WEBVTT
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Prof: Okay,
I left you guys thinking about
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inductors last time,
so I should start there.
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An inductor is very different
from the resistor when you start
00:17.539 --> 00:22.359
doing circuit theory,
because when you have a
00:22.363 --> 00:28.473
resistor and you connect some
voltage to it,
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the current is determined by
this equation.
00:35.330 --> 00:37.310
This is what I call an
algebraic equation.
00:37.310 --> 00:40.070
It's an equation in algebra,
one unknown.
00:40.070 --> 00:45.190
You solve for it by dividing
and you get I at time
00:45.187 --> 00:50.667
t is V at time
t divided by R.
00:50.670 --> 00:53.830
And you can make the network
more complicated,
00:53.829 --> 00:57.059
put a few more resistors,
put a few more there,
00:57.059 --> 01:00.429
a few more here,
resistors within resistors.
01:00.429 --> 01:02.849
It doesn't matter,
because we can always in the
01:02.853 --> 01:06.123
end combine the ones in series
then lump them with the stuff in
01:06.122 --> 01:08.792
parallel,
till the whole thing to the
01:08.786 --> 01:12.206
right of this is one single
effective resistor.
01:12.209 --> 01:15.299
Then you find the current
coming out of the sources as
01:15.295 --> 01:17.095
V divided by that guy.
01:17.099 --> 01:20.319
Then every time it comes to a
branch, we sort of know how to
01:20.323 --> 01:20.983
divide it.
01:20.980 --> 01:23.880
So resistor circuits are very
easy.
01:23.879 --> 01:26.739
But now when you bring
inductors, things are different,
01:26.735 --> 01:29.745
so I'm just going to tell you a
little more about them.
01:29.750 --> 01:31.000
Here is an inductor.
01:31.000 --> 01:33.550
That's the symbol for an
inductor.
01:33.550 --> 01:39.470
It's got some inductance
L measured in henries.
01:39.470 --> 01:42.460
And the implication of that is
the following:
01:42.459 --> 01:46.059
if you have a current going
through this inductor,
01:46.060 --> 01:50.750
there is going to be
necessarily a voltage with this
01:50.745 --> 01:54.415
polarity if the current is
increasing.
01:54.420 --> 01:59.690
And the voltage you need is
LdI/dt.
01:59.690 --> 02:00.700
This is the bottom line.
02:00.700 --> 02:05.950
Even if you skip all the other
stuff about conservative forces,
02:05.954 --> 02:10.624
this, that, this is something
you need to know how to do
02:10.616 --> 02:11.716
problems.
02:11.718 --> 02:15.158
The difference you notice is
that the relation between
02:15.155 --> 02:18.265
voltage and current is not an
algebraic equation,
02:18.265 --> 02:20.335
but a differential equation.
02:20.340 --> 02:22.960
This is called a differential
equation.
02:22.960 --> 02:26.320
Now you don't have to worry,
because I will tell you how to
02:26.317 --> 02:29.727
solve it, because I don't know
if it's a prerequisite or not
02:29.732 --> 02:30.892
for this course.
02:30.889 --> 02:35.009
I once had a kid in summer
school who when I started using
02:35.006 --> 02:38.176
partial derivatives,
got very upset and said,
02:38.184 --> 02:40.644
"It's not a prerequisite.
02:40.639 --> 02:43.169
Why are you using partial
derivatives?"
02:43.169 --> 02:45.899
So I told him,
"When you come to class,
02:45.901 --> 02:49.271
there's always a danger you'll
learn something new.
02:49.270 --> 02:50.840
You just have to learn to live
with that."
02:50.840 --> 02:53.170
So I'm not going to leave you
in the cold.
02:53.169 --> 02:57.579
If I do something you've not
done, I will tell you how to do
02:57.578 --> 03:01.088
it, but I cannot avoid using
certain notions.
03:01.090 --> 03:02.990
So this one,
even though it's called a
03:02.987 --> 03:05.957
differential equation and so on,
if V is a constant,
03:05.962 --> 03:07.812
we can easily find the
integral.
03:07.810 --> 03:11.830
It will be V over
L times t.
03:11.830 --> 03:14.810
Okay, but now we're going to be
interested in cases where
03:14.810 --> 03:19.260
V is not simply constant;
it could be varying with time
03:19.258 --> 03:21.168
in an arbitrary way.
03:21.169 --> 03:23.909
So you take this guy,
the first thing you notice is
03:23.914 --> 03:26.994
that the relation between
current and voltage is given by
03:26.991 --> 03:27.871
derivatives.
03:27.870 --> 03:31.220
Second thing is,
when a current flows through a
03:31.223 --> 03:35.523
resistor, whatever energy you
provide is gone in the form of
03:35.522 --> 03:36.182
heat.
03:36.180 --> 03:37.370
It has dissipated.
03:37.370 --> 03:39.760
The light bulb glows and that's
the end.
03:39.758 --> 03:44.008
With an inductor,
when you drive a current,
03:44.008 --> 03:47.118
you are building a magnetic
field inside the inductor and
03:47.122 --> 03:50.182
there's an energy associated
with the magnetic field.
03:50.180 --> 03:55.650
And that stored energy will be
given back to you later on.
03:55.650 --> 03:58.010
So it's like a capacitor.
03:58.008 --> 04:00.148
It takes a lot of work to
charge a capacitor,
04:00.151 --> 04:02.971
because you've got to take
charges from this plate and keep
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on piling them in that plate.
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It's a lot of work,
but then if you connect that to
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a bulb and you squeeze the
trigger in your camera,
04:10.287 --> 04:12.937
this charge is giving you the
energy back.
04:12.938 --> 04:18.188
So the inductor and capacitor
are different from the resistor
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in two ways: one is,
the voltage across the
04:21.891 --> 04:26.741
capacitor is the charge on the
capacitor divided by C,
04:26.740 --> 04:29.770
which if youlike is the
integral of the current up to
04:29.766 --> 04:31.626
that time divided by C.
04:31.629 --> 04:34.679
Again, calculus comes in.
04:34.680 --> 04:36.640
So these are the two
differences, the relation
04:36.644 --> 04:39.274
between current and voltage and
voltage is differentiating or
04:39.266 --> 04:41.976
integrating,
and these are energy storing
04:41.982 --> 04:44.802
circuit elements compared to
this one,
04:44.800 --> 04:47.680
where energy is dissipated.
04:47.680 --> 04:52.340
So let's start with a simple
problem where I have a voltage
04:52.336 --> 04:53.056
source.
04:53.060 --> 04:55.470
I'm going to take a fixed
voltage, V_0.
04:55.470 --> 04:59.310
There is a switch,
there is a resistor R,
04:59.312 --> 05:02.422
and there is this inductor
L.
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I'm going to assume the current
is flowing this way,
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in which case--I'm sorry.
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You know, this picture's wrong.
05:11.310 --> 05:17.410
What's wrong with this picture?
05:17.410 --> 05:19.260
What is wrong?
05:19.259 --> 05:21.289
Student: The switch is
open.
05:21.290 --> 05:23.110
Prof: The switch is
open, okay.
05:23.110 --> 05:25.500
That's what's wrong,
so nothing is flowing yet.
05:25.500 --> 05:29.610
But when I close the switch,
then it's fine.
05:29.610 --> 05:32.340
Now it's flowing.
05:32.339 --> 05:34.329
And the question is,
what does it do?
05:34.329 --> 05:36.489
How does it flow?
05:36.490 --> 05:39.720
After all, the inductor is a
resistance free wire,
05:39.720 --> 05:43.030
so you may think current =
V_0/R will
05:43.029 --> 05:46.319
start flowing immediately,
but that's wrong,
05:46.317 --> 05:49.877
because if such a current were
flowing,
05:52.670 --> 05:53.530
current in it.
05:53.529 --> 05:56.909
And that energy cannot come
from 0 to all of that in no
05:56.911 --> 05:57.351
time.
05:57.350 --> 05:59.360
There's no way to do that.
05:59.360 --> 06:02.430
Or if you like,
if the current really grows
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rapidly or instantaneously from
0 to some value,
06:05.867 --> 06:09.667
that's a current,
dI/dt is infinite here.
06:09.670 --> 06:12.490
dI/dt is infinite,
LdI/dt is infinite.
06:12.490 --> 06:15.360
There's an infinite voltage
needed somewhere in the circuit
06:15.362 --> 06:17.792
and we don't have it,
so that would not happen.
06:17.790 --> 06:22.580
So current will have to be
continuous, start from 0 and it
06:22.584 --> 06:25.114
will have to start climbing.
06:25.110 --> 06:28.480
The minute the current starts
climbing up it starts flowing
06:28.476 --> 06:29.576
through this guy.
06:29.579 --> 06:34.289
There is the resistance times
current voltage drop across
06:34.291 --> 06:34.881
this.
06:34.879 --> 06:38.369
Therefore the amount of voltage
available to drive current
06:38.367 --> 06:42.157
through this one or to increase
the current through this one is
06:42.160 --> 06:45.460
reduced from V_0
to V_0 -
06:45.464 --> 06:46.264
RI.
06:46.259 --> 06:49.589
That = LdI/dt.
06:49.589 --> 06:52.229
So you can see,
as the current increases,
06:52.230 --> 06:54.430
the motivation to increase the
current decreases,
06:54.430 --> 06:58.670
because this resistor starts
swallowing up more and more of
06:58.672 --> 06:59.772
this voltage.
06:59.769 --> 07:03.009
So the growth in current will
be very, very slow,
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and we expect that after a very
long time, it will settle down
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to some value.
07:08.339 --> 07:09.379
What is that value?
07:09.379 --> 07:11.929
We can find that value by
saying, "Wait until the
07:11.927 --> 07:13.657
current has stopped
growing."
07:13.660 --> 07:16.540
That means dI/dt =0,
that current will be
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V_0/R
and I'm going to give it the
07:20.024 --> 07:22.354
name
I_infinity,
07:22.350 --> 07:25.960
because it will turn out that
you will reach the value only
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after infinite time.
07:27.329 --> 07:30.249
But you can get very,
very close to that by waiting
07:30.245 --> 07:33.795
some reasonable amount of time,
and part of what we want to do
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is, how long should I wait?
07:35.610 --> 07:38.470
Suppose I'm settling for 90
percent of maximum.
07:38.470 --> 07:39.480
When does that happen?
07:39.480 --> 07:42.130
Is it after 1 second or 10
seconds or 10 hours?
07:42.129 --> 07:44.089
What's going to decide that?
07:44.089 --> 07:47.279
Well, once you have the right
equations, you can figure
07:47.283 --> 07:48.293
everything out.
07:48.290 --> 07:49.820
So we're going to do that.
07:49.819 --> 07:54.279
So basically we are going to
solve the equation
07:54.279 --> 07:55.829
LdI/dt.
07:55.829 --> 07:58.559
So let me show you one more
time how it is done,
07:58.560 --> 07:59.490
this equation.
07:59.490 --> 08:01.010
You should know how to write it.
08:01.009 --> 08:04.509
Start anywhere you like,
go round this guy.
08:04.509 --> 08:07.689
You gain voltage
V_0.
08:07.689 --> 08:11.909
Then go around this guy,
you drop by an amount
08:11.913 --> 08:13.043
RI.
08:13.040 --> 08:14.600
You drop by an amount RI.
08:14.600 --> 08:17.450
And here you jump from this end
to this end.
08:17.449 --> 08:19.999
Remember, you never go into the
inductor.
08:20.000 --> 08:22.000
That's a bad zone.
08:22.000 --> 08:24.780
That's a zone where the
potential is not even defined,
08:24.778 --> 08:27.358
but I've told you multiple
times that even though there are
08:27.362 --> 08:29.502
non conservative things going on
inside this,
08:29.500 --> 08:32.450
once you come outside,
you just find a regular
08:32.452 --> 08:35.932
electrostatic potential
difference = LdI/dt.
08:35.928 --> 08:38.728
So you drop again,
and you come back to where you
08:38.734 --> 08:41.154
start,
the whole thing should add up
08:41.153 --> 08:46.053
to 0,
which means the equation is
08:46.048 --> 08:51.638
LdI/dt RI =
V_0.
08:51.639 --> 08:54.909
That's what we want to solve.
08:54.908 --> 08:57.588
So when you have this thing,
as I told you,
08:57.587 --> 09:01.347
solving differential equations
is a matter of guess work.
09:01.350 --> 09:02.830
There is no algorithm.
09:02.830 --> 09:05.980
You rely on what you've seen
before and you guess the answer.
09:05.980 --> 09:08.900
You put some free parameters,
you fiddle with them till
09:08.900 --> 09:09.930
everything works.
09:09.928 --> 09:13.408
And in this kind of problem,
we can make everything work.
09:13.409 --> 09:14.859
Some problems we cannot solve.
09:14.860 --> 09:17.820
We can write the equations but
we cannot solve them.
09:17.820 --> 09:21.090
This is something you're not
used to when you take an
09:21.091 --> 09:23.231
elementary course like this one.
09:23.230 --> 09:26.270
But there are situations,
for example,
09:26.269 --> 09:29.419
you know inside the proton
there are these quarks,
09:29.418 --> 09:32.928
and the quarks interact with
each other with entities called
09:32.932 --> 09:33.472
gluons.
09:33.470 --> 09:36.730
They're called gluons because
they glue the quarks and form
09:36.730 --> 09:38.250
the protons and neutrons.
09:38.250 --> 09:43.100
We know the equations governing
the behavior of the gluons.
09:43.100 --> 09:45.400
We cannot solve them.
09:45.399 --> 09:46.029
It's really strange.
09:46.029 --> 09:48.039
You have an equation but you
cannot solve it.
09:48.038 --> 09:50.628
It's completely possible,
because certain
09:50.634 --> 09:54.594
equations--every equation has a
solution, but it does not mean
09:54.590 --> 09:57.120
you can write it down
analytically.
09:57.120 --> 09:59.570
So quite often,
one may be on top of the right
09:59.567 --> 10:03.047
answer, but one may not know how
to solve it, so one doesn't even
10:03.048 --> 10:04.788
know if the theory is right.
10:04.788 --> 10:07.278
So suppose this was the
equation you got,
10:07.275 --> 10:09.075
but you could not solve it.
10:09.080 --> 10:10.450
It happens to be the right
equation.
10:10.450 --> 10:13.960
You don't know what the answer
looks like, you can never be
10:13.961 --> 10:16.021
sure this is the right equation.
10:16.019 --> 10:18.699
Similarly when
Mr. Schr�dinger invented his
10:18.703 --> 10:21.633
quantum mechanics,
he wrote an equation that will
10:21.634 --> 10:24.384
tell you the energy levels of
hydrogen.
10:24.379 --> 10:27.049
But luckily,
he was able to solve it.
10:27.048 --> 10:29.458
By solving it,
he was able to show that his
10:29.457 --> 10:33.127
equation gives the energy levels
iand the spectra you expect from
10:33.128 --> 10:34.388
the hydrogen atom.
10:34.389 --> 10:37.229
So you have to remember that
not every interesting equation
10:37.227 --> 10:39.827
can be solved,
and I gave you the equation for
10:39.831 --> 10:44.091
these gluons,
which we can write down but
10:44.087 --> 10:45.847
cannot solve.
10:45.850 --> 10:48.560
There is no function whose
derivative we cannot take.
10:48.558 --> 10:50.808
There are lots of functions
whose integrals we cannot do.
10:50.808 --> 10:53.868
That's simply asymmetry between
those two.
10:53.870 --> 10:58.980
Here's a very simple function:
e^(−x2)dx from 0
10:58.977 --> 11:02.177
to 19, or some fixed number,
0 to some
11:02.182 --> 11:04.522
x_max.
11:04.519 --> 11:05.649
It has a definite value.
11:05.649 --> 11:09.139
It's area under the graph,
but no one knows how to write a
11:09.144 --> 11:12.274
formula for this in terms of
x_max,
11:12.269 --> 11:14.109
where it's in closed form.
11:14.110 --> 11:18.340
But anyway, this guy we can
trample to death as follows.
11:18.340 --> 11:21.040
First we say at very,
very long times,
11:21.044 --> 11:22.954
the current,
I told you, is
11:22.946 --> 11:24.916
V_0/R.
11:24.919 --> 11:27.889
That's when dI/dt is 0.
11:27.889 --> 11:32.749
So we're going to write the
actual current in our problem as
11:32.753 --> 11:37.043
the current at infinity some
difference I twiddle.
11:37.038 --> 11:39.928
Standard thing,
you take out the part you know
11:39.928 --> 11:41.918
and solve for the rest of it.
11:41.918 --> 11:46.268
So let's take that assumed form
and put it into the differential
11:46.269 --> 11:47.029
equation.
11:47.029 --> 11:49.739
So when I do LdI/dt,
this is a number,
11:49.740 --> 11:51.280
V over R.
11:51.279 --> 11:53.319
It has no d by dt.
11:53.320 --> 11:58.690
So dI/dt becomes
dI˜/dt R
11:58.693 --> 12:02.383
times I˜ R
times
12:02.383 --> 12:08.183
I_infinity =
V_0.
12:08.179 --> 12:09.739
Now you can see why we did that.
12:09.740 --> 12:14.550
We did that because now these
two guys cancel and you get
12:14.548 --> 12:17.208
LdI/dt RI = 0.
12:17.210 --> 12:28.330
That means dI˜/dt
is −R/L times
12:28.326 --> 12:30.506
I.
12:30.509 --> 12:33.409
So we are saying,
get me a function whose time
12:33.410 --> 12:36.510
derivative looks like the
function itself up to a
12:36.506 --> 12:38.426
constant,
and that's something we know
12:38.433 --> 12:40.943
from high school,
is an exponential function.
12:40.940 --> 12:45.320
So I˜
looks like some constant,
12:45.320 --> 12:48.580
I_0e^(-Rt/L).
12:48.580 --> 12:52.210
You can take the derivative of
this guy and out will come
12:52.207 --> 12:54.797
-R/L times the function
itself.
12:54.798 --> 12:58.938
And I˜_0
is completely arbitrary.
12:58.940 --> 13:01.730
It's not determined by this
equation.
13:01.730 --> 13:03.480
There's another useful property
to know.
13:03.480 --> 13:05.450
This is a linear equation.
13:05.450 --> 13:09.130
It means if you multiply both
sides by 19, you get another
13:09.129 --> 13:12.099
solution which is 19 times the
old solution.
13:12.100 --> 13:15.320
So the solution's overall scale
is not determined by the
13:15.316 --> 13:15.956
equation.
13:15.960 --> 13:17.700
If I give you one solution,
you can multiply it by any
13:17.703 --> 13:19.403
number you like,
it's also a solution,
13:19.397 --> 13:21.327
because if you multiply by a
number,
13:21.330 --> 13:23.950
say 9, the 9 goes into the
derivative,
13:23.950 --> 13:24.970
9 comes here.
13:24.970 --> 13:27.790
You can see 9 times I˜
satisfies the same
13:27.793 --> 13:28.383
equation.
13:28.379 --> 13:30.649
So we don't know this number
just from the equation,
13:30.649 --> 13:34.379
but we go back to what we know,
which is I of t,
13:34.379 --> 13:40.009
is I_infinity,
which is V_0/R
13:40.009 --> 13:44.959
I˜_0
e^(−Rt/L).
13:44.960 --> 13:50.000
But I know that at t =
0, this guy should vanish.
13:50.000 --> 13:52.030
t = 0,
this guy should vanish,
13:52.032 --> 13:55.422
because there was no current
when I just threw the switch.
13:55.419 --> 13:58.009
So that tells me that equals 0.
13:58.009 --> 14:01.759
That tells me
I˜_0 =
14:01.759 --> 14:06.669
-V_0/R and we
get our final result,
14:06.668 --> 14:16.558
which is I of t =
V_0/R times 1 -
14:16.557 --> 14:19.537
e^(-Rt/L).
14:19.538 --> 14:22.998
So at this level,
you are really doing physics
14:22.995 --> 14:25.065
the way we like to do it.
14:25.070 --> 14:28.270
We like to study things,
write down some equations and
14:28.270 --> 14:29.600
solve the equations.
14:29.600 --> 14:34.370
Then we get a very precise
prediction on what will happen.
14:34.370 --> 14:36.990
Because now we don't have to
guess when the current will come
14:36.985 --> 14:38.595
to 90 percent of its maximum
value.
14:38.600 --> 14:41.860
We can pick any number you
like, because we can now plot
14:41.859 --> 14:42.629
this graph.
14:42.629 --> 14:46.269
And let's plot this graph,
I(t) versus t.
14:46.269 --> 14:49.629
At t = 0,
e^(-0) is 1.
14:49.629 --> 14:51.889
You get 1 - 1, you get nothing.
14:51.889 --> 14:54.799
At t = infinity,
t to the minus infinity
14:54.796 --> 14:55.176
is 0.
14:55.179 --> 14:56.479
Yes?
14:56.480 --> 14:58.840
Student: Over on the
right where it says I˜
14:58.835 --> 15:01.425
sub twiddle = negative
V_0 over R,
15:01.429 --> 15:04.209
should that be 0?
15:04.210 --> 15:05.180
Prof: This one?
15:05.179 --> 15:07.319
Yeah, right. Thank you.
15:07.320 --> 15:12.260
Okay, so now this exponential
has a full strength of 1 at time
15:12.259 --> 15:13.149
equals 0.
15:13.149 --> 15:15.319
Goes to 0 at time equals
infinity.
15:15.320 --> 15:19.490
Therefore it takes away
exponentially small stuff at
15:19.486 --> 15:21.036
very large times.
15:21.038 --> 15:24.588
And if you want to know roughly
how long should I wait,
15:24.594 --> 15:27.294
the answer is,
the time you should wait is
15:27.293 --> 15:29.403
roughly of order L/R.
15:29.399 --> 15:35.589
Because you can write this as
e^(-t)/τ,
15:35.591 --> 15:39.131
where τ is L/R.
15:39.129 --> 15:42.069
Whenever we have an
exponential, it will always be e
15:42.067 --> 15:45.747
to the minus a physical quantity
divided by some number that sets
15:45.754 --> 15:48.064
the scale for the physical
quantity.
15:48.059 --> 15:49.799
Exponentials fall exponentially.
15:49.799 --> 15:50.679
We know that.
15:50.679 --> 15:52.049
But how long should I wait?
15:52.048 --> 15:55.478
Well, if you wait t =
τ seconds,
15:55.480 --> 15:59.230
then it will be 1/e,
which is roughly 1 over third
15:59.225 --> 16:02.895
of the starting value and 1
minus that will be roughly 2
16:02.903 --> 16:04.513
thirds of its value.
16:04.509 --> 16:10.409
If you wait t = 3τ,
e^(-3) is 1 over 20,
16:10.409 --> 16:13.929
then you'll be what,
95 percent.
16:13.928 --> 16:21.348
So you can calculate everything
you need in this simple problem.
16:21.350 --> 16:23.510
So now let's come to the same
problem.
16:23.509 --> 16:28.239
The switch has been closed for
a long time and the current has
16:28.240 --> 16:29.250
stabilized.
16:29.250 --> 16:30.530
So here is the switch.
16:30.529 --> 16:32.099
It has been closed.
16:32.100 --> 16:34.650
That's my V_0,
that's R.
16:34.649 --> 16:37.419
And the current,
you remember,
16:37.417 --> 16:40.087
is V_0/R.
16:40.090 --> 16:44.580
This is L.
16:44.580 --> 16:46.960
So now we want to do something
else.
16:46.960 --> 16:54.510
Now I want to open that switch.
16:54.509 --> 17:00.429
So what do you think will
happen if you open that switch?
17:00.429 --> 17:01.459
Yes?
17:01.460 --> 17:03.100
Student: The inductor
will resist the changing
17:03.100 --> 17:03.410
current.
17:03.408 --> 17:05.008
Prof: But how can it
resist?
17:05.009 --> 17:08.789
Student: By inducing
the magnetic field.
17:08.788 --> 17:11.548
Prof: It can do what it
wants, but how does it drive a
17:11.548 --> 17:13.158
current through the open
circuit?
17:13.160 --> 17:15.320
Student: From energy
stored in the magnetic field.
17:15.318 --> 17:17.898
Prof: But in what manner
will the current flow,
17:17.898 --> 17:19.698
given that I've broken the
circuit?
17:19.700 --> 17:22.500
Student: In the
direction--
17:22.500 --> 17:24.260
Prof: No,
but how will it manage to go
17:24.259 --> 17:24.619
around?
17:24.618 --> 17:26.978
It's like removing the bridge,
right?
17:26.980 --> 17:28.180
How will it go around?
17:28.180 --> 17:30.760
I agree with everything you
said, but how will it make the
17:30.757 --> 17:31.207
circuit?
17:31.210 --> 17:32.270
Yes?
17:32.269 --> 17:33.839
Student: There'll be a
higher concentration of charges
17:33.842 --> 17:34.592
on one side than the other.
17:34.588 --> 17:37.998
Prof: Yes,
but how will you really get rid
17:38.000 --> 17:41.270
of the current in a circuit that
seems open?
17:41.269 --> 17:44.609
Have you ever done this?
17:44.609 --> 17:47.429
Okay, that's just fine.
17:47.430 --> 17:50.750
But we know it has to get rid
of that current,
17:50.751 --> 17:54.371
because otherwise,
what happened to the energy?
17:54.369 --> 17:55.989
It has to go through.
17:55.990 --> 17:58.860
So you know what,
you guys don't know what will
17:58.856 --> 18:00.786
happen if I pull that switch.
18:00.789 --> 18:01.569
Yes?
18:01.568 --> 18:04.808
Student: Is the current
an electric field?
18:04.808 --> 18:08.518
The change in the magnetic
field will create an electric
18:08.518 --> 18:11.618
field which doesn't need the
closed circuit.
18:11.618 --> 18:13.388
Prof: Right,
no I agree, it will have an
18:13.385 --> 18:14.035
electric field.
18:14.038 --> 18:18.628
But how will the electrons flow
when the wire has been
18:18.626 --> 18:21.826
interrupted, is all I'm asking
you?
18:21.829 --> 18:23.279
Yes?
18:23.278 --> 18:24.138
Student: Will the
current oscillate,
18:24.140 --> 18:24.540
go back and forth?
18:24.539 --> 18:26.669
Prof: No.
18:26.670 --> 18:27.740
Yes?
18:27.740 --> 18:28.970
Student: You will have
a static discharge.
18:28.970 --> 18:29.760
Prof: You will have a
discharge.
18:29.759 --> 18:31.879
You will have an arc.
18:31.880 --> 18:36.490
You will have a flash,
and that's when you'll hear a
18:36.487 --> 18:39.917
zip, and then it will jump this
gap.
18:39.920 --> 18:43.170
So whenever you remove the
switch, you have to be careful
18:43.166 --> 18:45.716
that there is not stored energy
somewhere.
18:45.720 --> 18:47.450
You might think,
"Hey, I am pulling the
18:47.454 --> 18:49.074
plug on--"
well, I won't say who.
18:49.068 --> 18:53.418
You're pulling the plug,
okay, and what can it do to me?
18:53.420 --> 18:56.260
Well, it can zap you,
because this is a very
18:56.259 --> 18:58.569
dangerous place to pull the
plug.
18:58.569 --> 19:00.819
So you know what people do?
19:00.819 --> 19:02.219
They have another resistor.
19:02.220 --> 19:04.540
Let's call this guy
R_1.
19:04.538 --> 19:09.458
This is R_2
and R_2 is a
19:09.457 --> 19:12.647
huge number, 10,000 ohms let's
say.
19:12.650 --> 19:15.400
So most of the time,
R_2 doesn't do
19:15.402 --> 19:17.932
anything,
because when the current comes
19:17.932 --> 19:20.782
here in any situation,
it looks at the inductor and
19:20.777 --> 19:22.447
looks at 10,000 ohms and it
says,
19:22.450 --> 19:23.570
"I'm going this way."
19:23.568 --> 19:28.348
So primarily,
it will all go there,
19:28.354 --> 19:29.344
okay?
19:29.338 --> 19:33.258
In the end, when the switch has
been closed for a very long time
19:33.262 --> 19:36.772
in our old experiment,
when the current is stabilized,
19:36.772 --> 19:38.982
you remember LdI/dt 0
here.
19:38.980 --> 19:40.950
So there's 0 volts between
these two.
19:40.950 --> 19:42.990
That's the same 0 volt across
R_2.
19:42.990 --> 19:44.740
No current will flow in
R_2.
19:44.740 --> 19:47.310
Everything will flow through
L.
19:47.308 --> 19:49.938
That current in fact will be
V_0 over
19:49.944 --> 19:52.834
R, because this guy has
been cut out of the loop.
19:52.829 --> 19:54.309
This is how it's going.
19:54.308 --> 19:59.138
But now if you throw the switch
open, you have given it a path
19:59.141 --> 20:00.331
to discharge.
20:00.328 --> 20:05.238
So the inductor will discharge
through the resistor.
20:05.240 --> 20:07.590
The current will continue to
flow in this direction.
20:07.588 --> 20:10.308
Of course, it cannot flow
forever, because the resistor
20:10.308 --> 20:13.078
will burn up the energy,
but you're giving it a path.
20:13.078 --> 20:15.448
You're avoiding--in a way,
if you like,
20:15.450 --> 20:18.820
this is also a path with very,
very high resistance.
20:18.818 --> 20:21.488
The air is a path with very
high resistance.
20:21.490 --> 20:24.190
That means by and large,
there are no free carriers in
20:24.190 --> 20:24.700
the air.
20:24.700 --> 20:27.450
But if the two tips get really
charged, just like you said,
20:27.450 --> 20:29.680
there'll be an emf that will
pile up charges.
20:29.680 --> 20:31.690
They're jumping,
they're waiting to jump the
20:31.689 --> 20:31.969
gap.
20:31.970 --> 20:35.240
Eventually they'll polarize
air, which is electrically
20:35.240 --> 20:36.970
neutral, into and - parts.
20:36.970 --> 20:39.210
Then the - will go one way,
the will go the other way.
20:39.210 --> 20:43.030
You'll recognize that as a
discharge.
20:43.029 --> 20:45.849
But you don't have to worry
about that now,
20:45.851 --> 20:49.681
because R_2
will take up your current.
20:49.680 --> 20:54.540
So now the equation we write is
LdI/dt RI = 0
20:54.542 --> 20:58.992
because if you start anywhere
and you go around a loop,
20:58.993 --> 21:01.883
you have no change in anything.
21:01.880 --> 21:05.010
There's no voltage.
21:05.009 --> 21:09.409
And the answer to this one,
R_2,
21:09.414 --> 21:13.454
there I =
I_0e to the
21:13.452 --> 21:17.952
−R_2/L
times t.
21:17.950 --> 21:22.220
So now the current will decay
exponentially.
21:22.220 --> 21:25.310
Again, with the time constant,
R_2 over
21:25.306 --> 21:27.496
L,
which is the 1 over the time
21:27.501 --> 21:29.741
constant,
or the time constant is
21:29.740 --> 21:32.310
L over
R_2.
21:32.308 --> 21:34.698
So you put L in henries,
R in ohms,
21:34.695 --> 21:35.835
you'll get some time.
21:35.838 --> 21:39.058
That will give you an idea of
how many times that time you
21:39.059 --> 21:42.449
have to wait before the inductor
is completely discharged.
21:42.450 --> 21:44.770
Well, it's never going to be
fully discharged.
21:44.769 --> 21:46.649
It's going to take forever,
but if you say,
21:46.647 --> 21:49.597
"Look, 1 thousandth of the
original current is safe enough.
21:49.598 --> 21:51.498
How long do I have to
wait?"
21:51.500 --> 21:55.440
you put .001 here and see what
time you get.
21:55.440 --> 22:01.840
So you want
e^(−t/τ) to be
22:01.842 --> 22:02.912
.001.
22:02.910 --> 22:08.830
So every big inductor you have
in a circuit you will find has
22:08.833 --> 22:15.353
got a resistor in parallel with
it to take up the energy it has.
22:15.348 --> 22:18.888
So let's do a little energy
calculation.
22:18.890 --> 22:24.010
The energy calculation will be,
I had energy in the inductor in
22:26.561 --> 22:29.121
LI_0 ^(2).
22:29.119 --> 22:30.199
What happened to it?
22:30.200 --> 22:31.600
Well, we know what happened.
22:31.598 --> 22:34.808
There's a current in the
circuit, and when the current is
22:34.807 --> 22:37.787
flowing in the circuit,
it is dissipating heat in the
22:37.785 --> 22:38.525
resistor.
22:38.529 --> 22:42.569
Therefore the power in the
resistor will be
22:42.574 --> 22:45.274
I^(2)R_2.
22:45.269 --> 22:49.119
I^(2) is
I_0^(2) due to the
22:49.123 --> 22:52.753
minus twice R_2t/L
times another
22:52.750 --> 22:54.640
R_2.
22:54.640 --> 22:58.920
Now we integrate this power
from 0 to infinity,
22:58.916 --> 23:04.586
you will get I_0
^(2)R_2.
23:04.588 --> 23:07.928
The integral of e to the
minus something is e to
23:07.928 --> 23:10.498
the minus that divided by that
infinity to 0.
23:10.500 --> 23:15.320
I can promise you that what you
will get will be
23:15.315 --> 23:17.975
L/2R_2.
23:17.980 --> 23:26.610
It will give you then 1 over
2LI_0^(2).
23:26.608 --> 23:31.098
So this is not a surprise,
that the stored energy in the
23:31.095 --> 23:33.945
inductor is driving your
current.
23:33.950 --> 23:36.810
So if you want,
one way to store energy,
23:36.805 --> 23:40.245
it's a capacitor,
you charge it and it discharge
23:40.247 --> 23:43.027
when you want the current to
flow.
23:43.029 --> 23:46.519
Or you can have an inductor,
in which a current has been set
23:46.522 --> 23:46.822
up.
23:46.818 --> 23:50.078
Then when the power shuts off
in your house and somebody
23:50.084 --> 23:52.694
throws the switch,
that inductor can keep the
23:52.694 --> 23:55.074
light bulb burning,
in fact forever.
23:55.068 --> 23:57.148
It will be infinite time before
the bulb shuts down,
23:57.150 --> 23:59.810
but you won't be able to see
anything after a while,
23:59.808 --> 24:05.228
because this graph is
exponentially falling.
24:05.230 --> 24:09.630
This is a very simple problem
where we can do the math and we
24:09.630 --> 24:12.050
can understand what it's doing.
24:12.049 --> 24:13.019
Yes?
24:13.019 --> 24:17.269
Student: The graph,
what is that triangle thing?
24:17.269 --> 24:19.079
Prof: This one here?
24:19.079 --> 24:20.499
This is called τ.
24:20.500 --> 24:22.340
You mean that symbol?
24:22.338 --> 24:26.328
Tau is a unit of time whose
numerical value is
24:26.325 --> 24:31.635
R_2/L and we
can write I as I_0
24:31.637 --> 24:34.027
e^(−t/τ).
24:34.029 --> 24:35.909
We always like to write it in
this form,
24:35.910 --> 24:40.720
so that if τ = 100 seconds,
you will have to make many
24:40.718 --> 24:43.868
multiples of 100 seconds before
the current is negligible.
24:43.868 --> 24:46.128
Student: What are the
units on the axis?
24:46.130 --> 24:47.850
Prof: Oh,
in this graph here?
24:47.848 --> 24:52.228
This is time and this is the
current in the loop.
24:52.230 --> 25:01.400
It starts out at some initial
value.
25:01.400 --> 25:13.920
Okay, now I'm going to do the
next slightly more complicated
25:13.924 --> 25:21.784
circuit, and that is an LC
circuit.
25:21.778 --> 25:26.428
If you have a circuit like
that, it won't do anything.
25:26.430 --> 25:28.860
You can take these guys,
hook them up and wait all day.
25:28.858 --> 25:32.738
Nothing will happen because
it's got no energy.
25:32.740 --> 25:33.660
Why should it do anything?
25:33.660 --> 25:35.320
It's just going to sit there.
25:35.318 --> 25:40.688
But if you've charged up your
capacitor like this,
25:40.690 --> 25:43.430
and there's an open switch that
you then closed,
25:43.430 --> 25:47.700
then those charges are going to
leave and find their way around
25:47.703 --> 25:51.563
the other side and neutralize
them and the capacitor will
25:51.564 --> 25:52.534
discharge.
25:52.529 --> 25:57.549
But you know that once it
discharges, it's not the end of
25:57.545 --> 25:58.615
the story.
25:58.619 --> 25:59.439
So why is that?
25:59.440 --> 26:04.320
Why doesn't it stop when the
capacitor is discharged?
26:04.319 --> 26:05.129
Yes?
26:05.130 --> 26:06.990
Student: It's kind of
like the inductor has momentum
26:06.986 --> 26:07.706
and it keeps pulling.
26:07.710 --> 26:11.240
Prof: The inductor would
have had a current by then,
26:11.236 --> 26:14.576
so the inductor is not going to
suddenly stop having its
26:14.579 --> 26:15.309
current.
26:15.308 --> 26:17.958
So it's going to keep driving
more current for a while,
26:17.960 --> 26:22.380
until the capacitor's charged
up the opposite way to an extent
26:22.378 --> 26:25.238
that this fights that one,
that voltage,
26:25.240 --> 26:27.290
and then it will go the other
way.
26:27.289 --> 26:31.139
So it will go back and forth.
26:31.140 --> 26:33.700
So let's write down the
equation that tells you how it
26:33.698 --> 26:34.758
goes back and forth.
26:34.759 --> 26:36.389
So in other words,
you should draw pictures.
26:36.390 --> 26:39.460
Textbooks have nice pictures
and I won't even try it.
26:39.460 --> 26:41.990
In the beginning,
it may look like--I'm sorry,
26:41.991 --> 26:43.741
in my example,
in the beginning,
26:43.737 --> 26:47.167
there's an electric field here,
nothing in the capacitor.
26:47.170 --> 26:49.230
After a quarter cycle,
there's nothing here,
26:49.230 --> 26:51.290
there's a magnetic field in the
inductor.
26:51.288 --> 26:53.568
Another quarter cycle,
it's back here,
26:53.566 --> 26:57.316
but with the opposite polarity,
and it goes back and forth.
26:57.318 --> 27:05.538
So we can get all that,
by writing the equation
27:05.539 --> 27:11.079
LdI/dt Q/C = 0.
27:11.078 --> 27:15.768
So let's write everything in
terms of Q.
27:15.769 --> 27:21.519
Then dQ/dt is I,
because as the current flows
27:21.523 --> 27:24.703
this way, it builds up charge.
27:24.700 --> 27:26.120
That's my convention.
27:26.118 --> 27:36.488
So I can write it as
Ld^(2)Q/dt^(2) (1/C)Q =
27:36.490 --> 27:37.320
0.
27:37.318 --> 27:40.478
Now we have seen exactly this
equation before,
27:40.478 --> 27:41.038
right?
27:41.038 --> 27:49.998
We saw the equation for a mass
coupled to a spring the equation
27:50.000 --> 27:54.480
md^(2)x/dt^(2) kx = 0.
27:54.480 --> 27:56.140
Now we know what it does.
27:56.140 --> 28:01.290
We know it oscillates back and
forth.
28:01.288 --> 28:06.948
And L--this is an SAT
question--L is to
28:06.949 --> 28:11.519
m what 1/C is to
k.
28:11.519 --> 28:14.879
It's very useful to bear in
mind this analogy,
28:14.884 --> 28:18.924
because mathematically,
this equation and this equation
28:18.922 --> 28:21.542
have exactly the same solution.
28:21.538 --> 28:23.748
It's no more difficult to solve
this one than that one.
28:23.750 --> 28:25.100
This may involve electric
charges;
28:25.099 --> 28:26.219
that may involve masses.
28:26.220 --> 28:27.710
You don't care.
28:27.710 --> 28:31.670
So mathematically,
here's an equation:
28:31.671 --> 28:38.411
cow times d^(2) dog over
dt^(2) let's say elephant
28:38.414 --> 28:42.244
times dog = 0,
where dog is a function of
28:42.240 --> 28:44.790
time, has exactly the same
solution.
28:44.788 --> 28:47.978
What does it matter what you
call the unknown variables,
28:47.976 --> 28:48.436
right?
28:48.440 --> 28:51.600
But you've got to tell me that
cow and elephant are time
28:51.597 --> 28:55.157
independent and dog is the only
thing that depends on time,
28:55.160 --> 28:57.110
because otherwise the equation
is not the same.
28:57.108 --> 29:00.268
So here m and k don't change,
L and C don't change.
29:00.269 --> 29:01.969
So what's the answer to this
one?
29:01.970 --> 29:04.220
Once again, it's the
differential equation.
29:04.220 --> 29:08.750
You write dQ/dt squared
is − (1/LC) Q and
29:08.747 --> 29:12.307
you're asking yourself,
give me a function which,
29:12.310 --> 29:16.910
when I differentiate twice,
looks like itself.
29:16.910 --> 29:18.610
There's always the exponential.
29:18.608 --> 29:21.218
There's also sines and the
cosines and you can take any
29:21.218 --> 29:22.328
combination you like.
29:22.328 --> 29:31.888
And the one acceptable solution
is some constant A times
29:31.893 --> 29:40.843
cosine square root of 1 over
LC t. Is that right?
29:40.839 --> 29:41.299
Yes.
29:41.298 --> 29:43.358
If you take two derivatives of
Q,
29:43.358 --> 29:46.178
you'll pull one over root
LC each time,
29:46.180 --> 29:49.990
then you'll get a - sine when
cosine becomes - sine and one
29:49.990 --> 29:53.210
more derivative will bring it
back to - cosine.
29:53.210 --> 29:56.110
So this is called the
frequency, which is
29:56.114 --> 29:57.644
1/√LC.
29:57.640 --> 30:02.950
That's the analog of square
root of k over m.
30:02.950 --> 30:04.340
Anyway, this will oscillate.
30:04.338 --> 30:08.788
It's a mechanical analog--it's
the electrical analog of the
30:08.788 --> 30:10.398
mechanical problem.
30:10.400 --> 30:13.130
So I don't want to do too much
of this, because I think you
30:13.133 --> 30:15.213
should be quite familiar with
this, at least,
30:15.205 --> 30:16.945
the oscillatory behavior of
this.
30:16.950 --> 30:18.590
And the analogy is perfect.
30:18.588 --> 30:22.258
For example,
to say that I pulled the mass
30:22.259 --> 30:27.099
by 1 meter is to say that
x was given a non-zero
30:27.095 --> 30:30.045
value and dx/dt was 0.
30:30.048 --> 30:32.908
Analogous thing here will be
Q was given a non-zero
30:32.910 --> 30:35.370
value and dQ/dt which is
I, is 0.
30:35.368 --> 30:38.188
That means the capacitor was
charged, the inductor was not
30:38.191 --> 30:39.331
carrying any current.
30:39.329 --> 30:40.209
Then I let it go.
30:40.210 --> 30:41.140
What happens?
30:41.140 --> 30:43.190
The problems are identical.
30:43.190 --> 30:44.480
Yes?
30:44.480 --> 30:48.760
Student: When you solve
the differential equation,
30:48.758 --> 30:50.858
why did you omit the sine?
30:50.858 --> 30:52.598
Prof: Yes,
I should explain to you.
30:52.599 --> 30:55.009
That's an important point.
30:55.009 --> 30:58.249
Q(t) can be one number,
cosine
30:58.250 --> 31:00.950
ω_0t.
31:00.950 --> 31:04.450
ω_0 is
this guy another number sine
31:04.448 --> 31:06.288
ω_0t.
31:06.289 --> 31:08.649
And both are acceptable.
31:08.650 --> 31:11.410
And the property of this
differential equation is,
31:11.410 --> 31:13.870
if I have one answer,
cos ωt, and
31:13.865 --> 31:17.315
another possible answer,
sine ωt, then any
31:17.317 --> 31:21.467
constant times cos any constant
times sine is also a solution.
31:21.470 --> 31:23.030
You can verify that.
31:23.028 --> 31:25.458
If you have one solution,
Q_1,
31:25.460 --> 31:27.440
a second solution,
Q_2,
31:27.440 --> 31:29.680
you can add the two equations
and you can show
31:29.684 --> 31:32.734
Q_1 Q_2
obeys the same equation.
31:32.730 --> 31:36.480
Not only that,
suppose there's one problem you
31:36.480 --> 31:38.630
solve,
dQ_1/dt
31:38.630 --> 31:42.060
squared is -ω
on R squared
31:42.058 --> 31:43.738
Q_1.
31:43.740 --> 31:48.830
There is some function of time
and here's another function of
31:48.827 --> 31:51.707
time, obeying the same equation.
31:51.710 --> 31:55.120
Now you can multiply this by a
number A and you can
31:55.116 --> 31:58.216
multiply this by a number
B, take the A
31:58.224 --> 32:01.674
inside the derivative,
B inside the derivative
32:01.673 --> 32:03.083
and add the two sides.
32:03.078 --> 32:05.668
On the left hand side you will
find the second derivative of
32:05.673 --> 32:07.393
AQ_1
BQ_2.
32:07.390 --> 32:08.500
Right hand side,
you will find
32:08.502 --> 32:10.542
ω_0^(2)
times AQ_1
32:10.537 --> 32:11.417
BQ_2.
32:11.420 --> 32:14.510
That means AQ_1
BQ_2 obeys this
32:14.508 --> 32:16.078
same differential equation.
32:16.078 --> 32:19.718
This is called superposing two
solutions.
32:19.720 --> 32:21.580
You've got one solution,
another one,
32:21.584 --> 32:23.504
you can multiply one by a
constant.
32:23.500 --> 32:26.300
It's very important it's a
constant, because that's what
32:26.298 --> 32:28.588
will let you take it inside the
derivative.
32:28.588 --> 32:31.388
Therefore the correct answer
here is really
32:31.390 --> 32:35.190
Acosωd
Bsinωt and you can
32:35.190 --> 32:36.730
ask,
why did you write
32:36.732 --> 32:39.012
Acosω
_0t?
32:39.009 --> 32:40.869
First of all,
that's not a good solution.
32:40.868 --> 32:45.078
I can put a Φ
there which is arbitrary.
32:45.078 --> 32:49.038
And it's not hard to show--let
me call that C if you
32:49.038 --> 32:50.198
like, C.
32:50.200 --> 32:55.150
This can be written as
Ccosω
32:55.154 --> 32:58.384
_0t - Φ.
32:58.380 --> 33:00.730
In other words,
it's possible to take either a
33:00.733 --> 33:02.673
solution with a cosine and a
sine,
33:02.670 --> 33:06.430
with no extra phases inside,
to another one with one
33:06.432 --> 33:08.352
amplitude and one phase.
33:08.348 --> 33:11.378
The phase is the delay in the
cosine.
33:11.380 --> 33:15.180
Anybody here has trouble,
who does not know the details,
33:15.180 --> 33:18.360
I'll be happy to explain why
that's correct.
33:18.358 --> 33:20.458
If you don't know,
you should tell me now.
33:20.460 --> 33:25.410
You can check that if you take
this thing,
33:25.410 --> 33:30.200
make a right angled triangle
with A,
33:30.200 --> 33:34.680
B and C and an
angle Φ here,
33:34.680 --> 33:39.630
then A can be written as
CcosΦ and
33:39.625 --> 33:44.225
B can be written as
CsinΦ,
33:44.230 --> 33:46.360
and that just happens to be
cosωt -
33:46.363 --> 33:50.053
Φ,
this one, some trigonometry.
33:50.048 --> 33:53.868
So the differential equation,
with the second order in time,
33:53.865 --> 33:56.965
will always have two unknown
parameters in it.
33:56.970 --> 34:00.060
You can choose them to be these
numbers A and B,
34:00.055 --> 34:02.225
or this number C and
Φ.
34:02.230 --> 34:04.710
And the equation won't tell you
what they are,
34:04.710 --> 34:07.960
because once you tell me
there's a mass coupled to a
34:07.955 --> 34:10.375
spring and I say,
"Okay, this is the spring
34:10.382 --> 34:11.652
constant this is the mass?"
34:11.650 --> 34:13.780
and I ask you,
"Where is the mass right
34:13.775 --> 34:14.365
now?"
34:14.369 --> 34:15.069
I don't know.
34:15.070 --> 34:18.130
It depends on when you started
it and how you released it.
34:18.130 --> 34:20.930
So I need to know what's called
initial conditions,
34:20.929 --> 34:24.339
which is the value of Q
or the value of x at the
34:24.340 --> 34:27.640
initial time and the value of
the velocity or the current at
34:27.641 --> 34:28.761
an initial time.
34:28.760 --> 34:32.550
With those two pieces of
information, I can solve for
34:32.550 --> 34:34.300
A and B.
34:34.300 --> 34:38.470
This I think you should know,
this kind of stuff you should
34:38.469 --> 34:41.559
know, so I won't say too much
about that.
34:41.559 --> 34:46.479
All right, so now we do another
problem.
34:46.480 --> 34:52.880
We put an alternating voltage
on these guys.
34:52.880 --> 34:57.990
So this is V_0
cosωt and this
34:57.987 --> 35:01.067
is C and this is
L.
35:01.070 --> 35:05.080
Now we ask, what's the current?
35:05.079 --> 35:10.519
This omega is not the natural
frequency of oscillation.
35:10.518 --> 35:14.578
It is some externally given
omega, like 60 hertz from your
35:14.579 --> 35:16.929
power supply,
from your socket.
35:16.929 --> 35:20.049
So that's driving the circuit
and you can ask,
35:20.050 --> 35:21.370
what happens now?
35:21.369 --> 35:25.349
The answer will be,
you write the same equation.
35:25.349 --> 35:34.889
You can write Ld^(2)Q/dt^(2)
(1/C)Q =
35:34.894 --> 35:41.794
V_0cos
ωt.
35:41.789 --> 35:43.549
Now what are you going to do?
35:43.550 --> 35:46.840
You have to again guess the
solution.
35:46.840 --> 35:50.170
So I want a function,
Q of t,
35:50.170 --> 35:53.730
so that when I take two
derivatives and add it to some
35:53.731 --> 35:56.291
multiple of itself,
I get something,
35:56.289 --> 35:58.179
something times the cosine.
35:58.179 --> 36:01.809
So what should that function
look like?
36:01.809 --> 36:02.929
Yes?
36:02.929 --> 36:07.739
Student:
>
36:07.739 --> 36:09.939
Prof: No,
no, what function Q
36:09.940 --> 36:12.770
of t do you think will
satisfy this equation?
36:12.768 --> 36:15.798
I'm asking for a function of
time which, when I put into
36:15.797 --> 36:18.437
this, has some chance of obeying
the equation.
36:18.440 --> 36:23.480
What's the functional form?
36:23.480 --> 36:24.540
Yes?
36:24.539 --> 36:26.029
Student: Cosine times
some constant.
36:26.030 --> 36:26.760
Prof: Right.
36:26.760 --> 36:30.100
In other words,
I can take it to be some
36:30.101 --> 36:33.791
constant, C times
cosωt,
36:33.786 --> 36:34.726
period.
36:34.730 --> 36:37.830
Not even times Φ.
36:37.829 --> 36:41.609
Take this one, see what happens.
36:41.610 --> 36:46.230
Now when you take one
derivative, you get
36:46.226 --> 36:50.726
-ωLCcos
ωt.
36:50.730 --> 36:53.370
When you take another
derivative, you get
36:53.365 --> 36:55.735
−ω
^(2)L ^(2)--
36:55.739 --> 36:59.959
I'm sorry, this becomes sine
times cosωt,
36:59.960 --> 37:04.030
and this one is just oh my
god--this is my nightmare.
37:04.030 --> 37:06.560
You can see the nightmare now?
37:06.559 --> 37:08.969
How many people see the
nightmare?
37:08.969 --> 37:09.879
Yes?
37:09.880 --> 37:10.770
Student: Constant
C.
37:10.768 --> 37:13.758
Prof: Yes,
so I picked that C to be
37:13.757 --> 37:15.497
the same as this C.
37:15.500 --> 37:22.480
So we'll put a hat on this guy
so we can tell them about.
37:22.480 --> 37:27.730
So C˜/C times
C˜ is
37:27.728 --> 37:32.738
V_0
cosωt.
37:32.739 --> 37:35.879
And what this tells you is,
if you take two
37:35.880 --> 37:39.170
derivatives--I'm sorry,
this is so sloppy.
37:39.170 --> 37:47.470
If you take two derivatives
here, you will get
37:47.467 --> 37:51.337
ω^(2).
37:51.340 --> 37:59.300
cosω be also be
with the - sign.
37:59.300 --> 38:01.330
Q/C.
38:01.329 --> 38:02.799
Okay.
38:02.800 --> 38:03.820
You guys buy that?
38:03.820 --> 38:07.930
There is no L^(2),
just an L.
38:07.929 --> 38:14.239
That = V_0
cosωt.
38:14.239 --> 38:14.569
Right?
38:14.570 --> 38:18.610
Every derivative brings an
ω and there's a net -
38:18.614 --> 38:19.154
sign.
38:19.150 --> 38:22.910
So you find out that your
guess, C˜c
38:22.914 --> 38:27.624
osωt is going to
work, provided this equation is
38:27.621 --> 38:28.721
satisfied.
38:28.719 --> 38:30.529
And the cosωt can
be canceled.
38:30.530 --> 38:33.220
That's the whole point of doing
this thing.
38:33.219 --> 38:40.349
So then you get C˜
= V_0 -
38:40.351 --> 38:47.481
V_0 divided by
ω^(2)L −
38:47.481 --> 38:49.161
1/C.
38:49.159 --> 38:53.649
Or if you like,
-V_0/L divided
38:53.648 --> 38:58.528
by ω^(2) -
ω_0^(2).
38:58.530 --> 39:01.780
Don't worry about the numerator;
look at the denominator.
39:01.780 --> 39:05.950
The denominator says that if
your driving frequency is equal
39:05.951 --> 39:09.771
to the resonant frequency,
C˜ blows up.
39:09.768 --> 39:12.148
That means when a system has
got a resonance--yes?
39:12.150 --> 39:15.870
Student: Where did the
ω^(2) come from?
39:15.869 --> 39:19.069
Prof: I took two
derivatives--oh,
39:19.065 --> 39:19.635
here.
39:19.639 --> 39:20.999
Oh, I'm sorry.
39:21.000 --> 39:23.420
Yes, thank you.
39:23.420 --> 39:26.810
You know, it's good to do these
things slowly and not rely on
39:26.813 --> 39:28.343
what you heard somewhere.
39:28.340 --> 39:29.740
So let me do it for you again.
39:29.739 --> 39:30.739
Sorry about that.
39:30.739 --> 39:32.599
So this time I take two
derivatives.
39:32.599 --> 39:34.529
Each one brings an
ω.
39:34.530 --> 39:37.400
There is an L,
and the - sine comes because
39:37.398 --> 39:39.308
the cosine becomes - sine.
39:39.309 --> 39:41.809
Differentiate one more,
you get - the cosine.
39:41.809 --> 39:43.139
This guy is nothing.
39:43.139 --> 39:46.999
It just says "divide my by
C"
39:47.001 --> 39:49.321
and that's equal to that.
39:49.320 --> 39:52.800
And then I wrote 1/LC is
ω_0^(2).
39:52.800 --> 39:53.890
Thank you very much.
39:53.889 --> 39:55.379
Okay?
39:55.380 --> 39:57.400
I was really hung up on the
final result,
39:57.400 --> 39:59.590
which tells you,
you'd better not drive this at
39:59.585 --> 40:03.495
the resonant frequency,
because then current amplitude
40:03.503 --> 40:06.013
will build up indefinitely.
40:06.010 --> 40:07.900
That's also true in a swing.
40:07.900 --> 40:11.900
If you've got a swing and the
kid's coming back and forth,
40:11.900 --> 40:14.300
and you're reading a newspaper
just pushing the kid,
40:14.300 --> 40:17.800
you've got to push at the right
time to get the best result.
40:17.800 --> 40:20.620
And that kid's not going to fly
off, because there's one more
40:20.621 --> 40:23.161
term in the equation for the
kid, which is friction.
40:23.159 --> 40:25.749
But if you have a frictionless
swing and you're doing this,
40:25.750 --> 40:27.880
you've got to watch out,
because soon there'll be nobody
40:27.882 --> 40:31.142
around,
because the amplitude will keep
40:31.137 --> 40:31.967
growing.
40:31.969 --> 40:34.449
Well, these are unrealistic
problems, but I want you to
40:34.447 --> 40:35.317
notice one thing.
40:35.320 --> 40:37.050
Here's what I want you to
notice.
40:37.050 --> 40:41.120
The voltage you were given
looked like V_0
40:41.117 --> 40:45.257
cosωt and the
current that you got,
40:45.260 --> 40:50.570
I can obtain by taking the
derivative of this charge,
40:50.570 --> 40:53.700
which is like dQ/dt.
40:53.699 --> 40:56.639
That's going to be
C˜ω
40:56.637 --> 41:00.737
sinωt,
and C˜, we can
41:00.737 --> 41:04.987
write as V_0/L
divided by ω^(2) -
41:04.987 --> 41:08.137
ω_0
^(2)sinωt.
41:08.139 --> 41:10.559
This is I.
41:10.559 --> 41:14.019
Here's what I want you to
notice.
41:14.018 --> 41:17.548
It looks a lot like Ohm's law,
because the current looks like
41:17.552 --> 41:20.622
voltage divided by some number,
but that's a very big
41:20.615 --> 41:21.495
difference.
41:21.500 --> 41:27.000
The voltage is a cosine and the
current is a sine.
41:27.000 --> 41:29.060
That's something I want you to
think about.
41:29.059 --> 41:33.529
The current is not in step with
the voltage, whereas in a
41:33.529 --> 41:36.719
resistor circuit,
the current follows the
41:36.722 --> 41:37.682
voltage.
41:37.679 --> 41:40.549
It's the same profile as the
voltage, except you divide by
41:40.545 --> 41:41.095
R.
41:41.099 --> 41:43.449
Here one is a cosine,
one is a sine.
41:43.449 --> 41:47.589
That means when one guy is at
maximum, other is at minimum.
41:47.590 --> 41:50.110
It's called out of phase,
in fact out of phase by 90
41:50.114 --> 41:50.614
degrees.
41:50.610 --> 41:51.420
Yes?
41:51.420 --> 41:54.840
Student: ________ by
the C that you got,
41:54.844 --> 41:56.434
where did the omega go?
41:56.429 --> 41:57.299
Prof: Here?
41:57.300 --> 41:58.610
Student: On the other
side.
41:58.610 --> 42:02.990
When you multiply that through
by the C--
42:02.989 --> 42:04.759
Prof: Oh,
ω did not go,
42:04.755 --> 42:05.255
it's back.
42:05.260 --> 42:06.650
Right here. You're right.
42:06.650 --> 42:07.700
There's the ω.
42:07.699 --> 42:08.289
That came from...
42:08.289 --> 42:21.889
42:21.889 --> 42:25.969
Okay, these problems,
the only thing I'm focusing on
42:25.969 --> 42:28.239
right now,
I would say of all the things I
42:28.235 --> 42:30.795
wrote there,
to be the most important thing
42:30.797 --> 42:33.257
is number one,
this is what I want you to bear
42:33.257 --> 42:33.657
in mind.
42:33.659 --> 42:35.969
That's why sometimes I'm not
paying attention to some
42:35.967 --> 42:36.497
constants.
42:36.500 --> 42:41.050
What is important to notice is
that this is an equation you can
42:41.045 --> 42:42.655
solve by inspection.
42:42.659 --> 42:45.209
Why was it easier to solve it
by inspection?
42:45.210 --> 42:48.410
Because you are trying to get
in the end a cosine to balance
42:48.411 --> 42:51.251
the right hand side,
and you're trying to find a
42:51.253 --> 42:55.263
function who is itself a cosine,
or whose second derivative is a
42:55.259 --> 42:57.219
cosine,
and we know the answer to that
42:57.217 --> 43:00.037
is a cosine,
so we can guess the answer.
43:00.039 --> 43:02.649
Once you guess the answer,
you put it in and you analyze
43:02.652 --> 43:04.412
the solution,
you notice also that the
43:04.409 --> 43:06.499
current and the voltage are not
in step.
43:06.500 --> 43:07.730
One is a cosine.
43:07.730 --> 43:10.120
You know what a cosine does,
it does that.
43:10.119 --> 43:13.039
Other is a sine,
so they are out of step.
43:13.039 --> 43:15.649
When one is at maximum,
the other is at minimum and so
43:15.652 --> 43:15.902
on.
43:15.900 --> 43:20.230
That means a current as a
function of time is not equal to
43:20.233 --> 43:24.493
the voltage as a function of
time divided by anything.
43:24.489 --> 43:29.049
There is nothing you can divide
a cosine by to turn it into a
43:29.047 --> 43:29.577
sine.
43:29.579 --> 43:34.469
Whereas with resistors,
you just divide by R,
43:34.472 --> 43:36.922
you get the current.
43:36.920 --> 43:43.860
Now for the more realistic
problem, the realistic problem
43:43.860 --> 43:50.060
has got a capacitor,
an inductor and a resistor and
43:50.056 --> 43:53.276
no other power supply.
43:53.280 --> 44:03.000
The equation obeyed by this one
will be Ld^(2)Q/dt^(2)
44:03.003 --> 44:11.513
(that stands for
LdI/dt) RdQ/dt
44:11.510 --> 44:14.810
Q/C = 0.
44:14.809 --> 44:19.639
This is analogous to
md^(2)x/dt^(2) (I don't
44:19.639 --> 44:24.469
know how you guys wrote this
thing last semester.
44:24.469 --> 44:35.779
It doesn't matter)
γdx/dt kx =
44:35.784 --> 44:36.694
0.
44:36.690 --> 44:40.030
Notice that it's the same form.
44:40.030 --> 44:43.820
And I'm going to give this to
you as a homework problem to
44:43.817 --> 44:46.007
analyze the answer to this one.
44:46.010 --> 44:50.930
So this describes a problem
where you have a mass and a
44:50.927 --> 44:54.567
spring and some friction on the
table.
44:54.570 --> 44:57.950
This means if you pull it and
let it go, the oscillations will
44:57.952 --> 45:00.452
eventually get damped and it
will die down.
45:00.449 --> 45:06.419
And the general solution for
the case when it's oscillating
45:06.423 --> 45:11.993
will look like some number A
e to the minus some
45:11.985 --> 45:16.925
number αt times cosine
times some frequency
45:16.929 --> 45:19.299
ω't.
45:19.300 --> 45:20.530
Let me see.
45:20.530 --> 45:24.340
You can always add a phase
Φ,
45:24.335 --> 45:27.355
but I'm not going to do that.
45:27.360 --> 45:29.210
If you want,
you can put an extra Φ,
45:29.213 --> 45:32.163
but I'll choose my origin of
time so that I don't have that.
45:32.159 --> 45:34.759
This is what the answer's going
to look like,
45:34.760 --> 45:37.840
where ω' and
α are going to be
45:37.838 --> 45:41.218
controlled by L,
R and C.
45:41.219 --> 45:43.399
That's the thing I don't want
to do in class.
45:43.400 --> 45:45.670
Have you seen this before?
45:45.670 --> 45:48.230
Professor Harris tells me
you've seen this last time.
45:48.230 --> 45:52.110
And the hints in the homework
will guide you on how to do
45:52.114 --> 45:52.604
this.
45:52.599 --> 45:56.809
You just assume the solution
x at t looks like
45:56.811 --> 46:00.241
some A e to the - (I
don't want to call it
46:00.237 --> 46:04.017
α now)
β times t.
46:04.018 --> 46:07.478
You put it in the equation and
solve for β and
46:07.483 --> 46:10.523
you'll find β
will have a real part and
46:10.521 --> 46:11.861
an imaginary part.
46:11.860 --> 46:17.010
And you have to combine the two
to get this answer.
46:17.010 --> 46:21.140
Now that brings me to another
thing, so I don't know how
46:21.137 --> 46:25.117
prepared you guys are for what's
about to happen next,
46:25.117 --> 46:28.117
which is the use of complex
numbers.
46:28.119 --> 46:30.339
So everybody familiar with
complex numbers?
46:30.340 --> 46:34.190
Who doesn't know complex
numbers?
46:34.190 --> 46:36.440
How do you do your taxes?
46:36.440 --> 46:39.430
You don't know imaginary
numbers?
46:39.429 --> 46:42.809
So I will tell you.
46:42.809 --> 46:44.099
I'll give you a lightning
review.
46:44.099 --> 46:47.199
I'm assuming everybody has seen
them in high school.
46:47.199 --> 46:50.579
I will only tell you the part
you need, but I'm going to
46:50.581 --> 46:52.921
assume that I can use them
fluently.
46:52.920 --> 46:56.840
I don't want to stop every time
and worry about you guys.
46:56.840 --> 46:59.400
So I really need a show of
hands.
46:59.400 --> 47:04.370
Anybody never seen x iy
and x - iy?
47:04.369 --> 47:05.959
And how about
e^(iθ)?
47:05.960 --> 47:07.930
You know that guy?
47:07.929 --> 47:09.039
Okay, that's all you need.
47:09.039 --> 47:10.959
So I'm going to tell you what
the deal is.
47:10.960 --> 47:12.900
So everyone knows what a
complex number is.
47:12.900 --> 47:15.600
We know i is square root
of -1.
47:15.599 --> 47:19.929
Then we know that we can write
a complex number as x iy
47:19.934 --> 47:24.274
and visualize it in a complex
plane where you measure x
47:24.268 --> 47:27.678
this way and iy is
measured that way.
47:27.679 --> 47:34.309
That's usually called a generic
complex number z.
47:34.309 --> 47:39.119
But now let us find the length
of that--the complex number is a
47:39.123 --> 47:41.923
single point in the complex
plane.
47:41.920 --> 47:47.340
The length of that is square
root of x^(2) y^(2).
47:47.340 --> 47:52.480
So I can also write z as
x divided by square root
47:52.483 --> 47:57.633
of x^(2) y^(2) i
times y divided by square
47:57.626 --> 48:02.516
root of x^(2) y^(2) times
the square root of x^(2)
48:02.521 --> 48:03.851
y^(2).
48:03.849 --> 48:06.339
Just done nothing,
just rearranged stuff.
48:06.340 --> 48:11.480
This is going to be called the
modulus of z,
48:11.483 --> 48:13.443
and what is this?
48:13.440 --> 48:16.510
This angle is θ
here.
48:16.510 --> 48:19.420
This is really
cosθ
48:19.416 --> 48:24.476
isinθ times
modulus of z.
48:24.480 --> 48:28.670
And that, thanks to this great
identity by Euler,
48:28.670 --> 48:34.570
is e^(iθ). Now I
don't know how much you know
48:34.568 --> 48:40.668
about this great formula that
relates e^(iθ) to
48:40.673 --> 48:42.333
cosine sine.
48:42.329 --> 48:44.539
How many people know where it
comes from?
48:44.539 --> 48:49.029
What does it mean to raise
e to a complex power?
48:49.030 --> 48:51.040
You know where it really comes
from?
48:51.039 --> 48:51.809
Yes?
48:51.809 --> 48:52.739
Student:
>
48:52.739 --> 48:54.339
Prof: It comes from
power series.
48:54.340 --> 48:57.510
It turns out you can write a
power series for cosine.
48:57.510 --> 48:59.420
You can write a power series
for sine,
48:59.420 --> 49:03.750
then, for example,
cosine of θ = 1
49:03.751 --> 49:08.181
− θ^(2) over
2 factorial,
49:08.179 --> 49:10.329
θ to the 4
factorial,
49:10.329 --> 49:14.369
or 6 factorial, etc.
49:14.369 --> 49:16.959
And you can write a power
series for e^(θ) if
49:16.963 --> 49:19.463
you like,
which is 1 θ
49:19.458 --> 49:21.838
θ^(2)/2.
49:21.840 --> 49:25.510
This defines cosθ
in the sense that if you put
49:25.514 --> 49:29.194
θ = Π/2
in this infinite series,
49:29.190 --> 49:31.770
you will get 0.
49:31.768 --> 49:33.188
And if you put θ
= Π, you will get
49:33.188 --> 49:33.678
Π -1.
49:33.679 --> 49:36.479
In other words,
in spite of this funny looking
49:36.481 --> 49:38.601
form, it really is the same guy.
49:38.599 --> 49:40.189
It will oscillate,
it will have zeros,
49:40.186 --> 49:41.126
it will be periodic.
49:41.130 --> 49:43.680
None of it is obvious,
but this power series is
49:43.681 --> 49:47.181
numerically equal to this one if
you keep the infinite number of
49:47.175 --> 49:47.725
turns.
49:47.730 --> 49:51.000
Likewise, e^(θ) is
defined by this series.
49:51.000 --> 49:52.570
And similarly,
there's a formula for
49:52.570 --> 49:53.470
sinθ.
49:53.469 --> 49:57.889
Now once a power series is
defined, you can put e raised to
49:57.894 --> 49:59.884
anything you want there.
49:59.880 --> 50:03.410
So you know what I'm going to
put there.
50:03.409 --> 50:07.739
e raised to dog is 1 dog
dog squared.
50:07.739 --> 50:09.119
This is not a joke.
50:09.119 --> 50:10.339
This is really true.
50:10.340 --> 50:11.270
This is the definition.
50:11.268 --> 50:13.398
If someone says,
"How do I raise e
50:13.398 --> 50:14.528
to the power dog?"
50:14.530 --> 50:15.830
you do this.
50:15.829 --> 50:17.939
This will have all the
properties of the exponential.
50:17.940 --> 50:19.290
It doesn't matter what's in the
exponent.
50:19.289 --> 50:20.969
That's the key.
50:20.969 --> 50:24.109
People originally put real
numbers, then they put complex
50:24.108 --> 50:24.668
numbers.
50:24.670 --> 50:27.500
Now they put matrices,
operators, anything you want.
50:27.500 --> 50:30.760
e raised to anything,
you formally define to be this
50:30.762 --> 50:33.242
infinite series,
provided the infinite series
50:33.239 --> 50:35.939
converges and gives you a
meaningful answer.
50:35.940 --> 50:38.330
In that sense,
if you put e^(iθ)
50:38.331 --> 50:40.531
here,
and compare the result to
50:40.530 --> 50:42.840
cosθ
isinθ,
50:42.840 --> 50:44.440
it matches, that's all.
50:44.440 --> 50:48.140
I'm just going to keep using
that result.
50:48.139 --> 50:51.819
Then you should also know that
for every complex number,
50:51.822 --> 50:55.442
there's a complex conjugate,
which is x - iy.
50:55.440 --> 50:58.120
That means i goes to
-i.
50:58.119 --> 51:02.059
That means z,
the angle θ
51:02.061 --> 51:05.711
will change to
-θ.
51:05.710 --> 51:09.980
And given any complex number,
the real part of z,
51:09.980 --> 51:14.170
which is x,
is z z* over 2 and the
51:14.172 --> 51:20.022
imaginary part of z,
which is y,
51:20.019 --> 51:26.889
is z −
z* over 2i.
51:26.889 --> 51:29.329
This is what you need to know.
51:29.329 --> 51:32.219
Every complex number has a real
part and an imaginary part.
51:32.219 --> 51:34.919
In fact, z* looks like
this.
51:34.920 --> 51:37.820
And if you add z z*,
the vertical parts cancel
51:37.820 --> 51:40.110
and you get double the
horizontal part.
51:40.110 --> 51:41.840
That's why you divide by 2.
51:41.840 --> 51:45.650
You subtract and divide by
2i, you get y.
51:45.650 --> 51:49.180
Basically, that's all I want
you to know, but I want you to
51:49.181 --> 51:51.861
be able to manipulate them
rapidly enough.
51:51.860 --> 51:56.490
Here's something very useful
about complex numbers.
51:56.489 --> 52:02.719
Let's take a complex number
z_1 which is
52:02.717 --> 52:07.387
mod z_1
e^(iθ1),
52:07.389 --> 52:09.909
and another complex number,
z_2,
52:09.909 --> 52:15.419
which is mod z_2
e^(iθ2).
52:15.420 --> 52:16.840
So what do they look like?
52:16.840 --> 52:18.900
Well, z_1
looks like this,
52:18.900 --> 52:20.760
with some angle
θ_1,
52:20.760 --> 52:22.850
and z_2_
may look like that.
52:22.849 --> 52:25.019
It's got some angle
θ_2.
52:25.018 --> 52:28.118
So every complex number has a
length and an angle.
52:28.119 --> 52:29.239
You can think of it two ways.
52:29.239 --> 52:31.779
It's got an x part and a
y part,
52:31.784 --> 52:34.894
and a real and an imaginary,
or it's got a modulus and a
52:34.893 --> 52:35.463
phase.
52:35.460 --> 52:38.570
That's the Cartesian version of
the number and the polar number
52:38.567 --> 52:39.367
of the number.
52:39.369 --> 52:42.839
And the connection between them
comes from Euler's formula.
52:42.840 --> 52:45.910
But now look what happens when
I take the product
52:45.911 --> 52:48.281
z_1z
_2.
52:48.280 --> 52:51.260
I get mod z_1
mod z_2
52:51.260 --> 52:54.540
e^(iθ)_1^(
θ)_2,
52:54.539 --> 52:59.559
because exponentials add when
you multiply them.
52:59.559 --> 53:02.069
Well, this allows you to
immediately guess what the
53:02.065 --> 53:03.665
product is going to look like.
53:03.670 --> 53:06.050
The product is going to have a
length equal to the length of
53:06.048 --> 53:08.468
z_1 times
length of z_2.
53:08.469 --> 53:11.419
It's going to have an angle,
I've not done a good picture
53:11.418 --> 53:13.718
here,
which is the sum of the two
53:13.715 --> 53:14.995
angles,
because it's
53:15.000 --> 53:17.640
θ_1
θ_2.
53:17.639 --> 53:21.859
So listen to this statement
very carefully.
53:21.860 --> 53:24.420
When you take a complex number,
say number 1,
53:24.420 --> 53:27.100
and you multiply it by a second
complex number,
53:27.099 --> 53:29.369
you do two things to the first
guy.
53:29.369 --> 53:33.619
You rescale it and you rotate
it.
53:33.619 --> 53:36.789
So two operations are done in
one shot when you use complex
53:36.791 --> 53:37.341
numbers.
53:37.340 --> 53:38.930
The mod z_2
rescale is the mod
53:38.931 --> 53:40.831
z_1 and
θ_2 adds
53:40.833 --> 53:42.013
to θ_1.
53:42.010 --> 53:43.660
Likewise, if you divide,
if you take
53:43.655 --> 53:46.285
z_1 over
z_2, it is mod
53:46.286 --> 53:48.866
z_1 over mod
z_2 times
53:48.871 --> 53:51.881
e^(iθ)_1 ^(-
θ)_2,
53:51.880 --> 53:55.630
because the exponential
θ_2 is
53:55.632 --> 53:56.572
downstairs.
53:56.570 --> 53:59.620
So multiplication by real
numbers is very easy.
53:59.619 --> 54:02.579
You take a real number 4,
you multiply by 8,
54:02.577 --> 54:06.357
you'll get something in the
real axis 8 times longer.
54:06.360 --> 54:10.180
You multiply by -8,
you'll get something back here.
54:10.179 --> 54:12.089
With complex numbers,
you take a number,
54:12.088 --> 54:14.928
you multiply another number,
you rescale and you rotate.
54:14.929 --> 54:15.979
That's what I want you to know.
54:15.980 --> 54:19.920
That's going to be very
important for what I'm saying.
54:19.920 --> 54:22.890
Now I come to the problem I
really want to solve,
54:22.889 --> 54:31.639
which is an LCR circuit driven
by an alternating source,
54:31.639 --> 54:34.449
V_0
cosωt.
54:34.449 --> 54:37.599
This is R,
this is C and this is
54:37.599 --> 54:38.369
L.
54:38.369 --> 54:45.589
And the equation we have to
solve is L dI/dt
54:45.592 --> 54:52.672
RI 1/C integral
I dt =
54:52.672 --> 54:59.032
V_0
cosωt.
54:59.030 --> 55:03.510
Because Q is the
integral up to time t of the
55:03.512 --> 55:06.072
current,
that's just Q/C. It's
55:06.065 --> 55:09.075
the same equation,
but now, unlike in this
55:09.077 --> 55:12.507
problem, where I had no driving
voltage,
55:12.510 --> 55:14.990
I have a driving voltage,
and I've also written it in
55:14.987 --> 55:17.367
terms of current rather than in
terms of charge.
55:17.369 --> 55:19.509
Because in electrical
engineering, you don't really
55:19.514 --> 55:21.064
watch the charge in the
capacitor.
55:21.059 --> 55:24.809
You look at the current flowing
through the circuit.
55:24.809 --> 55:28.189
So you have to solve this
equation.
55:28.190 --> 55:31.220
The question is,
can you guess the answer?
55:31.219 --> 55:32.149
That's the only way we know.
55:32.150 --> 55:34.450
You've got to guess.
55:34.449 --> 55:38.459
You're trying to find a
function, I of t,
55:38.460 --> 55:41.740
so that when you differentiate
it, you get a cosine
55:41.735 --> 55:44.235
ωt,
when you integrate it,
55:44.237 --> 55:46.177
you get a cosine
ωt.
55:46.179 --> 55:47.659
So far we can do it.
55:47.659 --> 55:49.349
Sine ωt will do it.
55:49.349 --> 55:53.589
But when you leave it alone,
then also you get a cosine
55:53.588 --> 55:56.648
ωt. You cannot do
that,
55:56.650 --> 55:59.790
because sine will become a
cosine and sine will become a
55:59.789 --> 56:00.189
sine.
56:00.190 --> 56:03.840
In the LC circuit,
when you didn't have this guy,
56:03.840 --> 56:06.360
we were okay,
because if you pick a sine,
56:06.360 --> 56:10.080
this became a cosine and that
also became a cosine and you can
56:10.079 --> 56:11.969
combine them to get a cosine.
56:11.969 --> 56:14.479
But you cannot have
dI/dt,
56:14.478 --> 56:19.098
I and integral of I all in one
equation and all satisfied by
56:19.103 --> 56:21.223
trigonometric function.
56:21.219 --> 56:24.569
The only time we can guess the
answer is if it looks like
56:24.567 --> 56:26.717
V_0
e^(αt).
56:26.719 --> 56:29.789
Suppose that was the voltage.
56:29.789 --> 56:33.579
Then can you guess the answer,
what form the answer will have?
56:33.579 --> 56:36.539
It will also be e^(αt)
times some number,
56:36.539 --> 56:38.629
because then this will look
e^(αt), that will
56:38.630 --> 56:40.580
look like e^(αt),
that will look like
56:40.576 --> 56:41.386
e^(αt).
56:41.389 --> 56:43.759
e^(αt) has a great
property that whether you
56:43.764 --> 56:45.394
multiply it,
whether you integrate it,
56:45.394 --> 56:46.874
differentiate it or leave it
alone,
56:46.869 --> 56:48.629
it looks the same.
56:48.630 --> 56:50.580
Unfortunately,
no one's interested in this
56:50.577 --> 56:52.857
voltage, because it's growing
exponentially fast,
56:52.860 --> 56:55.380
or if you put a - sign,
it's dying exponentially.
56:55.380 --> 56:58.390
What we really want is this.
56:58.389 --> 57:02.379
So the question is,
how do we solve this problem
57:02.380 --> 57:04.590
with cosωt?
57:04.590 --> 57:08.130
So we're going to use a certain
trick.
57:08.130 --> 57:12.710
The trick I'm going to use is
the following:
57:12.708 --> 57:18.878
let me take a general case
where this is some function V
57:18.884 --> 57:23.444
of t,
where I'm not even assuming
57:23.436 --> 57:24.976
V is real.
57:24.980 --> 57:28.410
I'm not assuming I is
real, but L and R
57:28.413 --> 57:29.813
and C are real.
57:29.809 --> 57:33.119
Everything else can be complex
numbers.
57:33.119 --> 57:36.099
Now once you have an equation
and you've found a solution,
57:36.101 --> 57:38.561
if you take the complex
conjugate of both sides,
57:38.559 --> 57:39.919
they will still match.
57:39.920 --> 57:41.480
Do you understand that?
57:41.480 --> 57:44.750
If two things are equal,
their complex conjugates are
57:44.753 --> 57:47.323
equal,
because this thing is the real
57:47.318 --> 57:51.448
imaginary that balances the real
imaginary on the other side.
57:51.449 --> 57:54.339
Complex conjugate just reverses
the imaginary terms,
57:54.335 --> 57:57.275
so they will still match when
you flip both sides.
57:57.280 --> 58:05.220
So it means if you take L
dI* of dt R times
58:05.221 --> 58:13.871
I* 1/C integral I* of
t' dt' t = V* of
58:13.871 --> 58:15.291
t.
58:15.289 --> 58:18.129
So star is the rule by which if
V had a real and
58:18.125 --> 58:20.535
imaginary part,
you flip the imaginary part.
58:20.539 --> 58:25.499
So that solution for V
of t implies a second
58:25.501 --> 58:26.461
equation.
58:26.460 --> 58:29.090
In other words,
if the voltage V drives
58:29.090 --> 58:31.840
a current I,
the conjugate of the voltage
58:31.840 --> 58:34.180
drives the conjugate of the
current.
58:34.179 --> 58:35.929
That simply follows from the
equation.
58:35.929 --> 58:38.799
It comes from the fact that
when I conjugated it,
58:38.804 --> 58:40.964
L, R and C are
real.
58:40.960 --> 58:42.530
Do you understand that?
58:42.530 --> 58:45.680
Suppose L were imaginary
or had imaginary part,
58:45.682 --> 58:48.482
then I must also put a
conjugate on L.
58:48.480 --> 58:50.570
Then I* obeys a
different equation,
58:50.574 --> 58:53.644
because this had L in
it, this had L* in it.
58:53.639 --> 58:57.339
But if everything is real,
I* obeys the same
58:57.338 --> 59:00.888
equation when you have a
V* driving it.
59:00.889 --> 59:02.509
I'm almost done now.
59:02.510 --> 59:06.070
I want you to add the left hand
side to the left hand side and
59:06.067 --> 59:09.797
the right hand side to the right
hand side, and what do I get?
59:09.800 --> 59:13.870
Well, if I write it here,
you may not be able to see it.
59:13.869 --> 59:15.469
Let me put it here then.
59:15.469 --> 59:21.959
If you add these two,
you'd get L d/dt of
59:21.956 --> 59:29.546
I I* R
times I I* 1/C
59:29.547 --> 59:37.407
integral of I I* dt =
V V*. This again comes
59:37.414 --> 59:44.044
because the equation is a linear
equation.
59:44.039 --> 59:46.959
You can always add the left
hand side and the right hand
59:46.956 --> 59:49.866
side to get another problem
where the driving voltage is
59:49.871 --> 59:52.631
V V* and the driving
current is I I*,
59:52.630 --> 59:58.760
etc.
59:58.760 --> 1:00:05.800
But I I* is 2 times the
real part of I and V
1:00:05.795 --> 1:00:11.185
V* is 2 times the real part
of V.
1:00:11.190 --> 1:00:17.640
Therefore this implies that
L d/dt of the real part
1:00:17.644 --> 1:00:24.554
of I R plus the
real part of I 1/C
1:00:24.552 --> 1:00:31.462
times the integral of the real
part of I = real part of
1:00:31.460 --> 1:00:33.160
V.
1:00:33.159 --> 1:00:36.289
So let me say in words what the
equation means.
1:00:36.289 --> 1:00:40.659
If you by luck solve an
equation with a complex
1:00:40.655 --> 1:00:43.455
potential V,
and you get a current,
1:00:43.461 --> 1:00:47.301
then the answer to the problem
where the potential is only the
1:00:47.297 --> 1:00:50.437
real part of the actual
potential you applied,
1:00:50.440 --> 1:00:53.630
the current will be the real
part of the answer you got.
1:00:53.630 --> 1:00:55.270
That's all I want you to know.
1:00:55.268 --> 1:00:58.568
If you solve a complex problem
with a driving voltage which is
1:00:58.565 --> 1:01:00.285
complex--
which is just a mathematical
1:01:00.293 --> 1:01:02.263
fiction,
in real life you don't have
1:01:02.259 --> 1:01:05.609
that--then the current that you
get will also be complex.
1:01:05.610 --> 1:01:09.000
But its real part will be due
to the real part of your
1:01:09.000 --> 1:01:09.640
voltage.
1:01:09.639 --> 1:01:12.119
And you can also show the
imaginary part of the current
1:01:12.123 --> 1:01:14.473
will be due to the imaginary
part of the voltage.
1:01:14.469 --> 1:01:17.709
This is just the principle of
superposition.
1:01:17.710 --> 1:01:19.260
So here is the trick we are
going to use.
1:01:19.260 --> 1:01:22.480
We are going to go back to
original equation,
1:01:22.480 --> 1:01:29.470
LdI/dt RI 1/C integral
I dt = V_0cos
1:01:29.469 --> 1:01:33.849
ωt,
and we're going to solve a new
1:01:33.849 --> 1:01:34.489
problem.
1:01:34.489 --> 1:01:39.029
We're going to solve a problem
where this is V_0
1:01:39.027 --> 1:01:40.537
e^(iωt).
1:01:40.539 --> 1:01:42.829
And the current for that,
I'm going to call some
1:01:42.831 --> 1:01:43.711
I˜.
1:01:43.710 --> 1:01:49.800
It's a complex current,
because the driving voltage is
1:01:49.800 --> 1:01:52.100
a complex voltage.
1:01:52.099 --> 1:01:54.599
So think of
Ve^(iωt) as the
1:01:54.597 --> 1:01:56.837
V in the previous
problem.
1:01:56.840 --> 1:02:00.280
The actual problem I was given
had V_0cos
1:02:00.282 --> 1:02:02.552
ωt,
but you know that
1:02:02.554 --> 1:02:04.604
V_0
e^(iωt) is
1:02:04.597 --> 1:02:07.257
V_0
cosωt I
1:02:07.264 --> 1:02:10.164
times V_0
sinωt.
1:02:10.159 --> 1:02:13.559
Consequently,
this potential has a real part
1:02:13.559 --> 1:02:16.539
which is a cosine,
an imaginary part which is a
1:02:16.536 --> 1:02:19.306
sine, and the current will have
a real and imaginary part.
1:02:19.309 --> 1:02:22.739
The answer to the original
question is the real part of the
1:02:22.740 --> 1:02:24.280
answer to this question.
1:02:24.280 --> 1:02:25.760
This is what you should think
about.
1:02:25.760 --> 1:02:28.850
This is what you should
understand.
1:02:28.849 --> 1:02:33.589
So I've manufactured a problem
with an unphysical complex
1:02:33.588 --> 1:02:35.868
voltage,
and all I know is that at the
1:02:35.869 --> 1:02:38.529
end of the day,
if I take the real part of the
1:02:38.534 --> 1:02:40.564
current,
I'll get the answer to the real
1:02:40.556 --> 1:02:42.926
part of the potential,
which happens to be the actual
1:02:42.931 --> 1:02:44.571
potential,
V_0
1:02:44.574 --> 1:02:46.264
cosωt.
1:02:46.260 --> 1:02:48.290
So why would I do all this?
1:02:48.289 --> 1:02:50.709
Why would I take a problem
that's bad enough with the
1:02:50.713 --> 1:02:52.953
cosine and turn it into a
complex exponential?
1:02:52.949 --> 1:02:56.679
I think you can sort of guess
what the reason is.
1:02:56.679 --> 1:03:00.139
The reason is that it may be
complex, but it's still an
1:03:00.141 --> 1:03:01.041
exponential.
1:03:01.039 --> 1:03:04.049
That means its derivative is
going to look just the same.
1:03:04.050 --> 1:03:07.920
That means I can make a guess
that the current I˜
1:03:07.920 --> 1:03:11.590
is proportional to
e^(iωt). So you can
1:03:11.586 --> 1:03:15.246
make the assumption that
I˜ of t is
1:03:15.253 --> 1:03:19.063
some constant I˜
times e^(iωt).
1:03:19.056 --> 1:03:23.266
Make that assumption and put it
into this equation and see what
1:03:23.266 --> 1:03:24.486
you get.
1:03:24.489 --> 1:03:30.719
You will find you will get
IωL times
1:03:30.722 --> 1:03:37.812
I˜ R times
I˜ 1/IωC
1:03:37.809 --> 1:03:42.699
times I˜
times e^(Iωt) is
1:03:42.697 --> 1:03:46.727
V_0e^(iωt).
1:03:46.730 --> 1:03:51.430
This is because if you assume
the current is an exponential,
1:03:51.429 --> 1:03:55.259
every derivative = an
iω and every integral
1:03:55.264 --> 1:03:59.384
= 1/iω, and
multiplying is just multiplying.
1:03:59.380 --> 1:04:02.170
So d/dt has got to be
replaced by ω,
1:04:02.170 --> 1:04:05.730
and you get this equation
for I˜. That means
1:04:05.731 --> 1:04:10.891
you will satisfy the equation,
provided I˜ times
1:04:10.894 --> 1:04:13.454
all of this = V .
1:04:13.449 --> 1:04:16.359
And I'm going to write this as
Z times I˜
1:04:16.362 --> 1:04:18.972
= V where Z is the
name for all the guys
1:04:18.965 --> 1:04:20.625
multiplying I˜.
1:04:20.630 --> 1:04:21.780
So I'll start with that here.
1:04:21.780 --> 1:04:39.420
1:04:39.420 --> 1:04:43.470
Okay.
1:04:43.469 --> 1:04:50.499
So I wrote iωL
R 1 over iωC
1:04:50.501 --> 1:04:56.261
I˜ e^(iωt) is
V_0
1:04:56.255 --> 1:04:59.065
e^(iωt).
1:04:59.070 --> 1:05:02.960
Everybody get that?
1:05:02.960 --> 1:05:07.360
That's because differentiating
on an exponential is just
1:05:07.360 --> 1:05:09.760
multiplying by the exponent.
1:05:09.760 --> 1:05:12.220
So this is a complex number.
1:05:12.219 --> 1:05:17.199
R iωL −
i/ωC.
1:05:17.199 --> 1:05:20.969
I want you to know that i
in the bottom is like -i
1:05:20.965 --> 1:05:22.045
in the top.
1:05:22.050 --> 1:05:26.650
And this is called the
impedance Z,
1:05:26.653 --> 1:05:29.353
it's a complex number.
1:05:29.349 --> 1:05:32.069
You can visualize the complex
number as follows:
1:05:32.074 --> 1:05:34.514
it's got a real part,
which is R.
1:05:34.510 --> 1:05:38.270
It's got an imaginary part,
which is ωL −
1:05:38.268 --> 1:05:41.578
1/ωC,
and the impedance Z is a
1:05:41.577 --> 1:05:44.857
complex number with some modulus
and some phase.
1:05:44.860 --> 1:05:47.430
If you know the real and
imaginary parts,
1:05:47.427 --> 1:05:50.957
you can construct the modulus
in the phase like that.
1:05:50.960 --> 1:05:53.860
If you want,
it's the modulus of Z.
1:05:53.860 --> 1:06:00.980
Therefore canceling this,
I find I˜
1:06:00.983 --> 1:06:04.963
= V_0/Z.
1:06:04.960 --> 1:06:10.060
That becomes V_0 over
the modulus of Z times
1:06:10.063 --> 1:06:14.693
e^(i)^(Φ),
where Φ is this angle
1:06:14.688 --> 1:06:19.108
defined by tanΦ
= ωL −
1:06:19.112 --> 1:06:22.342
1/ωC divided by R.
1:06:22.340 --> 1:06:24.660
So don't worry too much about
writing all of this down,
1:06:24.655 --> 1:06:26.795
because this you're going to
find in every book.
1:06:26.800 --> 1:06:31.110
There's nothing novel or
different.
1:06:31.110 --> 1:06:34.300
So this is the formula for
I twiddle.
1:06:34.300 --> 1:06:38.540
Now I itself,
you remember,
1:06:38.539 --> 1:06:41.569
= I˜ of 0.
1:06:41.570 --> 1:06:44.070
I'm sorry, this is
I˜ of 0.
1:06:44.070 --> 1:06:48.460
Did I define it that way?
1:06:48.460 --> 1:06:49.850
Yeah, I should write it--I'm
sorry.
1:06:49.849 --> 1:06:52.819
I should write it as
I˜ of 0.
1:06:52.820 --> 1:06:59.000
There's a subscript 0 there,
because the full I twiddle has
1:06:58.996 --> 1:07:02.826
got an e^(iωt) in
it.
1:07:02.829 --> 1:07:10.329
That looks like
V_0 over mod
1:07:10.331 --> 1:07:19.881
Z times e^(iωt -
)^(Φ).
1:07:19.880 --> 1:07:23.330
But this is not the physical
current, because it's the result
1:07:23.327 --> 1:07:24.647
of a complex voltage.
1:07:24.650 --> 1:07:26.770
The physical current is a real
part of this,
1:07:26.768 --> 1:07:30.428
without the twiddle,
that we write as real part of
1:07:30.427 --> 1:07:34.007
I˜. Where do I get
the real part?
1:07:34.010 --> 1:07:36.530
V_0 and
absolute value of Z are
1:07:36.530 --> 1:07:37.090
both real.
1:07:37.090 --> 1:07:41.300
Real part of e to the
i something is cosine of
1:07:41.302 --> 1:07:43.732
something and that's our answer.
1:07:43.730 --> 1:07:51.210
That's our answer,
the final answer you get.
1:07:51.210 --> 1:07:54.300
This tells you that the current
has a magnitude which is
1:07:54.304 --> 1:07:57.684
V_0 over the
absolute value of Z,
1:07:57.679 --> 1:08:00.749
and it's got a phase
Φ by which it is
1:08:00.753 --> 1:08:02.653
behind the driving voltage.
1:08:02.650 --> 1:08:05.070
The magnitude of Z for
as in any complex number,
1:08:05.070 --> 1:08:07.090
if someone says,
"What's the magnitude of
1:08:07.088 --> 1:08:07.938
Z?"
1:08:07.940 --> 1:08:18.990
it is simply square root of
real part squared imaginary part
1:08:18.987 --> 1:08:20.857
squared.
1:08:20.859 --> 1:08:26.369
So I want you to think about
the magic of complex numbers.
1:08:26.368 --> 1:08:30.608
Your final answer has a current
which is a cosωt -
1:08:30.613 --> 1:08:31.643
Φ.
1:08:31.640 --> 1:08:34.720
Your driving voltage is a
cosωt.
1:08:34.720 --> 1:08:38.340
There is no way you can divide
the voltage by anything to get
1:08:38.336 --> 1:08:39.236
that current.
1:08:39.238 --> 1:08:44.618
But in the complex language,
the complex current is the
1:08:44.623 --> 1:08:49.713
complex voltage divided by the
complex impedance.
1:08:49.710 --> 1:08:50.710
How does the happen?
1:08:50.710 --> 1:08:54.550
That happens because impendence
Z has got a magnitude and
1:08:54.548 --> 1:08:56.768
a phase,
therefore if you had a voltage
1:08:56.765 --> 1:08:59.235
V_0
e^(iωt) and you divide
1:08:59.240 --> 1:09:01.820
by mod Z e^(i)
^(Φ) , that
1:09:01.819 --> 1:09:04.499
becomes e^(-i)
^(Φ) upstairs.
1:09:04.500 --> 1:09:08.770
You're able to change the
magnitude and you're able to
1:09:08.770 --> 1:09:11.270
change the phase by dividing.
1:09:11.270 --> 1:09:14.380
So if you want to take the
voltage and you want to rescale
1:09:14.380 --> 1:09:16.710
it and shift its phase,
all in one shot,
1:09:16.711 --> 1:09:19.561
you can do it if you divide by
a complex number,
1:09:19.560 --> 1:09:23.970
because a complex number will
rescale it and also shift its
1:09:23.970 --> 1:09:24.580
phase.
1:09:24.579 --> 1:09:28.079
And then you take the real part.
1:09:28.078 --> 1:09:30.198
In other words,
cosωt divided by
1:09:30.203 --> 1:09:31.993
nothing will give you
cosωt -
1:09:31.988 --> 1:09:35.098
Φ,
but e^(iωt),
1:09:35.100 --> 1:09:37.010
when divided by
e^(i)^(Φ),
1:09:37.010 --> 1:09:38.670
will in fact give you
e^(iωt -
1:09:38.670 --> 1:09:39.720
i)^(Φ).
1:09:39.720 --> 1:09:43.060
So that phase shift we can
produce only by dividing with a
1:09:43.061 --> 1:09:44.061
complex number.
1:09:44.060 --> 1:09:47.630
So I'll come back and explain
to you a little bit more on how
1:09:47.627 --> 1:09:50.897
you do circuit problems using
the complex impedance,
1:09:50.899 --> 1:09:53.889
but I think it will be helpful
for those of you who don't know
1:09:53.887 --> 1:09:56.237
complex numbers to get use to
this part of it.
1:09:56.239 --> 1:10:01.999