830.
Magnetic flux through a surface in a magnetic field B is
 dA

∯ B.
1 Lu Hsu-Chang, Department of Physics and Astronomy, Rutgers University
2 http://www.iee.org/TheIEE/Research/Archives/faraday1.cfm
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Faraday's Law
where dA is the magnitude of a surface element and its direction is perpendicular, and the
“dot” is a standard notation indicating a cosine factor involving the angle between the two
vectors..
If B is constant and the surface is a plane with area A, this reduces to
=B A cos 
A conducting loop which has an ammeter attached to it will register a current if the
magnetic flux through the loop changes in time. The change may arise from motion:
Figure 1
The fundamental magnetic source – a dipole. If cut, two magnetic
monopoles do not appear but, rather, two dipoles. Corresponding to the absence of
magnetic monopoles, the magnetic field lines close.
Or the change in flux may be due to increasing current in a circuit. In (a) below there is
no induced emf in loop 2. But when the battery is connected, the increasing current in
loop 1 produces a changing magnetic field and hence induces an emf in loop 2.
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Faraday's Law
Figure 2
Closing the switch in circuit 1 produces changing current I 1, and
corresponding changing magnetic field (concentrated by the ferromagnetic coupling
bar). The changing magnetic field within the loops of the right hand circuit induces
a current I2. When the current in I1 becomes constant, there is still a magnetic field
within loops of circuit 2, but not a changing field. Correspondingly, there is no
longer an induced current I2.
The action is reciprocal – the changes in I2 affect current I1 – the two circuits are
coupled. The magnetic flux (integral of field strength over the loop) is additive, loop
by loop – i.e., proportional to the number of loops N1 or N2.
Faraday noted that the emf (electromotive force) induced in a loop is proportional to the
rate of change of magnetic flux though it:
E i =−N pickup
d
, for our case of N turns, where φ is the flux through a single turn.
dt
Notice the negative sign. Lenz's Law states that the induced emf will be in a direction
such that any induced magnetic current would itself produce a magnetic field opposing
(reducing) the original magnetic flux change.
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Faraday's Law
Figure 3
Faraday study apparatus. Application of changing voltage to the two
Helmholtz coils on the outside of the frame (black wrapping) generates a changing
magnetic field at the small signal pickup coil near the center. Alternatively, moving
a permanent magnet in the vicinity of the pickup coil can also provide a changing
magnetic flux and induce a Faraday current in it (motional EMF).
Examine the apparatus picture. There are three sets of coils, two of which are fixed with
respect to each other (wrapped in black tape) and which share the same number of turns
and diameter. These two coils (total number of turns N=20) act in concert with each
other, each producing a magnetic field in the same direction. These outer coils produce a
fairly uniform magnetic field inside the apparatus, near the center. They are similar in
geometry to the larger Helmholtz Coils in the E&M Forces lab, which generate a field
that remains approximately constant throughout the path of the electron beam.
The third inner coil (total number of turns N=11) is more rectangular than circular,
smaller than the other two, and can be rotated with respect to them using the knob on the
side of the apparatus. This is the signal “pickup” coil. There are protractor markings
around the knob to measure the angle between inner and outer coils. The FFTScope
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Faraday's Law
function generator, driving the outer coils with a sinusoidal waveform, will produce the
changing (sinusoidal) flux, inducing a current (also sinusoidal) in the inner coil. Current
can also be induced manually by moving a small magnet close to the inner coil. The
d
equations above imply (for sine drive) that the pickup voltage −
will also be
dt
sinusoidal in form, will be 90 0 out of time phase with the drive voltage and will be
proportional to drive frequency and to drive amplitude. You will test these predictions.
Note, in particular, the twisted-pair lead wires to the Helmholtz coil. These are partially
twisted together to for management but, also, so that the twisting will minimize unwanted
signal pickup by a) minimizing the area enclosed by the lead pair, and b) reversing many
times the orientations of the twist-loops. This procedure to avoid unwanted signals or
voltages may have application in hospital and other instrumentation.
The magnetic field of the outer drive coils (Helmholtz) is loosely air coupled to the small
inner coil (small fraction of outer coils field passes through the inner coil). The picture
below shows a closely coupled step-up transformer arrangement, with ferromagnetic
field coupling.
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Faraday's Law
Figure 4
Transformer action, shown in step-up winding configuration.
(N 2 > N 1). If all the field generated by the primary passes inside the secondary
winding and there is no energy loss, the turns ratio should give the voltage ratio:
(987/141) . The ferromagnetic core concentrates the magnetic field lines. See Figure
2 discussion.
Procedure
Setup
Usually, this will already be done for you. The description is below is provided for
completeness and for checking if FFTScope does not act as expected 3. Remember that
3 There are some internal computer settings. (Only TA's have access.) Double click on the small speaker
icon at lower right. In Play Control window, select Options. In Properties, select Recording. (Line in and
microphone should both be selected). OK. Then, in Play Control, select line in.
The Faraday circuits are connected to a standard interface board, which connects in turn to the Labtec Spin
70 speaker, and thence to the PC sound card. The interface board has five banana jacks, in two sections:
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Faraday's Law
FFTScope may malfunction if two windows are open simultaneously.
NOTE: DO NOT DISCONNECT OR CHANGE THE APPARATUS WIRING!
1. Qualitative Study of Faraday's Law - Motional EMF
READ (three jacks: G(nd), L(eft) and R(ight)), and DRIVE ( two jacks: Signal and G(nd)).
Looking down on the Faraday plastic frame, there are five banana jacks, grouped into a functional doublet
and a triplet.
Two (easily identified as the doublet) go the the small, rotatable pickup coil. (These will be to the left or
the right, depending on which frame face is down.) The black pickup-coil jack connects to Read Gnd, the
other to Read L(eft). These read the induced pickup coil voltage.
The other three frame jacks (triplet) involve the Helmholtz coils current. The center of the set of five frame
jacks connects to the board Drive Signal jack. The outside triplet frame jack connects to the Drive Gnd.
And the center of the three frame jack triplet connects to the board Read R(ight) jack.
From the board to the Labtec Spin 70 speaker: The twisted pair from the board goes to the speaker
earphone input; the single wire goes to the mic input.
From the Labtec speaker to the Sound Card connections at rear of computer: Green pin to green sound card
socket, pink pin to blue sound card socket. Line in.
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Faraday's Law
Figure 5
Motional EMF. The magnetic field lines run from north (+) to south
(-) pole. The meter is registering a negative current. The loop is moving to the
right. In the applet, the loop moves. This is equivalent to the magnet moving
oppositely, as in your experiment.
One side of a small refrigerator magnet corresponds to magnetic North, the other to
South. See the previous picture of the bar magnet and (incomplete) associated magnetic
field lines. In contrast to the electrical situation, where radial E field lines can begin or
end on + and – isolated electrical charge sources (electric monopoles), magnetic field
lines close because there are no magnetic source monopoles (in our epoch of the
universe, at least) - a dipole magnetic pattern is the simplest available. Field lines crowd
together when the field is strong, and are well separated where the field is weaker. And,
in contrast to the simpler electrical case, the magnetic force on a moving charge is not in
the direction of the magnetic field, but perpendicular to it, and to the direction of motion..
Nevertheless, the magnetic field line description is the most economical and useful.
Moving a paper clip around a magnet will give an idea of where the field lines are close
and far from each other. (The force is between the permanent magnet (permanently
aligned microscopic magnetic domains in the magnet) and induced magnetism in the clip
(microscopic magnetism temporarily aligned by the action of the permanent magnet).
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Faraday's Law
Again study the coil frame and the three windings shown above. The inner (pickup) coil
is connected to one input of the PC sound card, the outer (drive) coil to another. Lay the
coil frame flat on the table, and turn the knob that rotates the inner coil such that the plane
of that coil, just like the two outer Helmholtz coils, is horizontal.
Figure 6
Motional EMF with small, rod mounted magnet. [Enter:Enter]*:Down,
then Up, with delay. Scope mode, Function Generator Silent, time window 2
seconds (2000 milliseconds). How would this graph change if your magnet were
mounted in reverse?
* (First Enter starts, second Enter stops after specified time window.)
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Set FFTScope Function Generator to Silent. Select Scope mode, mono. Set Time to 4
seconds. Set the pickup coil horizontal. Observe pictures below. Using a small
refrigerator magnet, quickly start and stop data acquisition (Enter, Enter) then
quickly pull the magnet up from the pickup coil. (The Off will not become effective
until completion of the sweep time.) Double click, if necessary to re-expand the x-axis
time display. Autoscale as needed. Observe polarity of pickup coil voltage induced by
magnet motion. Turn over coil frame and repeat, Does the spike polarity reverse? Now
repeat, moving magnet down toward coil. Is spike polarity opposite of immediate
previous. What polarity do you expect if you return coil frame to initial orientation and
move magnet toward coil? Try it.
In this section we have physically moved a magnet around a coil, inducing a current in it.
In order to do this we have performed work on the magnet because we had to oppose a
force to keep the magnet moving, even at constant velocity. (Relative to the work done
against the gravitational force, this magnetic work is not perceptible.) What is the source
of this force? According to Lenz's Law, the induced current produces an induced
magnetic field, which is represented by a magnetic moment (denoted by m on the
diagram below, which opposes the motion of the magnet. This magnetic moment can be
thought of as "virtual magnet" whose poles either oppose or attract the real magnet,
depending on the direction of motion. In either case, the direction of induced current can
be identified using your right hand: point your extended thumb in the North direction of
the magnetic moment; your fingers will curl in the direction the current is going through
the loop.
Figure 7
Another example of motional EMF. The magnetic flux through the loop
changes as the bar magnet moves, and a current is induced in the sense prescribed
by Lenz Law.
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Faraday's Law
If there were no conducting loop, there would still be an EMF generated.
Suppose there is no pickup coil to carry an induced current. An EMF is still induced
around any path through which the magnetic flux changes, but there is then no Lenz' Law
work done. So we interpret Lenz law in terms of the direction induced current would
flow to reduce the driving flux change
Doing work on a magnet to create a current is the basis behind an electrical generator.
In hydroelectric dams, falling water turns large paddles connected to electrical generators
that convert mechanical energy to electrical energy. In a steam generator steam turns the
turbine blades. Conversely, we can use changing flux to do work. This is the principle
behind an electric motor.
2. Qualitative Study of Faraday's Law – Phase Relations; Vector Nature of B
A sinusoidal current in the Faraday coils will produce a sinusoidally changing magnetic
field in the pickup coil region. In turn, this will induce (magnetic induction) a sinusoidal
EMF in the pickup circuit, with corresponding current of magnitude depending on the
pickup circuit impedance. No motion of coils is involved – a Faraday EMF can be
produced however the change in magnetic field is produced. The FFTScope function
generator will provide the outer drive coils with a sinusoidally fluctuating current to vary
the magnetic drive field, and hence the magnetic flux through the inner coil.
(And this pickup current will produce its own magnetic field acting back to induce an
EMF in the Helmholtz coil circuit, etc., etc. There are not really two independent coil
circuits (Helmholtz and pickup), but a single coupled circuit, which can be so analyzed.
However, the currents induced into the pickup circuit will be relatively small, and we will
treat the two circuits as independent, neglecting all but the primary interaction.)
d
a sine drive function will result in a cosine pickup voltage: the
dt
two wave forms should be 90o out of phase. When the sine is zero and increasing, the
cosine should be max and decreasing, but the – sign means the cosine should be - |max|
and increasing. Further, the relative phase should reverse when the pickup coil
orientation is inverted by 180 0 , relative to the Helmholtz drive coils .
Since E = −
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And, because
Faraday's Law
d
{ Asin (  t ) } =  A cos (  t ) ,
dt
induced pickup voltage will increase linearly with drive frequency f ( f = ω / 2 π )
and also with amplitude A.
Observe the frequency and phase relations between drive and pickup sines with
FFTScope.
Figure 8
Phase relation between the Helmholtz drive currents and the induced
pickup signal.
Oscilloscope, stereo; sine drive at 2100 Hz. Red leads blue by 90o in time(¼ cycle) at
180o coil space orientation (for this coil). Phase flips suddenly by 180o at 90o space
orientation, (blue leads red) as pickup coil is rotated. (Red-blue lead/lag may be
reversed, depending on initial pickup coil orientation.)
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Faraday's Law
a) Pickup orientation 0 0 . Scope mode , sine function generator drive , f = 2000, mono,
0.1 second, stereo. Enter:Enter. Box and expand; autoscale. Estimate phase from L – R
offset, relative to period. Remember whether red precedes blue, or vice versa. Record.
Rotate pickup orientation slowly to 180 o , observing amplitude and phase variation.
Does phase change? When? Record phase at 180 o pickup orientation. What is phase
change from the 0o coil orientation? (The amplitude changes show that relative
directions of pickup coil and B field matter – B is a vector. The phase flip at 90
degrees shows this also – the field lines suddenly enter the other side of the pickup
coil.)
Figure 9
Expanded frequency region showing FFT stereo record of relative
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Faraday's Law
phase (blue) and amplitude (red) variation vs. Helmholtz coil drive frequency, with
white noise (all frequencies) excitation. Pickup coil orientation 180 degrees.
Phase flips suddenly by 180o as pickup coil orientation vector passes 90o (cosine
factor goes from + to -). Phase change between 0 and 180 0 coil orientations is 180 0 ,
at any particular frequency. (Ignore small phase slope.)
b) Pickup orientation 0 0 . FFT, averaging on, white noise (all frequencies within sound
card capability), 1 second, stereo. Record phase , (select phase-stable frequency range),
at 0o coil orientation. Scale amplitude – note approximate linearity up to 6 kHz or so.
Note absence of any resonant RLC amplitude peaks.
Rotate pickup coil to 180 o, observing phase along the way. Does the phase change
smoothly or suddenly with orientation? Describe phase vs. orientation. Record phase at
180 o pickup orientation.
Record difference in phase between 0 and 180 o orientations at a particular frequency.
( 0 and 180 o pickup orientations may not produce exactly +- 90 o phases, but should be
close.)
c) Scope mode. Swept-sine drive, f = 4000 Hz (1/3–5/3 = > 700–3500 Hz valid
frequency range), 2 – 4 seconds, stereo. Observe variation in relative L/R amplitudes
d
with frequency (
Faraday effect).
dt
( The oscilloscope display is not static – which is the point. Therefore it is not feasible to
provide a picture of the expected behavior. But the FFT display below also shows the
linear increase of pickup coil voltage with swept drive frequency, within the 1/3 to 5/3
valid frequency range, just as the white noise FFT display showed for the entire sound
card FFT frequency range. Also, the phase can be seen to flip by 180 0 if the pickup coil
is rotated past 90o, again just as with the white noise drive.
The FFT graph below for swept sine drive is thus just a redundant subset of the white
noise FFT shown above. But the oscilloscope display with swept sine drive is new and
instructive.)
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Faraday's Law
Figure 10 FFT stereo, swept sine, 3 seconds (1/3 – 5/3 of 2100 Hz frequency setting
– 700 – 3500 Hz is limit of valid frequency range). Pickup coil orientation is 180o.
Observe phase flip as pickup coil area orientation vector is turned through 90o
(similar to white noise, over more limited frequency range.)
3. Quantitative Study of Faraday's Law
Time rate of change aspects of Faraday's law: Induced pickup coil voltage
amplitude vs. drive frequency (orientation fixed)
Examine the proportionality of induced emf to time rate of B field change. Re-orient the
coils (drive and pickup) so that their planes are parallel to each other (should read 0 or
180 0).
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Faraday's Law
Scope mode. Time = 0.1, stereo, sine. Expand and auto scale. Drive waves should
never be flat-topped. Speaker volume control speaker controls drive amplitude.
Set the FFT function generator for sine, adjust drive amplitude (speaker knob). Take data
in stereo mode from 200 to 8000 Hz. (Note that VR will probably not remain constant
over this frequency range.) After each short run:
In the view menu: select: “View peak to peak amplitudes” This will show first VL,
then hit OK and VR will appear. But there is no phase information – in the next section,
you will have to input phase information manually.
(Our convention is that the pickup voltage VL is less than the drive voltage VR. If you
find the opposite, interchange VL and VR with the colored semicircular arrows button.)
Enter the VL1 and VR1 voltages into Vernier GA page 1. The program will calculate the
ratio. Plot VL1/VR1 vs. f . Analyze: Linear (select linear portion). Record slope for
linear region, and slope error.
Vector Aspects of Faraday's Law: Induced pickup coil voltage amplitude vs.
relative coil orientation angle (fixed drive frequency and amplitude)
Scope mode: Set for a fixed frequency near 5 kHz (the sensitivity is greater at higher
frequencies)). Set the FG for sine drive with the max amplitude w/o flat topping. Check
first that the pickup coil is parallel to the Helmholtz B-field drive coils. Call this setting
0 0, whatever the pickup pointer says.
Vary the angle of the pickup coil in 10 degree steps recording L2 voltages as data in GA
page 2. (We are only interested in the shape of the curve – the value of VR2 is not of
interest as long as it remains constant.)
The coil pointer is marked + to – 180 degrees. Choose the initial parallel
orientation as 0 and enter your data from 0 to 360 degrees.)
Copy the VL2 data into the VL (phased) column; then change signs to - for pickup
coil orientation angles between 90 and 270o . (Remember that the phase flips
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suddenly at 90 and at 270 degrees.) When you have an initial plot fit, reexamine the
90 and 270 degree points to determine whether + or – values lie closer – there may
be some angular zero errors in the pointer scale. Change if appropriate.
Theory suggests a cosine shape. Analyze: Curve Fit, sinusoid. Try the automatic fit first,
but it may not work with the default (1,1,1,1) starting parameter values.
You will probably need to manual fit initially, to find good starting parameters for the
final auto fit. Starting values could be:
> Amplitude: (Look at your curve)
> Frequency: 0.017
This is an spatial angular frequency not the temporal
frequency of the drive voltage.
(The angular period is expected to be 360 degrees, the frequency is inverse to the period
(1/360 ~ 0.0028, and the angular frequency is 2π times the frequency, or about 0.0175 ,
> Phase: π/2 radians ~ 1.57 (90 degrees, since you are fitting a presumed cosine to a
sine function)
> Offset:
0 (No reason to expect any offset)
If auto fit balks, given a reasonable starting manual fit, try turning off auto step size
(accept default) – this might keep auto from jumping completely out of a local minimum
in chi square. You can also try selecting data on the graph with the mouse, especially if
you get a nonsense message that there are insufficient points to fit.
But you might have to settle for a decent manual fit. When you have a satisfactory fit,
OK it to transfer fit and parameters to the main graph.
The sinusoidal voltages you have been measuring in this experiment have been peak to
peak voltages (twice the conventional amplitude), which vary from positive to negative.
If you averaged these over time, these would add up to zero! Instead, AC voltages are
usually expressed as root-mean-square voltages, or V rms = V max / sqrt(2) .
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Report: Print the two GA graphs showing pickup voltage response to drive amplitude
and to coil angle (phase modified data).
WARNING: GA has been known to crash! Copy, Name and Save frequently.
Some applications
High-voltage closely-coupled coil transformers are used in conveying electricity from
your electrical company to your home. Since power loss on the lines is equal to i2 R, it
makes sense to use a high voltage and low current when transporting power over great
distances. (High voltages require insulation, provided mainly by air and ceramic pole
standoffs. Nevertheless, micro discharges in damp air can produce discernible radio
noise. Smoky fires in the West have caused arc discharges which have killed cattle,
presumably initiated by small particles, with high fields at sharp points initiating
cascading electrical breakdown.)
AC transmission is done in stages. Over long distances, transmission voltage amplitudes
may be hundreds of kilovolts. Sub stations reduce this for shorter range local distribution
and still smaller transformers mounted on telephone poles reduce it further to relatively
safe house voltage. European house voltage is 50 Hz and higher than U.S., requiring
converters for portable devices designed for 60 Hz, 120 V AC. PCB's once provided
excellent transformer insulation between the two coils, but have been found to have
undesirable environmental effect.
Figure 11 A practical transducer from mechanical motion to electrical signal via
motional EMF Faraday effect.
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Faraday's Law
Michael Faraday
1791 - 1865
Michael Faraday's scientific work laid the foundations of all subsequent electrotechnology. From his experiments came devices which led directly to the modern electric
motor, generator and transformer. Faraday was also the greatest scientific lecturer of his
day, who did much to publicize the great advances of nineteenth-century science and
technology through his articles, correspondence and the Friday evening discourses which
he established at the Royal Institution. The Royal Institution Christmas lectures for
children, begun by Faraday, continue to this day.
Michael Faraday was born on 22nd September 1791. At the age of fourteen he was
apprenticed to a London bookbinder. Reading many of the books in the shop, Faraday
became fascinated by science, and wrote to Sir Humphry Davy at the Royal Institution
asking for a job. On 1st March 1813, he was appointed laboratory assistant at the Royal
Institution. There Faraday immersed himself in the study of chemistry, becoming a skilled
analytical chemist. In 1823 he discovered that chlorine could be liquefied and in 1825 he
discovered a new substance known today as benzene.
However, his greatest work was with electricity. In 1821, soon after the Danish
chemist, Oersted, discovered the phenomenon of electromagnetism, Faraday built
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Faraday's Law
two devices to produce what he called electromagnetic rotation: that is a continuous
circular motion from the circular magnetic force around a wire. Ten years later, in
1831, he began his great series of experiments in which he discovered
electromagnetic induction. These experiments form the basis of modern
electromagnetic technology.
On 29th August 1831, using his "induction ring", Faraday made one of his greatest
discoveries - electromagnetic induction: the "induction" or generation of electricity
in a wire by means of the electromagnetic effect of a current in another wire.
The induction ring was the first electric transformer. In a second series of experiments in
September he discovered magneto-electric induction: the production of a steady electric
current. To do this, Faraday attached two wires through a sliding contact to a copper disc.
By rotating the disc between the poles of a horseshoe magnet he obtained a continuous
direct current. This was the first generator.
Although neither of Faraday's devices is of practical use today they enhanced
immeasurably the theoretical understanding of electricity and magnetism. He described
these experiments in two papers presented to the Royal Society on 24th November 1831,
and 12th January 1832. These were the first and second parts of his "Experimental
researches into electricity" in which he gave his ,"law which governs the evolution of
electricity by magneto-electric induction". After reading this, a young Frenchman,
Hippolyte Pixii, constructed an electric generator that utilized the rotary motion between
magnet and coil rather than Faraday's to and fro motion in a straight line. All the
generators in power stations today are direct descendants of the machine developed by
Pixii from Faraday's first principles.
Faraday continued his electrical experiments. In 1832 he proved that the electricity
induced from a magnet, voltaic electricity produced by a battery, and static
electricity were all the same. He also did significant work in electrochemistry,
stating the First and Second Laws of Electrolysis. This laid the basis for
electrochemistry, another great modern industry.
Faraday's descriptive theory of lines of force moving between bodies with electrical
and magnetic properties enabled James Clerk Maxwell to formulate an exact
mathematical theory of the propagation of electromagnetic waves. In 1865, Maxwell
proved mathematically that electromagnetic phenomena are propagated as waves
through space with the velocity of light, thereby laying the foundation of radio
communication confirmed experimentally in 1888 by Hertz and developed for
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practical use by Guglielmo Marconi at the turn of the century.
In 1865, Faraday ended his connection with the Royal Institution after over 50 years of
service. He died at his house at Hampton Court on 25th August 1867. His discoveries
have had an incalculable effect on subsequent scientific and technical development. He
was a true pioneer of scientific discovery.
http://www.iee.org/TheIEE/Research/Archives/faraday1.cfm