In the 1820s and 1830s, a guy named Michael
Faraday of England was playing around
with current. It was back when scientists
licked batteries for the fun of it. So
if I have a loop of wire here with a
power supply and a switch. So that I can
connect the switch to make the current
flow, take the switch off to make it stop.
And he had another loop of wire over
here connected to an ammeter. He would
connect the switch, and therefore a
current would start flowing through here,
and they noticed that when he connected
to the switch this ammeter, probably a
galvanometer since they tend to be a
little bit more sensitive in my
experience, that needle would flick that
a current was suddenly induced in here.
When he disconnected it on this side
over here the needle flicked in the
other direction. And I say flicked, they'll
go off, and then come back, go the other
way, and then come back. He played around
with a little bit. He sort of
made this skinnier, and it didn't get quite
as effective. He, you know, went this way -
not as effective, and far away not as
effective. What really was effective was
when it was close to here, or matter of
fact if you could actually put this wire
inside that, so I've got my circuit
outside circuit here, and then I've got
my galvanometers set up there. And that's
a hint to a lab that you do later, or
watch later. Then he realized that what
really was the effect, since this was
staying constant, what affected was how
much of the magnetic field created by
this was affecting that. And since the
sort of the area here seemed to affect
how much the needle flicked, he
eventually connected this voltage that's
induced in this wire right here. This is
a symbol for the EMF. This is an ideal
voltage. That is proportional to the
change in the magnetic flux with respect
to time, or a calculus base d Phi sub B /dt.
The next step, this was 1832, comes out
with that 1833. In 1835 there's a guy
named Emil Lenz, who is Russian, comes
out with two papers related to this, and
he actually formalizes it a little bit
more. Generally he's credited with the
negative sign, but that negative sign is
critical, or Delta, but the negative sign
up front is the key. This negative sign
is the part of the model that basically
describes why nature hasn't killed us.
Let's think about that. This minus sign
is telling me that the voltage that is
produced is opposing the change in the
magnetic flux. So if I connect the wire
here - let's look at this scenario in this
situation right here. If I connect the
wire, I suddenly create a current going
in a counterclockwise direction, so
because it's going counterclockwise it
is producing a magnetic field that is
coming out of the page inside and into
the page on the outside, which means the
magnetic field that is going through
this loop here with the ammeter,
it's going in. So there is a flux that is
increasing in the middle of this going
down. The minus sign is telling us is
that the current that is produced here
wants to oppose that change, and so when
I connect it, the magnetic flux is
suddenly getting bigger going down that
way - I guess magnetic flux doesn't have a
direction, but it wants to reduce that
change, and so it's going to create a
current that creates the magnetic field
going to the other direction. So the
current induced, this is the induced
current, not inducted, induced current
will be going counterclockwise also, so
that it produces a magnetic field that
opposes this magnetic field. In this
situation here when I connect the switch
that I have a current suddenly going in
that direction, it creates a magnetic
field coming out of the page here. Since
the magnetic field is coming out, I have -
a - my change in flux is increasing, sort
of in that direction, and
the wording is off, but the spirit is
good there. So the induced current is
going to want to oppose that. The
induced current is going to go in the
opposite direction here. Because it's
creating a magnetic field that opposes
this external one. So we have these two
competing magnetic fields. We have the
magnetic field from the outer wire here,
or the one that connected the power
supply; and then we have a magnetic field
created by the induced current in the
secondary wire. Sort of illustrating that
a bit more, I have an area here with the
magnetic field coming out of the page.
From an experimental point of view we
actually can get pretty darn close to
this. We can sort of in a sense trap the
magnetic field to a specific region. And
then I take a loop of wire here, so it's
not a square, it's just a loop of
wire and the loop of wire is going inside there.
I'm sorry it's sort of empty, it's just
four wires, or one wire shaped as a
square. As this enters, suddenly the
magnetic flux inside here will be
increasing. It wants to oppose the
increasing magnetic flux. It has
two choices: one it can change the area
of the wire, which is not going to be easy, or
it can induce a current which changes
the magnetic field. So since this
magnetic field is coming out, as this
enters, as the square enters into the
space, it is going to create a current, so
when it gets to here - it will create a
current that creates a magnetic
field going into the page, so the induced
current will be clockwise. When the
square is completely inside, there's no
change in the magnetic flux, and so the
current is zero. And then as it leaves, my
magnetic flux is suddenly decreasing, so
it's going to create a current that
wants to bolster the collapsing magnetic
flux, and so it's going to create a
current, since this is coming out it
wants to create a current going around
like that. So this will have a current
flowing in that direction.
So the induced current will try to
either bolster or hamper the magnetic
field that is created. All right, actually
the change in the magnetic flux. So if the
magnetic flux increases, it wants to
decrease it; if the magnetic flux is
decreasing, it wants to increase it.
Let's take it to the next step. If I take a bent
piece of wire here, and I'm going to make
this distance here l. I have a magnetic
field that is bathing it. The magnetic
field is coming out, so let's set up a
coordinate system i-hat, j-hat,
and k-hat coming out, and on top of
this I'm going to put a wire. This wire is
not connected, it's touching it so we can
get a complete circuit here, but the wire
is not physically connected; it's just
sitting on top. So we start out with this
distance here, which we're gonna make x,
and my magnetic flux is B dot A. My
magnetic field we'll just make B coming
out. It's a uniform magnetic field - just
make things not as complicated as they
could be, but this is B times A. Let's
make my area vector also come up and so
that's just BA which is B times lx.
Now I take this wire, and I move the wire
out at some speed v. So now my magnetic
flux is changing. It's not changing
because of a changing magnetic field
like the earlier examples; it's changing
because I have a changing area. So my
change in the magnetic flux would
be the change in BA. Now B is uniform,
and it's not changing with time, so
that stays constant. So this is just B
times the change in the area. Well area
is just the length times x. So this is B
times the change in lx, but the length
isn't changing. The only thing is
changing is this distance here. So this
becomes Bl times the change in x.
So the change in the magnetic flux over
time is equal to Bl delta x over delta
t, and that change in distance over
change of time is just the speed. So this
is the induced EMF. I'm not worrying
about sign right now, and this is
actually assuming almost every one of
you has seen "Princess Bride", and if you
haven't seen it, then that is a homework
assignment for you. That you need to
watch "Princess Bride" before the end of
the summer. It is the perfect movie, but
when Wesley is being treated by the
Billy Crystal character, and he pumps air
into the lungs,
what does Wesley say? What does Billy
Crystal think he says? Billy Crystal
thinks he says blev, the blev. Princess
Bride reference right there. So that is
the induced EMF. That is the induced
voltage, so we can now sort of make some
predictions here. Now in terms of the
direction of the current, that I have my
magnetic flux is increasing, because I'm
moving this away, so it's going to induce
a current that wants to oppose that. If
my current runs counterclockwise, along
this circuit here, then it's going to
increase the magnetic field in the
middle, which we don't want, so the the
current is going to be running clockwise,
because that'll create a magnetic field
going into the page which is going to
slow down the increase in the magnetic
flux. If I have a resistor here, now let's
actually do some numbers here, let's make
this 0.5 meters, let's make the initial
distance here this is an irrelevant number,
but make this 7 meters, make this 2 meters
per second, make the magnetic field - we'll
make it huge, 3 Tesla, and that's all the
information we need. And I have a
resistor right here and we'll make it a
10, no, let's make it a 9 ohm resistor.
So I know that the induced voltage is
going to be 3 Tesla times 0.5 meters
times 2 meters per second, so that
becomes 3 Tesla meters squared per
second, which obviously is a volt. Now how
do I know that? Because I know this is
voltage, and the equation has to work, the
units have to match. So that's 3 volts.
That's going to be equal to the current
times the resistance, which is 9 ohms.
This is IR, and so the current that's
induced is 1/3 amperes, and it is going
clockwise, counter clockwise would be CCW.
Recognize that A there is amperes;
that A right there is area. Now it works
in just, in more than just this case. So
if I have a metal bar here, of length 4
meters, and I drop it. Let's make the
magnetic field somewhat realistic. Let's
make that 0.4 meters to make it
realistic. This is about 0.5 times 10 to
the negative 4th Tesla, which is about on
par with Earth's magnetic field, and
let's say that it's coming out at this
(book) limit. When I first drop it, it
doesn't have any speed, when I first let
go. But it does pick up speed, so let's
assume it's going at some point, 5 meters
per second, then the induced EMF is going
to be 0.5 times 10 to the negative
fourth times, 0.4, times 5. That's 2,
so we end up with 0.1 millivolts. Now
which way will the current flow, or which
is the higher. Well if I've got a this
falling, so think about it this way. I
have a - I have positive charges falling
going down, and the magnetic field is
coming up, so V cross B
means the positive charges will flow
this way, but we know positive charges
aren't normally flowing, so actually
electrons will be flowing to the right,
which means that the higher potential is
on the right, and the lower potential is
on the left, because positive charge
flows from high potential to low
potential, negative charge flows from low
potential to high potential.
