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PROFESSOR: So welcome back.
00:00:48.310 --> 00:00:54.280
And today, I'm going to
do some examples of what
00:00:54.280 --> 00:00:58.790
happens when you have more
than one charge radiating.
00:00:58.790 --> 00:01:04.129
This is a very
important example,
00:01:04.129 --> 00:01:11.720
because whenever you have many
charges radiating, if it's
00:01:11.720 --> 00:01:16.720
coherent, you get the
phenomenon of interference,
00:01:16.720 --> 00:01:20.870
and it plays a very important
role in many fields of physics,
00:01:20.870 --> 00:01:23.250
in particular, in optics.
00:01:23.250 --> 00:01:26.330
You may have heard terms
like interference patterns,
00:01:26.330 --> 00:01:28.230
diffraction, et cetera.
00:01:28.230 --> 00:01:36.600
All that is to do with when
you have more than one source
00:01:36.600 --> 00:01:38.130
radiating.
00:01:38.130 --> 00:01:42.990
So the example I'm going to
consider is the following.
00:01:42.990 --> 00:01:49.090
Suppose you have
two charges, both q.
00:01:49.090 --> 00:01:50.810
Each one is oscillating.
00:01:55.020 --> 00:01:59.190
It's oscillating with an angular
frequency 2 pi over lambda.
00:01:59.190 --> 00:02:02.340
This tells you the
position of the charge
00:02:02.340 --> 00:02:04.010
as a function of time.
00:02:04.010 --> 00:02:07.380
So it's centered to
distance d from here
00:02:07.380 --> 00:02:11.380
and oscillating
within amplitude a
00:02:11.380 --> 00:02:15.730
as a cosine 2 pi over lambda t.
00:02:15.730 --> 00:02:18.750
Well, lambda is, of
course, the wavelength
00:02:18.750 --> 00:02:21.330
of the radiated
electromagnetic wave.
00:02:29.630 --> 00:02:33.490
So this one is oscillating
up and down like that,
00:02:33.490 --> 00:02:37.990
and there is another one over
here oscillating just for fun
00:02:37.990 --> 00:02:39.940
in the other direction.
00:02:39.940 --> 00:02:44.350
This one is oscillating
in along the x-axis,
00:02:44.350 --> 00:02:49.530
and this one is oscillating
along the y-axis.
00:02:49.530 --> 00:02:52.600
And here is the
formula, which tells you
00:02:52.600 --> 00:02:57.970
what is the position of this one
at the given instant of time.
00:02:57.970 --> 00:03:00.940
They're both given in
terms of the same time,
00:03:00.940 --> 00:03:02.830
with the same time.
00:03:02.830 --> 00:03:03.850
t equals 0.
00:03:03.850 --> 00:03:08.290
Therefore, these are
two coherent sources.
00:03:08.290 --> 00:03:12.850
The question is, what will
be the resultant field
00:03:12.850 --> 00:03:14.800
anywhere in space?
00:03:14.800 --> 00:03:20.640
At some point we want to know
what is the resultant field.
00:03:20.640 --> 00:03:25.770
Now, each one of these--
you know what it does.
00:03:25.770 --> 00:03:32.230
It radiates a spherical
wave with a certain angular
00:03:32.230 --> 00:03:34.060
distribution.
00:03:34.060 --> 00:03:34.560
Alright?
00:03:37.280 --> 00:03:41.400
And the waves overlap in space.
00:03:41.400 --> 00:03:45.210
Now because the
electromagnetic waves
00:03:45.210 --> 00:03:48.920
are solutions of the
electromagnetic wave
00:03:48.920 --> 00:03:55.890
equation, which is a
linear equation, if we have
00:03:55.890 --> 00:04:00.010
a coherent source of
radiation at any point,
00:04:00.010 --> 00:04:06.300
we simply add vectorially
the electromagnetic waves.
00:04:06.300 --> 00:04:09.610
The electric fields
you add vectorially
00:04:09.610 --> 00:04:11.145
all the magnetic field.
00:04:14.320 --> 00:04:25.910
Now in principle, we should
go-- to first principles.
00:04:25.910 --> 00:04:27.920
Consider Maxwell's equations.
00:04:27.920 --> 00:04:32.190
Consider what happens when
a charge is oscillating.
00:04:36.740 --> 00:04:41.110
And in fact, consider
both charges,
00:04:41.110 --> 00:04:43.260
what they're doing at
any instant of time,
00:04:43.260 --> 00:04:47.380
and solve in all of
space Maxwell's equation
00:04:47.380 --> 00:04:50.790
to see what the electric
field is everywhere.
00:04:50.790 --> 00:04:56.400
We build on our experience,
and what we will do
00:04:56.400 --> 00:05:02.170
is we will assume that we've
solved the problem of what
00:05:02.170 --> 00:05:06.710
happens when this single
charge is accelerating.
00:05:06.710 --> 00:05:09.100
We've done this several times
in the [? PAR sensor ?].
00:05:09.100 --> 00:05:11.940
Professor Walter Lewin
has done it in the course.
00:05:11.940 --> 00:05:18.640
Whenever a charge is
accelerating, it radiates.
00:05:18.640 --> 00:05:21.210
And this is the
formula, which tells
00:05:21.210 --> 00:05:26.370
you what is the electric field
from the given charge, which
00:05:26.370 --> 00:05:31.640
is accelerating at
some position, r and t.
00:05:31.640 --> 00:05:36.090
I'm just reminding
you that what we find
00:05:36.090 --> 00:05:44.120
is that the electric field at
any position at some time t
00:05:44.120 --> 00:05:50.870
is given by, or related
to, the perpendicular
00:05:50.870 --> 00:05:56.050
component of the
acceleration of the charge.
00:05:56.050 --> 00:06:01.650
It's perpendicular to the
direction of propagation,
00:06:01.650 --> 00:06:09.010
but at a point t in space,
the electric field there
00:06:09.010 --> 00:06:15.420
is related to the acceleration
at an earlier time.
00:06:15.420 --> 00:06:20.320
At the time, t prime, which
is earlier than the time when
00:06:20.320 --> 00:06:27.390
we are considering
by the amount r/c.
00:06:27.390 --> 00:06:31.450
All that this does--
it reflects the fact
00:06:31.450 --> 00:06:34.710
that if I have some with
an accelerated charge,
00:06:34.710 --> 00:06:37.610
at that instant, the
electromagnetic wave starts
00:06:37.610 --> 00:06:46.780
to propagate, and it takes a
time r/c to reach a distance r.
00:06:46.780 --> 00:06:50.250
That's all this is telling us.
00:06:50.250 --> 00:06:53.680
The other thing this is
telling us-- the wave front
00:06:53.680 --> 00:06:56.260
is a sphere.
00:06:56.260 --> 00:07:01.470
The amplitude of the electric
field drops off like 1/r.
00:07:01.470 --> 00:07:05.920
And it is not uniform
in all direction,
00:07:05.920 --> 00:07:08.480
as we've seen in the past.
00:07:08.480 --> 00:07:09.450
Right?
00:07:09.450 --> 00:07:12.450
For example, if I
take this charge
00:07:12.450 --> 00:07:18.350
as it's oscillating up and down,
the maximum amplitude radiated
00:07:18.350 --> 00:07:22.400
is in this plane
where we are looking,
00:07:22.400 --> 00:07:24.710
where the
perpendicular component
00:07:24.710 --> 00:07:28.190
of the acceleration of
this is the maximum.
00:07:28.190 --> 00:07:34.000
And its minimum upwards is
0, because this acceleration
00:07:34.000 --> 00:07:37.550
has no component
of x perpendicular
00:07:37.550 --> 00:07:41.000
component to this direction.
00:07:41.000 --> 00:07:44.470
And similarly, for this.
00:07:44.470 --> 00:07:46.900
The other thing which
I wish to point out
00:07:46.900 --> 00:07:53.730
is that the polarization of
the electromagnetic wave,
00:07:53.730 --> 00:07:57.640
the electric vector, is
parallel to the perpendicular
00:07:57.640 --> 00:07:59.560
component of a.
00:07:59.560 --> 00:08:03.630
So from this, the
electric vector,
00:08:03.630 --> 00:08:06.490
when it gets to
here or here, will
00:08:06.490 --> 00:08:10.630
be in the direction of
x, while from this one,
00:08:10.630 --> 00:08:14.540
it'll polarized in
the direction y.
00:08:14.540 --> 00:08:16.540
The other thing which
I want to stress,
00:08:16.540 --> 00:08:19.830
that we will in the
other examples--
00:08:19.830 --> 00:08:22.520
always consider
a situation where
00:08:22.520 --> 00:08:27.670
the point where we're looking
at the electromagnetic field
00:08:27.670 --> 00:08:32.500
is far away from
where the charges are.
00:08:32.500 --> 00:08:36.720
The formula, which I
just showed you here,
00:08:36.720 --> 00:08:39.809
applies only in the far field.
00:08:39.809 --> 00:08:42.840
Nearby, the electric field
and the magnetic field
00:08:42.840 --> 00:08:44.580
are very complicated.
00:08:44.580 --> 00:08:47.900
They can be calculated, but they
cannot be summarized in that
00:08:47.900 --> 00:08:48.980
simple form.
00:08:48.980 --> 00:08:54.030
But if I'm far away from the
accelerating charge then,
00:08:54.030 --> 00:08:58.380
it is possible to write
it in this simple form.
00:08:58.380 --> 00:09:05.330
So our discussion is only valid
if the wave length radiated
00:09:05.330 --> 00:09:09.060
is much smaller
than the distance
00:09:09.060 --> 00:09:10.365
from where we're considering.
00:09:13.590 --> 00:09:18.700
This distance is also
much smaller than that.
00:09:18.700 --> 00:09:21.950
And this is to allow us
to approximate sine theta
00:09:21.950 --> 00:09:23.710
by theta, et cetera.
00:09:23.710 --> 00:09:25.730
And that's the case, alright?
00:09:28.380 --> 00:09:29.760
OK.
00:09:29.760 --> 00:09:34.680
With that, we can now
immediately solve the problem.
00:09:34.680 --> 00:09:40.040
Simply consider the
acceleration of each charge
00:09:40.040 --> 00:09:43.630
from our knowledge of
where the charge is.
00:09:43.630 --> 00:09:46.140
We can differentiate
it twice, and we
00:09:46.140 --> 00:09:47.570
can get the acceleration.
00:09:47.570 --> 00:09:52.785
So we know that the acceleration
of the first charge,
00:09:52.785 --> 00:09:55.360
which is oscillating
along the x-axis,
00:09:55.360 --> 00:09:58.290
is given by minus
a omega squared
00:09:58.290 --> 00:10:01.400
cosine omega in the x-direction.
00:10:01.400 --> 00:10:03.480
I want to emphasize,
because this
00:10:03.480 --> 00:10:09.760
is in the same direction as the
perpendicular component of a1.
00:10:14.240 --> 00:10:15.820
And, I'm sorry.
00:10:15.820 --> 00:10:17.110
I misspoke.
00:10:17.110 --> 00:10:18.950
The answer-- what
I said is correct
00:10:18.950 --> 00:10:20.120
but for the wrong reason.
00:10:20.120 --> 00:10:25.760
In this case, by definition, the
acceleration of the first one
00:10:25.760 --> 00:10:31.450
is this because the
displacement of the first one
00:10:31.450 --> 00:10:33.260
is in the x-direction.
00:10:33.260 --> 00:10:36.310
Similarly, the acceleration
on the second one
00:10:36.310 --> 00:10:39.390
is in the y-direction,
because the displacement
00:10:39.390 --> 00:10:41.800
is in the y-direction.
00:10:41.800 --> 00:10:42.370
Alright?
00:10:42.370 --> 00:10:44.970
So these are the
two accelerations.
00:10:44.970 --> 00:10:49.170
This is just some--
occasionally I'll use omega,
00:10:49.170 --> 00:10:52.580
occasionally lambda just
to remind you of that.
00:10:52.580 --> 00:10:53.080
OK.
00:10:53.080 --> 00:10:59.736
So all I now have to do
for each of the charges
00:10:59.736 --> 00:11:05.170
is calculate this
quantity at the position
00:11:05.170 --> 00:11:08.720
where I want to calculate
the electric and the magnetic
00:11:08.720 --> 00:11:10.040
field.
00:11:10.040 --> 00:11:13.040
So, first of all
in this problem,
00:11:13.040 --> 00:11:15.170
we're asked to do
it in two places.
00:11:15.170 --> 00:11:18.120
We're asked to do
it at position p1
00:11:18.120 --> 00:11:21.290
and at p2. p1 is
along the z-axis,
00:11:21.290 --> 00:11:24.450
and p2 is off the z-axis.
00:11:24.450 --> 00:11:27.570
First, let's do it for p1.
00:11:27.570 --> 00:11:33.740
So, the electric field,
due to the first charge
00:11:33.740 --> 00:11:40.780
at position p1 at time t,
is given by this formula
00:11:40.780 --> 00:11:49.610
if I insert in this for the
distance r, which is L. OK?
00:11:49.610 --> 00:11:54.050
And I put in the
acceleration of this one
00:11:54.050 --> 00:11:59.400
and the perpendicular
component of the acceleration
00:11:59.400 --> 00:12:04.870
perpendicular to the line
drawing my two charges.
00:12:04.870 --> 00:12:08.300
And the position
p1 is-- in fact,
00:12:08.300 --> 00:12:10.250
that line is along the z-axis.
00:12:10.250 --> 00:12:13.790
So the perpendicular
direction is, in fact,
00:12:13.790 --> 00:12:15.650
in the x-direction.
00:12:15.650 --> 00:12:16.150
OK.
00:12:16.150 --> 00:12:21.670
So I don't even have to take
the-- the actual acceleration
00:12:21.670 --> 00:12:23.870
is the perpendicular component.
00:12:23.870 --> 00:12:24.580
OK.
00:12:24.580 --> 00:12:30.570
And I have to calculate
it at a time which
00:12:30.570 --> 00:12:40.230
is t minus the distance p1 is
from in my charges, which is L.
00:12:40.230 --> 00:12:43.730
So it's minus omega L over c.
00:12:43.730 --> 00:12:47.660
I calculate t prime in here.
00:12:47.660 --> 00:12:51.620
Notice, I have not
tried to calculate
00:12:51.620 --> 00:12:56.110
the distance between the
charge and p1 exactly,
00:12:56.110 --> 00:13:00.060
because I told you that
the distance d is very much
00:13:00.060 --> 00:13:05.900
smaller than L. And so the
overall distance is negligently
00:13:05.900 --> 00:13:11.250
different from being
just L. So that's
00:13:11.250 --> 00:13:13.540
the electric field
due to the first one.
00:13:13.540 --> 00:13:17.250
Similarly, I can do the
same for the second one.
00:13:17.250 --> 00:13:19.150
Everything is the same.
00:13:19.150 --> 00:13:22.650
The only difference
in that case is
00:13:22.650 --> 00:13:27.720
that now the electric
vector-- the acceleration
00:13:27.720 --> 00:13:32.470
is along the y-axis and
therefore, the electric field
00:13:32.470 --> 00:13:35.710
is polarized along
the y-direction.
00:13:35.710 --> 00:13:42.290
So at the point p1, the two
electric fields are the same.
00:13:42.290 --> 00:13:44.830
They're both
oscillating in phase
00:13:44.830 --> 00:13:47.260
with the same
frequency, et cetera.
00:13:47.260 --> 00:13:50.270
But this one is pointing
in the x-direction.
00:13:50.270 --> 00:13:52.940
This in the y-direction.
00:13:52.940 --> 00:13:56.870
Now, as I mentioned at the
beginning, if at any point
00:13:56.870 --> 00:14:01.450
you have electric field
due to two sources,
00:14:01.450 --> 00:14:05.780
they simply add because the
system is a linear system.
00:14:05.780 --> 00:14:08.800
So we simply have to
add the electric fields.
00:14:08.800 --> 00:14:11.060
Electric fields are
vectors, therefore
00:14:11.060 --> 00:14:16.390
we don't add them
algebraic as scalars,
00:14:16.390 --> 00:14:19.750
but we have to add
them vectorially.
00:14:19.750 --> 00:14:24.220
And so, we take this vector
and that vector and add it.
00:14:24.220 --> 00:14:28.910
And that gives us the total
electric field at point p1.
00:14:28.910 --> 00:14:32.390
And simply from here,
I get d0 like that,
00:14:32.390 --> 00:14:38.840
which I can rewrite in
terms of a unit vector.
00:14:38.840 --> 00:14:39.750
Alright?
00:14:39.750 --> 00:14:45.980
This is a unit vector,
which is at 45 degrees.
00:14:45.980 --> 00:14:51.680
In other words, if that's
the z-direction x and y,
00:14:51.680 --> 00:14:56.180
it's at 45 degrees to
both the x and y-axes.
00:14:56.180 --> 00:14:57.830
That's the unit vector.
00:14:57.830 --> 00:14:59.780
This is the amplitude of it.
00:14:59.780 --> 00:15:05.280
And this is d-- tell us what
is the oscillating frequency
00:15:05.280 --> 00:15:08.730
and what is the phase.
00:15:08.730 --> 00:15:13.670
Notice, although each
one of these sources
00:15:13.670 --> 00:15:19.430
radiated there an electric
field of magnitude E0,
00:15:19.430 --> 00:15:25.080
when I add them, I did not get
to 2E0, because as I mentioned,
00:15:25.080 --> 00:15:26.980
electric fields
are vectors, and we
00:15:26.980 --> 00:15:28.550
have to add them vectorially.
00:15:28.550 --> 00:15:31.770
And that's why you
get here the root 2.
00:15:31.770 --> 00:15:36.940
So this is the first
example, which I've done.
00:15:36.940 --> 00:15:40.870
Now I want to take
the same geometry
00:15:40.870 --> 00:15:46.050
but calculate what is
the electric field.
00:15:46.050 --> 00:15:49.170
What we've just done,
we've calculated here.
00:15:49.170 --> 00:15:51.750
Now I want to do
the same situation--
00:15:51.750 --> 00:15:57.530
two sources
oscillating coherently.
00:15:57.530 --> 00:16:01.240
But I want to look at what is
the electric field at the point
00:16:01.240 --> 00:16:02.560
p2.
00:16:02.560 --> 00:16:09.620
And I've taken p2 to be
in that the xz-plane,
00:16:09.620 --> 00:16:14.986
but in a direction that's
an angle lambda over a to d.
00:16:14.986 --> 00:16:16.900
d is this distance.
00:16:16.900 --> 00:16:22.490
Lambda is the wavelength of the
radiated electromagnetic wave.
00:16:22.490 --> 00:16:25.760
And I've taken this
crazy number simply
00:16:25.760 --> 00:16:29.690
to make the arithmetic
come out easier at the end.
00:16:29.690 --> 00:16:34.020
And so we now want to calculate
the electric field here.
00:16:34.020 --> 00:16:37.320
And we do exactly the same
except we go a little faster.
00:16:37.320 --> 00:16:41.330
What I will do is I will
calculate the electric field
00:16:41.330 --> 00:16:42.310
here.
00:16:42.310 --> 00:16:45.610
Due to this charge
oscillating, I'll
00:16:45.610 --> 00:16:47.990
then calculate the
electric field here.
00:16:47.990 --> 00:16:51.120
Due to this charge, this one
is oscillating like that.
00:16:51.120 --> 00:16:54.210
I'm reminding you this one
is oscillating like that.
00:16:54.210 --> 00:16:57.740
And we will vectorially
add the two.
00:16:57.740 --> 00:17:02.790
Now, from the point of
view of the amplitude,
00:17:02.790 --> 00:17:10.630
the difference in distance
between this and here or this
00:17:10.630 --> 00:17:15.940
and here or this and
here is insignificant.
00:17:15.940 --> 00:17:19.270
And so I have ignored that
in the previous calculations,
00:17:19.270 --> 00:17:21.599
and I'll do the same here.
00:17:21.599 --> 00:17:28.470
But when you add
to the two, we have
00:17:28.470 --> 00:17:30.990
to worry about
the relative phase
00:17:30.990 --> 00:17:34.050
of the alternating
the electric fields.
00:17:34.050 --> 00:17:38.370
And the difference of
this distance compared
00:17:38.370 --> 00:17:44.360
to that distance is no
longer insignificant
00:17:44.360 --> 00:17:50.700
when it comes to the relative
phase of the two radiations.
00:17:50.700 --> 00:17:58.490
So, in this formula when I'm
calculating the electric field,
00:17:58.490 --> 00:18:03.630
this r-- it doesn't
matter whether I
00:18:03.630 --> 00:18:07.390
call this L or L plus
a little bit or less.
00:18:07.390 --> 00:18:09.730
This doesn't change very much.
00:18:09.730 --> 00:18:19.330
But in calculating the
phase of the radiation,
00:18:19.330 --> 00:18:21.990
I cannot ignore that.
00:18:21.990 --> 00:18:25.350
In the previous case,
the phase was the same
00:18:25.350 --> 00:18:28.890
because this distance
and that distance
00:18:28.890 --> 00:18:32.050
are exactly the
same by symmetry.
00:18:32.050 --> 00:18:37.660
For this position, that
distance and that distance
00:18:37.660 --> 00:18:39.380
are not the same.
00:18:39.380 --> 00:18:45.670
And so I what I will do is I'll
use as reference that distance,
00:18:45.670 --> 00:18:48.420
and when I'm calculating
this distance,
00:18:48.420 --> 00:18:52.870
I will calculate by how much
I have to subtract from here
00:18:52.870 --> 00:18:57.690
to end up with this length
and for this one, by how much
00:18:57.690 --> 00:19:01.750
I've add to it to get to here.
00:19:01.750 --> 00:19:05.590
Pay attention to that when
I'm doing that in a second.
00:19:05.590 --> 00:19:11.410
And the rest is very
straightforward and similar
00:19:11.410 --> 00:19:12.740
to what I've just done.
00:19:17.960 --> 00:19:21.650
So now, as I say, I want to
calculate the electric field
00:19:21.650 --> 00:19:25.510
at the point p2, which is this.
00:19:25.510 --> 00:19:28.350
This is the x-coordinate,
y, and z-coordinate.
00:19:28.350 --> 00:19:31.950
And I pointed it out
to you a second ago.
00:19:31.950 --> 00:19:32.450
Alright.
00:19:36.050 --> 00:19:41.210
So I don't have to rewrite
all those q's and omegas,
00:19:41.210 --> 00:19:44.290
et cetera, by
analogy, what we've
00:19:44.290 --> 00:19:52.290
done-- the electric field due to
the first charge at position p2
00:19:52.290 --> 00:19:55.845
will be that the
amplitude that we've got
00:19:55.845 --> 00:19:59.530
is the same for the
previous case of P1.
00:19:59.530 --> 00:20:02.340
So it's-- E0 is the amplitude.
00:20:02.340 --> 00:20:07.800
It's pointing in the
x-direction as before.
00:20:07.800 --> 00:20:18.780
And we know that the description
of the electric field
00:20:18.780 --> 00:20:28.745
will be given by cosine omega
t1 prime where t is-- so far
00:20:28.745 --> 00:20:30.660
I've always talked
about t prime,
00:20:30.660 --> 00:20:34.740
because I only had one when
I was considering just one
00:20:34.740 --> 00:20:35.490
charge.
00:20:35.490 --> 00:20:42.680
But here the t1 is
different to t2.
00:20:42.680 --> 00:20:51.920
So quickly, this term comes
from the perpendicular component
00:20:51.920 --> 00:20:55.630
of a right as before.
00:20:55.630 --> 00:21:00.770
And t1 prime is
equal to the time
00:21:00.770 --> 00:21:11.120
when I'm looking at the electric
field minus the time it takes
00:21:11.120 --> 00:21:17.030
for this signal from the
charge to get to the point
00:21:17.030 --> 00:21:20.850
where I'm looking at the
electric field to point p2.
00:21:20.850 --> 00:21:26.320
So, it will be-- as I
told you a second ago,
00:21:26.320 --> 00:21:29.440
I'll take L as the reference.
00:21:29.440 --> 00:21:35.940
And I'm subtracting from it that
distance d sine lambda over 8d.
00:21:35.940 --> 00:21:38.380
For a second, let me
go back to that picture
00:21:38.380 --> 00:21:42.080
because you can
easily get confused.
00:21:42.080 --> 00:21:44.790
So I'm coming back here.
00:21:44.790 --> 00:21:53.160
What we're trying to calculate--
the distance from here to here.
00:21:53.160 --> 00:21:58.400
So I'm taking this
distance, which is L,
00:21:58.400 --> 00:22:04.680
and I'm subtracting from
it this distance, which
00:22:04.680 --> 00:22:11.500
is d sine this angle here,
which is the same as this angle.
00:22:11.500 --> 00:22:15.150
So that's what I'm
calculating there.
00:22:15.150 --> 00:22:21.840
So this is L minus d sine
lambda over 8d divided by c.
00:22:21.840 --> 00:22:26.570
Then the electric field is
then immediately followed
00:22:26.570 --> 00:22:33.540
from this d0 in the x-direction,
cosine omega t minus this.
00:22:33.540 --> 00:22:37.990
And if you assume that
this angle is small,
00:22:37.990 --> 00:22:43.470
and I told you that that's
given in the problem,
00:22:43.470 --> 00:22:46.150
the sine of an angle
is equal to an angle.
00:22:46.150 --> 00:22:48.030
Approximately, they're small.
00:22:48.030 --> 00:22:52.500
And if I multiply this out,
I simply get an angle here.
00:22:52.500 --> 00:22:54.320
That's pi/4.
00:22:54.320 --> 00:22:59.680
So I find that the
electric field at point p2
00:22:59.680 --> 00:23:03.620
is pointing, due to the
first charge, is pointing
00:23:03.620 --> 00:23:05.630
in the x-direction,
as you'd expect,
00:23:05.630 --> 00:23:08.350
because of the direction
which the charge is moving
00:23:08.350 --> 00:23:14.085
times E0 times cosine omega
t minus omega L over c
00:23:14.085 --> 00:23:20.120
plus a phase like that.
00:23:20.120 --> 00:23:22.320
How about from the other charge?
00:23:22.320 --> 00:23:25.800
If I take the other
charge, everything
00:23:25.800 --> 00:23:30.720
will be the same except now
this charge is oscillating
00:23:30.720 --> 00:23:34.670
in the y-direction, so
the electric field will
00:23:34.670 --> 00:23:36.960
be in the y-direction.
00:23:36.960 --> 00:23:42.070
And this time,
it's omega t2 prime
00:23:42.070 --> 00:23:46.890
where t2 will be
very similar to t1
00:23:46.890 --> 00:23:52.180
but now the distance is
greater by this amount.
00:23:52.180 --> 00:23:55.350
For a second, let's go
back to this picture
00:23:55.350 --> 00:23:57.810
so you see what
I'm talking about.
00:23:57.810 --> 00:24:02.360
If I calculate
this distance, it's
00:24:02.360 --> 00:24:12.400
the same as that distance
plus d sine this angle, which
00:24:12.400 --> 00:24:17.340
is this lambda over 8d,
and I'm subtracting.
00:24:17.340 --> 00:24:20.390
And I want to
emphasize that what
00:24:20.390 --> 00:24:23.740
I'm focusing on here,
what is important,
00:24:23.740 --> 00:24:28.640
is the difference
between this and that
00:24:28.640 --> 00:24:31.040
and not the absolute value.
00:24:31.040 --> 00:24:35.640
So when I'm
approximating-- if you
00:24:35.640 --> 00:24:37.160
do this calculation
for yourself,
00:24:37.160 --> 00:24:39.540
you'll see I'm doing
a slight approximation
00:24:39.540 --> 00:24:40.750
to calculate that.
00:24:40.750 --> 00:24:44.530
But this and this is
the same quantity.
00:24:44.530 --> 00:24:47.610
But the important thing is
the difference of this phase
00:24:47.610 --> 00:24:49.320
here is plus pi/4.
00:24:49.320 --> 00:24:52.330
Here is minus pi/4,
because here it's
00:24:52.330 --> 00:24:54.460
minus that little
bit of a distance,
00:24:54.460 --> 00:24:58.960
and here it's plus a
little bit of the distance.
00:24:58.960 --> 00:25:00.470
So we finished.
00:25:00.470 --> 00:25:05.820
The total electric
field at p2 and time t
00:25:05.820 --> 00:25:10.700
is the sum of the electric field
due to the first charge plus
00:25:10.700 --> 00:25:12.000
due to the second challenge.
00:25:15.170 --> 00:25:19.810
And as I mentioned a second ago,
electric fields are vectors.
00:25:19.810 --> 00:25:23.230
Thus, we have to add
these two vectorially.
00:25:23.230 --> 00:25:26.140
This is in the x-direction.
00:25:26.140 --> 00:25:28.350
This is in the y-direction.
00:25:28.350 --> 00:25:30.620
And so what we
have at that point
00:25:30.620 --> 00:25:34.850
is two electric oscillating
electric fields.
00:25:34.850 --> 00:25:39.430
They're oscillating
coherently but out of phase.
00:25:39.430 --> 00:25:41.700
This is a plus pi/4 phase.
00:25:41.700 --> 00:25:44.600
This is a minus pi/4 phase.
00:25:44.600 --> 00:25:48.900
So the difference of phase
between those two is pi/2.
00:25:48.900 --> 00:25:50.680
It's 90 degrees.
00:25:50.680 --> 00:25:55.860
So this is out of phase with,
one with respect to the other,
00:25:55.860 --> 00:25:57.280
by 90 degrees.
00:25:57.280 --> 00:26:01.740
They're coherent, but
90 degrees out of phase.
00:26:01.740 --> 00:26:04.510
The magnitudes are
the same, but this one
00:26:04.510 --> 00:26:07.830
is pointing in the x-direction
and this in the y-direction,
00:26:07.830 --> 00:26:10.900
and you know what
that corresponds to.
00:26:10.900 --> 00:26:15.480
If I have at the location
two electric fields, one
00:26:15.480 --> 00:26:20.410
doing that and the
other one doing this,
00:26:20.410 --> 00:26:27.960
if they're out of phase by 90
degrees, and I add the two,
00:26:27.960 --> 00:26:29.010
what do I get?
00:26:29.010 --> 00:26:34.010
I get a constant
size of a radius,
00:26:34.010 --> 00:26:35.900
and the thing is rotating.
00:26:35.900 --> 00:26:42.100
That is the description
of a rotating
00:26:42.100 --> 00:26:45.340
vector of magnitude E0.
00:26:45.340 --> 00:26:53.310
In the other case, at P1, there
was not this phase difference,
00:26:53.310 --> 00:26:59.280
so the two were-- one was
oscillating like this.
00:26:59.280 --> 00:27:02.140
The other one was
oscillating like that.
00:27:02.140 --> 00:27:05.330
But they were oscillating
at the same phase.
00:27:05.330 --> 00:27:07.230
So this is what they were doing.
00:27:07.230 --> 00:27:12.200
And if you add those up, clearly
you get a line diagonally.
00:27:12.200 --> 00:27:17.700
And that's why before, the
result was a linearly polarized
00:27:17.700 --> 00:27:21.990
electric field there at 45
degrees to the x and y-axis.
00:27:21.990 --> 00:27:28.740
In this case, these two are
out of phase by 90 degrees.
00:27:28.740 --> 00:27:32.440
So when one is doing
this, the other one
00:27:32.440 --> 00:27:33.790
is also but out of phase.
00:27:33.790 --> 00:27:35.420
And so when this
one's at the maximum,
00:27:35.420 --> 00:27:38.670
this one's at the minimum, and
so that's what's happening.
00:27:38.670 --> 00:27:39.810
So they're out of phase.
00:27:39.810 --> 00:27:42.030
And then if you
add the two up, you
00:27:42.030 --> 00:27:43.710
get something doing
this, which we
00:27:43.710 --> 00:27:46.000
called circularly
polarized light.
00:27:46.000 --> 00:27:48.600
And that's why I chose
that crazy angle,
00:27:48.600 --> 00:27:51.920
because it came out like this.
00:27:51.920 --> 00:27:53.910
So that's the end
of that problem.
00:27:53.910 --> 00:27:57.840
I will now do another
one, which to emphasize
00:27:57.840 --> 00:28:04.110
some technique-- a different
technique of doing it.
00:28:04.110 --> 00:28:04.850
OK.
00:28:04.850 --> 00:28:06.170
Let's move.
00:28:06.170 --> 00:28:12.300
So the next problem
is the following.
00:28:12.300 --> 00:28:20.670
We're again dealing with several
charges oscillating coherently.
00:28:20.670 --> 00:28:22.790
Their cohering sources.
00:28:22.790 --> 00:28:26.440
And we're asking, what
are the electric fields?
00:28:26.440 --> 00:28:27.920
Somewhere in space?
00:28:27.920 --> 00:28:29.880
And if we found the
electric field, of course,
00:28:29.880 --> 00:28:33.990
we could always calculate
the magnetic field
00:28:33.990 --> 00:28:37.420
at that location using Maxwell's
equations or our knowledge
00:28:37.420 --> 00:28:39.960
of the relation between
electromagnetic field
00:28:39.960 --> 00:28:43.790
in a progressive
electromagnetic way.
00:28:43.790 --> 00:28:47.540
Now I'll take three charges.
00:28:47.540 --> 00:28:50.630
But that's not the hope.
00:28:50.630 --> 00:28:55.780
It's more different, but
what this problem adds
00:28:55.780 --> 00:29:03.060
is I'll use it to show a
technique that is often used,
00:29:03.060 --> 00:29:05.910
which helps in the
solution of such problems.
00:29:05.910 --> 00:29:09.190
So the problem now
is the following.
00:29:09.190 --> 00:29:11.010
I have three charges.
00:29:11.010 --> 00:29:13.540
Each of the same magnitude.
00:29:13.540 --> 00:29:17.482
They're located along
the x-axis at position 0
00:29:17.482 --> 00:29:20.300
minus d and minus 2d.
00:29:20.300 --> 00:29:25.100
So these distances are the
same-- each distance d.
00:29:25.100 --> 00:29:31.260
And what the problem
is-- up to t equals 0,
00:29:31.260 --> 00:29:37.080
these three charges are
displaced from equilibrium.
00:29:37.080 --> 00:29:39.000
I put y is 0.
00:29:39.000 --> 00:29:40.715
So, at the height, y0 here.
00:29:44.060 --> 00:29:49.740
Then, at t0-- don't ask me how.
00:29:49.740 --> 00:29:56.190
Magically, I get these three
charges to start oscillating.
00:29:56.190 --> 00:30:00.720
Such that at time
yt, the displacement
00:30:00.720 --> 00:30:04.600
is y0 cosine omega t.
00:30:04.600 --> 00:30:09.590
And they continue-- this
continues like that forever.
00:30:09.590 --> 00:30:16.910
The question is, as a result
of this oscillation of charges,
00:30:16.910 --> 00:30:23.180
what will be the electric
field a long way away from here
00:30:23.180 --> 00:30:28.530
at position L along the x-axis,
so we call that position p.
00:30:28.530 --> 00:30:35.720
So the problem is, calculate
E at position p for all times.
00:30:35.720 --> 00:30:38.730
Once we've calculated, we could
calculate the magnetic field,
00:30:38.730 --> 00:30:43.060
But to save you time--I mean,
you could do it for yourself.
00:30:43.060 --> 00:30:45.220
You know if you calculate
the electric field,
00:30:45.220 --> 00:30:47.390
there is a progressive
wave over here.
00:30:47.390 --> 00:30:49.790
if E is like this,
then b will be
00:30:49.790 --> 00:30:52.250
perpendicular to it
of the amplitude which
00:30:52.250 --> 00:30:54.300
is just a dE over c.
00:30:54.300 --> 00:30:57.350
So that will be straightforward,
so I'm not asking it.
00:31:00.030 --> 00:31:02.280
Now, let's just
think for a second
00:31:02.280 --> 00:31:04.070
what goes on in this problem.
00:31:04.070 --> 00:31:07.820
Initially, the charges
are stationary,
00:31:07.820 --> 00:31:10.290
so there will be a
Coulomb field around.
00:31:10.290 --> 00:31:14.210
But we are specifically asked
to ignore static fields.
00:31:14.210 --> 00:31:18.440
So in the last problem too,
I ignored static fields.
00:31:18.440 --> 00:31:22.590
We were only considering
the time-dependent fields.
00:31:25.410 --> 00:31:29.200
You can always have superimposed
on the time-dependent field
00:31:29.200 --> 00:31:33.690
some static field that
doesn't add anything.
00:31:33.690 --> 00:31:39.620
So what we have here is at
time up to time t equals 0,
00:31:39.620 --> 00:31:41.030
the charges are here.
00:31:41.030 --> 00:31:43.450
So let's take one
of the charges here.
00:31:43.450 --> 00:31:47.920
Then it starts moving.
00:31:47.920 --> 00:31:51.880
It will have an
acceleration, which
00:31:51.880 --> 00:31:54.750
is perpendicular to the
direction in which I'm
00:31:54.750 --> 00:31:58.990
interested the propagation
of the electromagnetic wave.
00:31:58.990 --> 00:32:02.840
So it will certainly
radiate in this direction.
00:32:02.840 --> 00:32:08.860
So this charge, which was
oscillating, will radiate.
00:32:08.860 --> 00:32:12.470
Over here at the point
p, the electric field
00:32:12.470 --> 00:32:14.690
will be initially 0.
00:32:14.690 --> 00:32:23.380
And it will continue being 0
until the electric field, which
00:32:23.380 --> 00:32:28.530
is generated here,
propagates that distance.
00:32:28.530 --> 00:32:42.970
So, only after a time, L/c, will
the electric vector get here?
00:32:42.970 --> 00:32:49.440
So up to the time L/c, there
will be no electric field here.
00:32:53.670 --> 00:32:55.230
How about this charge?
00:32:55.230 --> 00:32:59.520
This charge is initially
is displaced at y0.
00:32:59.520 --> 00:33:02.990
Then it starts
oscillating the same.
00:33:02.990 --> 00:33:05.670
Initially, it's stationary.
00:33:05.670 --> 00:33:08.450
It'll produce a Coulomb field,
which is a static field.
00:33:08.450 --> 00:33:10.330
We're not interested in it.
00:33:10.330 --> 00:33:16.010
But once it starts accelerating,
it starts radiating.
00:33:16.010 --> 00:33:18.800
And that radiating progresses.
00:33:18.800 --> 00:33:23.410
So that will also produce
a field over here.
00:33:23.410 --> 00:33:29.660
But since these two
distances are different,
00:33:29.660 --> 00:33:32.700
the radiation from here, first
of all, will get there a little
00:33:32.700 --> 00:33:33.970
later.
00:33:33.970 --> 00:33:39.240
But also, once it gets here,
it will have a different phase
00:33:39.240 --> 00:33:43.780
because the radiation
traveled a different distance.
00:33:43.780 --> 00:33:45.420
And same for the last one.
00:33:48.190 --> 00:33:50.520
So now, how do we do this?
00:33:54.030 --> 00:33:58.350
I almost sound like
a broken record.
00:33:58.350 --> 00:34:02.240
As always, I can calculate
the electric field
00:34:02.240 --> 00:34:06.940
from each charge, and it'll be
given by this formula, which
00:34:06.940 --> 00:34:11.620
we've seen over and over
again where t prime is
00:34:11.620 --> 00:34:17.469
the distance from the
charge to the point p.
00:34:17.469 --> 00:34:21.880
Furthermore, I know the
perpendicular acceleration
00:34:21.880 --> 00:34:24.719
and also the perpendicular
component of it,
00:34:24.719 --> 00:34:27.560
because in this problem,
the perpendicular component
00:34:27.560 --> 00:34:29.404
is the same as the
actual acceleration.
00:34:32.429 --> 00:34:37.380
After time t equals
0, the acceleration
00:34:37.380 --> 00:34:42.630
of every one of these charges
a1, a2, a3, is the same.
00:34:42.630 --> 00:34:44.179
It's the same direction.
00:34:44.179 --> 00:34:46.980
And it's given by
that simply because we
00:34:46.980 --> 00:34:50.620
know what is the
displacement y of t.
00:34:50.620 --> 00:34:54.389
If I differentiate it twice,
I get the acceleration.
00:34:54.389 --> 00:34:58.460
If I plug this into
here for each charge,
00:34:58.460 --> 00:35:02.400
I will know what is the electric
fields from each charge.
00:35:02.400 --> 00:35:06.550
I can then add the electric
fields from each charge,
00:35:06.550 --> 00:35:09.730
and I'll get the
total electric field.
00:35:09.730 --> 00:35:12.380
You have to add
them vectorially.
00:35:12.380 --> 00:35:18.900
So, once again, for
t less than L/c,
00:35:18.900 --> 00:35:23.880
the electric field at
position p will be 0.
00:35:23.880 --> 00:35:27.820
There was no earlier
accelerated charge
00:35:27.820 --> 00:35:30.830
which radiated an
electric field which
00:35:30.830 --> 00:35:35.110
got here before this time.
00:35:35.110 --> 00:35:37.330
So that's nice and easy.
00:35:37.330 --> 00:35:45.110
How about for a later period,
a period between L/c or just
00:35:45.110 --> 00:35:47.010
after L/c.
00:35:47.010 --> 00:35:53.320
For t just after
L/c, the acceleration
00:35:53.320 --> 00:36:01.870
of the charge at x equals 0
would produce an electric field
00:36:01.870 --> 00:36:07.720
which propagated and would
have got to my point p.
00:36:07.720 --> 00:36:14.220
So between that time
L/c and the time when
00:36:14.220 --> 00:36:20.420
the radiation from the second
charge got to the point p,
00:36:20.420 --> 00:36:24.050
I will have an
electric field but only
00:36:24.050 --> 00:36:28.470
from the first radiation
due to the first charge.
00:36:28.470 --> 00:36:32.630
I can forget the second
and third charge at minus d
00:36:32.630 --> 00:36:38.530
and a minus 2d but not
the charge at x equals 0.
00:36:38.530 --> 00:36:40.980
So that will be I
use this formula.
00:36:40.980 --> 00:36:41.830
I plug it in.
00:36:41.830 --> 00:36:43.880
I use the acceleration of a1.
00:36:43.880 --> 00:36:50.771
I put it in here, and I know
that t prime is t minus r/c.
00:36:50.771 --> 00:36:55.300
r in this case is
L. So this formula
00:36:55.300 --> 00:37:00.180
immediately tells me that's
what it is and lets me save--
00:37:00.180 --> 00:37:02.640
I don't want to write
this over and over again.
00:37:02.640 --> 00:37:04.400
This-- I'll call that E0.
00:37:04.400 --> 00:37:08.310
So from the first
charge, this is
00:37:08.310 --> 00:37:12.600
what the electric field-- it's
an oscillating electric field--
00:37:12.600 --> 00:37:15.440
what it looks like
at position p.
00:37:15.440 --> 00:37:24.350
But I'm reminding you, this is
only true for this time window
00:37:24.350 --> 00:37:27.410
when the radiation
from the charge at q
00:37:27.410 --> 00:37:33.700
equals 0 has got to point p,
but from the other one's not.
00:37:33.700 --> 00:37:36.880
How about from a little later?
00:37:36.880 --> 00:37:42.040
From the time L/c plus d
over c-- in other words,
00:37:42.040 --> 00:37:49.390
now radiation over
a distance L plus d
00:37:49.390 --> 00:37:52.570
has had time to
reach my point p.
00:37:52.570 --> 00:38:00.190
So now, the second charge,
the one that minused d,
00:38:00.190 --> 00:38:05.190
has had enough time
to reach my point p.
00:38:05.190 --> 00:38:11.370
But if I limit to the window,
this and that, in other words,
00:38:11.370 --> 00:38:16.950
there's still a difference
of d/c in time between those.
00:38:16.950 --> 00:38:21.630
I will get at point
p the radiation as
00:38:21.630 --> 00:38:23.970
from the first charge.
00:38:23.970 --> 00:38:27.010
That's exactly the same as this.
00:38:27.010 --> 00:38:28.830
That's obviously got there.
00:38:28.830 --> 00:38:32.850
But now from the second
charge, it has got there.
00:38:32.850 --> 00:38:35.110
And they're very similar.
00:38:35.110 --> 00:38:41.800
They are both-- the
charges are accelerating
00:38:41.800 --> 00:38:50.060
along y-direction in phase.
00:38:50.060 --> 00:38:56.150
So they have the same
frequency, and the amplitude
00:38:56.150 --> 00:39:00.790
is going to be the same,
because in this formula,
00:39:00.790 --> 00:39:05.790
this r is the total
distance between the charge
00:39:05.790 --> 00:39:07.570
and the point p.
00:39:07.570 --> 00:39:11.210
Now, there is a tiny
difference in that distance
00:39:11.210 --> 00:39:14.130
for the first and second charge.
00:39:14.130 --> 00:39:17.420
But if I take them
here a tiny difference,
00:39:17.420 --> 00:39:19.465
this will not change very much.
00:39:19.465 --> 00:39:21.680
I'm ignoring that difference.
00:39:21.680 --> 00:39:24.790
So I'm calling both of
them E0 and ignoring
00:39:24.790 --> 00:39:26.210
that tiny difference.
00:39:26.210 --> 00:39:31.140
But I cannot ignore the
difference on the phase.
00:39:31.140 --> 00:39:36.920
So the t prime into
two cases is different.
00:39:36.920 --> 00:39:40.580
In one case, it's t minus L/c.
00:39:40.580 --> 00:39:44.640
And the other case,
it's minus L/c
00:39:44.640 --> 00:39:49.510
minus this little extra
time where the radiation
00:39:49.510 --> 00:39:54.070
takes from the second
to the first charge.
00:39:54.070 --> 00:39:57.140
And the third one,
the radiation could
00:39:57.140 --> 00:39:59.360
not to reach the point p yet.
00:39:59.360 --> 00:40:04.805
So that is now the total
electric field from the two.
00:40:08.400 --> 00:40:11.380
And finally-- and
I have to do this.
00:40:11.380 --> 00:40:12.870
I haven't done the addition.
00:40:12.870 --> 00:40:20.030
And finally, if we take a time
which is greater than this,
00:40:20.030 --> 00:40:26.140
then we are in a situation
where the oscillation
00:40:26.140 --> 00:40:29.900
on the last charge
has had enough time
00:40:29.900 --> 00:40:31.880
to get to the point p.
00:40:31.880 --> 00:40:37.650
Let me go back to this
picture and just repeat this
00:40:37.650 --> 00:40:39.030
so that you don't get lost.
00:40:39.030 --> 00:40:45.480
So these charges have started
to move all at the same time.
00:40:45.480 --> 00:40:51.460
After a time L/c, the
radiation from this one
00:40:51.460 --> 00:40:52.580
will have got to here.
00:40:55.220 --> 00:41:00.740
If I add to the time
d/c, that's how long
00:41:00.740 --> 00:41:04.380
the electromagnetic radiation
takes to get to from here
00:41:04.380 --> 00:41:05.060
to here.
00:41:05.060 --> 00:41:07.970
So a little bit
later, the radiation
00:41:07.970 --> 00:41:12.200
from this plus the radiation
from this is getting here.
00:41:12.200 --> 00:41:15.140
And a little bit later,
the radiation from this
00:41:15.140 --> 00:41:16.660
also gets there.
00:41:16.660 --> 00:41:19.900
So at the end, there
is now radiation
00:41:19.900 --> 00:41:23.140
from all three charges
getting to this point.
00:41:23.140 --> 00:41:26.010
And from then on,
it'll continue forever
00:41:26.010 --> 00:41:27.550
as long as these
are oscillating.
00:41:37.620 --> 00:41:42.100
So from then on, you have
the radiation from the three.
00:41:42.100 --> 00:41:44.110
And the only
difference between them
00:41:44.110 --> 00:41:47.675
is they all have a
slightly different phase.
00:41:50.370 --> 00:41:54.020
So you might be satisfied
with just knowing
00:41:54.020 --> 00:41:56.690
that these are the
electric fields as sums
00:41:56.690 --> 00:42:01.180
of the algebraic
sum between those
00:42:01.180 --> 00:42:03.300
all pointing the same direction.
00:42:03.300 --> 00:42:05.640
So I don't have to worry
about the vector addition,
00:42:05.640 --> 00:42:10.760
but I have to worry about the
addition of the amplitudes.
00:42:10.760 --> 00:42:14.630
If we want to find
out what this sum is--
00:42:14.630 --> 00:42:23.410
what happens when you add one
or two or three oscillatory
00:42:23.410 --> 00:42:24.400
functions like this?
00:42:24.400 --> 00:42:28.890
What is the resultant
oscillation?
00:42:28.890 --> 00:42:33.830
We have to algebraically,
or trigonometrically, add
00:42:33.830 --> 00:42:36.780
these three or these two.
00:42:36.780 --> 00:42:42.850
And I do this example
in order to introduce
00:42:42.850 --> 00:42:47.780
a mathematical technique, which
in situations of this kind,
00:42:47.780 --> 00:42:51.716
makes life much, much easier
than just going and adding
00:42:51.716 --> 00:42:52.215
cosines.
00:42:54.750 --> 00:43:02.190
And it is by using the
so-called complex amplitudes.
00:43:02.190 --> 00:43:10.560
So the issue is, how do we add
these three cosine functions
00:43:10.560 --> 00:43:16.000
and where they each have a
slightly different phase?
00:43:16.000 --> 00:43:20.110
One way is brute force.
00:43:20.110 --> 00:43:23.610
Do it by using trigonometrical
[? formulae ?].
00:43:23.610 --> 00:43:26.980
For example, you could
add the first and second
00:43:26.980 --> 00:43:30.530
by using the formula cosine
a plus cosine b equals twice
00:43:30.530 --> 00:43:32.880
cosine half the
sum of the angles
00:43:32.880 --> 00:43:36.830
cosine half the
difference of the angle.
00:43:36.830 --> 00:43:40.171
And you could do that for
adding the first to the second,
00:43:40.171 --> 00:43:41.920
and then later, once
you've got an answer,
00:43:41.920 --> 00:43:45.240
you could add to that
the third, et cetera.
00:43:45.240 --> 00:43:48.530
But you can see that if
you have five, six, seven,
00:43:48.530 --> 00:43:54.045
eight sources, or many, many
more, this becomes cumbersome.
00:43:56.930 --> 00:44:00.150
There's a nice
mathematical trick.
00:44:00.150 --> 00:44:04.210
And that is by using
complex numbers.
00:44:04.210 --> 00:44:07.580
You know that this is
De Moivre's theorem
00:44:07.580 --> 00:44:12.480
that E to the j
theta can be written
00:44:12.480 --> 00:44:18.640
as the cosine of an angle
plus j sine of an angle.
00:44:18.640 --> 00:44:22.130
I can use this
mathematical trick
00:44:22.130 --> 00:44:25.960
to solve the problem
of adding these.
00:44:25.960 --> 00:44:31.140
Remember that at this stage,
this is pure mathematics.
00:44:31.140 --> 00:44:35.140
As always, we've converted
an experimental situation
00:44:35.140 --> 00:44:38.140
into a mathematical problem.
00:44:38.140 --> 00:44:41.630
And we've got to solve
this using mathematics.
00:44:41.630 --> 00:44:43.320
You don't have to
ask yourself what's
00:44:43.320 --> 00:44:46.500
the meaning of j
sine theta something.
00:44:46.500 --> 00:44:48.915
It is a mathematical expression.
00:44:48.915 --> 00:44:51.120
And we're going
to use mathematics
00:44:51.120 --> 00:44:54.200
to solve this problem.
00:44:54.200 --> 00:44:58.200
Using this, I could
always write--
00:44:58.200 --> 00:45:01.590
that suppose I have the
cosine of some function,
00:45:01.590 --> 00:45:10.030
I can always write it
as the real part of E
00:45:10.030 --> 00:45:11.890
to the j that angle.
00:45:11.890 --> 00:45:14.760
So if cosine, for
example, of omega
00:45:14.760 --> 00:45:20.470
t minus kL, and you'll see it
somewhere up there for example.
00:45:20.470 --> 00:45:30.690
It can be written as the real
part of E to the j omega t
00:45:30.690 --> 00:45:35.100
minus j to the kL, but that is
the same as multiplying these
00:45:35.100 --> 00:45:41.170
as E to the j omega t times E
to the minus jkL, where here I'm
00:45:41.170 --> 00:45:43.875
just reminding you of k
omega c, et cetera, is.
00:45:46.630 --> 00:45:50.940
I can do this for
every-- if I want
00:45:50.940 --> 00:45:54.920
to d-- let's say we're
doing this one first.
00:45:54.920 --> 00:45:56.850
I want this to that.
00:45:56.850 --> 00:46:02.900
I can write this as the real
part of a complex number.
00:46:02.900 --> 00:46:09.090
And I can write this as a real
part of the complex number.
00:46:09.090 --> 00:46:15.500
What I will then do-- I
will first solve this.
00:46:15.500 --> 00:46:21.030
I will do the addition by
adding the two complex numbers,
00:46:21.030 --> 00:46:27.610
knowing full well that if I
added the complex numbers,
00:46:27.610 --> 00:46:31.910
I will have in the process
added the real parts and also
00:46:31.910 --> 00:46:34.160
the imaginary parts.
00:46:34.160 --> 00:46:40.510
And since you cannot have a real
number equal to an imaginary
00:46:40.510 --> 00:46:48.890
number in any way, if I take
during the process of addition,
00:46:48.890 --> 00:46:53.740
I would have continuously kept
separate the real and imaginary
00:46:53.740 --> 00:46:55.210
part.
00:46:55.210 --> 00:47:02.000
So for example, for this third
part, this addition here,
00:47:02.000 --> 00:47:11.082
I can write the first term as
the real part of each of the E
00:47:11.082 --> 00:47:16.600
to the j omega t, E to the minus
jkL in the y-direction times
00:47:16.600 --> 00:47:17.510
E0.
00:47:17.510 --> 00:47:19.840
This is, of course,
nothing other
00:47:19.840 --> 00:47:24.680
than E cosine omega t minus Lc.
00:47:24.680 --> 00:47:27.720
The second one I
can do the same.
00:47:27.720 --> 00:47:29.300
This is the same.
00:47:29.300 --> 00:47:31.730
And the only difference
between those two
00:47:31.730 --> 00:47:41.850
is that phase minus d/c,
which is kd [? using ?] here.
00:47:41.850 --> 00:47:47.880
And so, the real
part of this will
00:47:47.880 --> 00:47:50.815
be the answer to
the sum of those.
00:47:55.450 --> 00:47:58.580
So I'll do this in
a second, but then
00:47:58.580 --> 00:48:01.530
let me immediately go to the
third one so I do both of them
00:48:01.530 --> 00:48:02.790
at the same time.
00:48:02.790 --> 00:48:07.330
And in this third case,
the answer I'll want
00:48:07.330 --> 00:48:10.410
is the real part.
00:48:10.410 --> 00:48:12.040
This is the same as before.
00:48:12.040 --> 00:48:14.850
And here the three terms.
00:48:14.850 --> 00:48:19.260
The first term is just E0,
because all the phases are out
00:48:19.260 --> 00:48:20.370
here.
00:48:20.370 --> 00:48:25.380
The second one differs from
that by minus d/c, which
00:48:25.380 --> 00:48:29.410
is minus kd, so it's
E to the minus jkd.
00:48:29.410 --> 00:48:34.150
And the last one is E to
the minus j times 2 kd,
00:48:34.150 --> 00:48:35.510
because here we have a 2d.
00:48:42.600 --> 00:48:45.530
Now, why did I
bother to do this?
00:48:45.530 --> 00:48:51.480
Because this addition is
trivial while the other one
00:48:51.480 --> 00:48:52.660
was not trivial.
00:48:52.660 --> 00:48:54.960
It needed hard labor.
00:48:54.960 --> 00:48:57.180
Why do I say this is trivial?
00:48:57.180 --> 00:49:00.830
Because I've converted
this algebraic problem
00:49:00.830 --> 00:49:02.580
into a geometry one.
00:49:02.580 --> 00:49:07.990
I can represent each
one of these terms
00:49:07.990 --> 00:49:10.830
on an Argand diagram.
00:49:10.830 --> 00:49:16.920
So for example, here to here.
00:49:16.920 --> 00:49:22.450
Let's take the third case.
00:49:22.450 --> 00:49:25.940
We're adding these two vectors.
00:49:25.940 --> 00:49:30.310
This is only a real
part, so it's E0.
00:49:30.310 --> 00:49:34.180
That's a vector of length
E0 along the real axis.
00:49:34.180 --> 00:49:40.510
I'm reminding you on
an Argand diagram,
00:49:40.510 --> 00:49:46.880
this direction is the real axis,
and this is the imaginary axis.
00:49:46.880 --> 00:49:53.460
With that, so the
E0 is only real,
00:49:53.460 --> 00:50:00.670
and I'm adding to it E to
the minus jkd, which is what?
00:50:00.670 --> 00:50:03.420
Has a magnitude of E0.
00:50:03.420 --> 00:50:09.820
And this is the angle,
theta, with respect
00:50:09.820 --> 00:50:16.170
to the real axis, which
in this case is minus kd.
00:50:16.170 --> 00:50:18.005
So this is the
angle, so it's minus.
00:50:18.005 --> 00:50:20.250
So I'm going down.
00:50:20.250 --> 00:50:25.910
So it's a vector,
which is length E0,
00:50:25.910 --> 00:50:31.944
and it's pointing in
an angle of minus kd.
00:50:31.944 --> 00:50:32.610
It's this angle.
00:50:32.610 --> 00:50:38.220
I suppose, not to confuse
you, I'll call it minus kd.
00:50:38.220 --> 00:50:41.160
This is minus and minus,
just so not to confuse you.
00:50:51.110 --> 00:50:56.374
With the distance
between the charges of d,
00:50:56.374 --> 00:51:01.800
kd is 2 pi over 3,
which is a 120 degrees.
00:51:01.800 --> 00:51:06.400
So this term is a
vector like that.
00:51:06.400 --> 00:51:09.240
And we have to add
those two vectors.
00:51:09.240 --> 00:51:12.180
Well, this is easy to do.
00:51:12.180 --> 00:51:14.880
This angle is 60 degrees.
00:51:14.880 --> 00:51:17.030
This is 120.
00:51:17.030 --> 00:51:17.780
OK?
00:51:17.780 --> 00:51:20.680
And this length
is equal to that,
00:51:20.680 --> 00:51:25.050
so the result of
this plus of that
00:51:25.050 --> 00:51:31.780
will be this red vector
here, which has magnitude E0.
00:51:31.780 --> 00:51:33.550
And at what angle is it?
00:51:33.550 --> 00:51:37.685
This angle here 240 degrees.
00:51:47.300 --> 00:51:47.800
I'm sorry.
00:51:47.800 --> 00:51:50.970
I need that later in the
second for the next part.
00:51:50.970 --> 00:51:52.590
I don't need it at this stage.
00:51:52.590 --> 00:51:53.910
I'm sorry.
00:51:53.910 --> 00:51:57.020
I'm adding this to that.
00:51:57.020 --> 00:52:00.110
The result is this vector.
00:52:00.110 --> 00:52:03.610
And this vector
has a magnitude E0.
00:52:03.610 --> 00:52:09.000
And this angle
here is, of course,
00:52:09.000 --> 00:52:13.070
60 degrees, which
is 2 pi over 6.
00:52:13.070 --> 00:52:22.330
And so, so adding those two
will give me just the real part
00:52:22.330 --> 00:52:25.740
on each of E to the j
omega t to the minus
00:52:25.740 --> 00:52:28.960
jkL in the y-direction.
00:52:28.960 --> 00:52:37.300
And adding those two
is E0 in the direction
00:52:37.300 --> 00:52:43.330
of 60 degrees, which
is minus 2 pi over 6.
00:52:43.330 --> 00:52:45.450
Bracket right there.
00:52:45.450 --> 00:52:48.920
So this describes this addition.
00:52:48.920 --> 00:52:53.860
And this, of course, I
can now take the real part
00:52:53.860 --> 00:52:58.140
of this whole thing,
and there is the answer.
00:52:58.140 --> 00:53:00.680
E0 in the y-direction
cosine omega
00:53:00.680 --> 00:53:05.670
t minus omega over c and
with a phase of minus 2 pi
00:53:05.670 --> 00:53:12.240
over 6, which is of
course, 60 degrees.
00:53:12.240 --> 00:53:15.570
For the first one, so
this is the answer.
00:53:15.570 --> 00:53:18.810
And we didn't have
to add to any cosine.
00:53:18.810 --> 00:53:25.090
We just do a vector addition
on the Argand diagram.
00:53:25.090 --> 00:53:27.040
Let's do the last case.
00:53:27.040 --> 00:53:31.115
In the last case we have
three terms-- one, two, three.
00:53:34.150 --> 00:53:44.810
If I write these as the real
part of a complex amplitudes,
00:53:44.810 --> 00:53:49.020
by analogy with this, the
first term is the same.
00:53:49.020 --> 00:53:51.290
The second term is the same.
00:53:51.290 --> 00:53:59.750
And the third one is almost
the same as the second time,
00:53:59.750 --> 00:54:02.610
except the d has now become 2kd.
00:54:02.610 --> 00:54:04.210
This was d/c.
00:54:04.210 --> 00:54:05.940
Here it's 2d over c.
00:54:05.940 --> 00:54:09.150
So here we have minus j 2kd.
00:54:09.150 --> 00:54:14.110
So now I have to add these
three complex numbers.
00:54:14.110 --> 00:54:19.220
And again, this is much easier
to do it geometrically then
00:54:19.220 --> 00:54:21.370
trigonometrically.
00:54:21.370 --> 00:54:25.970
The first one I can represent
by this on the Argand diagram.
00:54:25.970 --> 00:54:28.270
The second one by this.
00:54:28.270 --> 00:54:34.000
And the third one, of course,
is at 240 degrees to here.
00:54:34.000 --> 00:54:35.670
This was 120.
00:54:35.670 --> 00:54:41.110
The next one is 240, which
means that the last one, all
00:54:41.110 --> 00:54:44.230
that it is, I'd
remove this vector,
00:54:44.230 --> 00:54:46.210
and it's in this direction.
00:54:46.210 --> 00:54:49.440
So now we're adding
this to that to this,
00:54:49.440 --> 00:54:52.280
and even I can solve
that in my head.
00:54:52.280 --> 00:54:55.430
If I add this vector to
that to that, I get 0.
00:54:55.430 --> 00:54:57.170
I'm back where I started.
00:54:57.170 --> 00:55:00.140
So this quantity is 0.
00:55:00.140 --> 00:55:05.050
And so the electric
field will be 0.
00:55:05.050 --> 00:55:07.980
And see how
relatively easy it is
00:55:07.980 --> 00:55:13.700
to do if you use this
complex amplitude method?
00:55:13.700 --> 00:55:19.980
One can do many problems to
do with waves and vibrations
00:55:19.980 --> 00:55:22.390
using complex amplitudes.
00:55:22.390 --> 00:55:25.810
In general, it makes
the algebra easier.
00:55:25.810 --> 00:55:32.103
But for most cases, I have
not done that in order
00:55:32.103 --> 00:55:38.200
to-- that you don't have the
double difficulty of trying
00:55:38.200 --> 00:55:41.630
to understand the physics and
struggling with the mathematics
00:55:41.630 --> 00:55:44.200
that you may not be so familiar.
00:55:44.200 --> 00:55:48.500
By the time we come to many
sources of the radiation,
00:55:48.500 --> 00:55:52.980
it is so much easier to
do using complex amplitude
00:55:52.980 --> 00:55:57.780
that I would urge you to learn
it on simple cases like this,
00:55:57.780 --> 00:56:02.390
and then use it in more
complicated situation.
00:56:02.390 --> 00:56:06.260
Finally, I just want
to save one word,
00:56:06.260 --> 00:56:09.670
and that is the following.
00:56:09.670 --> 00:56:11.210
Some of you may be surprised.
00:56:11.210 --> 00:56:16.110
How is it that I got--
I have three charges.
00:56:16.110 --> 00:56:22.980
Three charges
oscillating, radiating.
00:56:22.980 --> 00:56:26.130
How is it that when
you get to here,
00:56:26.130 --> 00:56:31.070
you get after a certain
time, you get to [INAUDIBLE].
00:56:31.070 --> 00:56:33.530
And the answer is--
pictorially you
00:56:33.530 --> 00:56:37.390
can see what happens--
at this point,
00:56:37.390 --> 00:56:41.960
the radiation from one of
the charges looks like that.
00:56:41.960 --> 00:56:45.770
From the second one,
it's out the phase
00:56:45.770 --> 00:56:50.790
by 1/3 of the wavelength,
and the next one by 2/3.
00:56:50.790 --> 00:56:56.550
And so you're adding three
waves, three oscillating
00:56:56.550 --> 00:56:59.270
motions on top of each other.
00:56:59.270 --> 00:57:02.670
And if you add these,
the result is 0.
00:57:02.670 --> 00:57:07.110
That is, pictorially,
the same as what
00:57:07.110 --> 00:57:08.940
I did here in this diagram.
00:57:08.940 --> 00:57:14.470
You simply-- you do have three
waves arriving from the three
00:57:14.470 --> 00:57:18.510
charges, but they
add to 0 because each
00:57:18.510 --> 00:57:22.620
has a different phase
from the previous one.
00:57:22.620 --> 00:57:24.720
And you can see it
here what happens.
00:57:24.720 --> 00:57:28.010
Here I'm plotting as
a function of time.
00:57:28.010 --> 00:57:30.410
The amplitude at the point p.