WEBVTT

00:02.130 --> 00:05.710
Prof: Okay,
normally I would ask in a small

00:05.708 --> 00:10.308
class if there is something you
didn't follow from last time.

00:10.310 --> 00:13.840
I'm afraid to do that now,
because it's a big class and I

00:13.838 --> 00:17.428
don't know how many things you
follow or didn't follow.

00:17.430 --> 00:21.600
What I will do first is write
down a very quick summary of the

00:21.598 --> 00:23.578
main points from last time.

00:23.580 --> 00:27.160
So you should ask yourself,
"Did I follow all those

00:27.155 --> 00:28.125
things?"

00:28.130 --> 00:30.630
And if your answer is yes,
then you are fine.

00:30.630 --> 00:32.870
Because I talked about many,
many things,

00:32.865 --> 00:34.705
but you don't need all of that.

00:34.710 --> 00:38.960
So what I'm going to write down
right now is the absolute

00:38.956 --> 00:41.606
essentials of last lecture,
okay?

00:41.610 --> 00:47.810
That's going to be needed for
what I do next.

00:47.810 --> 00:52.420
First point was:
everything is made of atoms.

00:52.420 --> 00:53.760
You know that.

00:53.760 --> 00:57.630
And the atom has a nucleus.

00:57.630 --> 01:00.940
In the nucleus are some things
called protons,

01:00.939 --> 01:04.979
some things called neutrons,
and outside are some things

01:04.984 --> 01:06.534
called electrons.

01:06.530 --> 01:11.600
That's all the atomic structure
we need.

01:11.599 --> 01:18.909
Then we say certain entities
have a property called electric

01:18.905 --> 01:20.015
charge.

01:20.019 --> 01:24.479
The symbol for electric charge
is q, and you can put a

01:24.482 --> 01:27.832
subscript to say who you are
talking about.

01:27.830 --> 01:33.560
So you can say q for the
neutron is 0.

01:33.560 --> 01:40.410
A q for the electron is
-1.6 times 10 to the -19,

01:40.412 --> 01:44.402
and it's measured in coulombs.

01:44.400 --> 01:52.360
The q for the proton is
really--let me put it this way,

01:52.364 --> 01:58.114
q for the proton is a
positive number,

01:58.111 --> 02:02.291
so these minus signs cancel.

02:02.290 --> 02:06.510
Now, the importance of the
coulomb is that if anything has

02:06.510 --> 02:10.220
some coulombs on it,
it will interact with anything

02:10.215 --> 02:13.025
else that has some coulombs on
it.

02:13.030 --> 02:17.530
So that if you have two
entities and this one has a

02:17.534 --> 02:21.414
charge of q_1
coulombs,

02:21.408 --> 02:24.318
that one has a charge of
q_2 coulombs,

02:24.318 --> 02:27.828
and the distance between them
is r,

02:27.830 --> 02:30.530
then the force is
q_1q

02:30.532 --> 02:34.432
_2_ over
4Πε

02:34.426 --> 02:37.046
_0r^(2).

02:37.050 --> 02:40.180
I'm purposely not putting all
the vector signs on F

02:40.180 --> 02:42.980
because it takes too long,
but you all know what the

02:42.980 --> 02:43.750
answer is.

02:43.750 --> 02:46.560
Namely, if you want the force
on 2,

02:46.560 --> 02:50.070
due to 1, will be repulsive if
q_1 and

02:50.066 --> 02:53.766
q_2 are of the
same sign and point to the

02:53.765 --> 02:55.515
direction joining them.

02:55.520 --> 02:56.440
Yep?

02:56.440 --> 02:59.110
Student:  Shouldn't it
be 1 over r^(2)?

02:59.110 --> 03:02.560
Prof: Yes, thank you.

03:02.560 --> 03:04.970
There is another force law
which is r^(2),

03:04.967 --> 03:06.617
not so famous,
called Hooke's Law,

03:06.623 --> 03:08.233
but you're absolutely right.

03:08.229 --> 03:11.889
That's the difference between
being Newton and being Hooke.

03:11.889 --> 03:13.839
Hooke is known for the
r^(2)^( )law.

03:13.840 --> 03:18.050
Newton is known for the 1 over
r^(2)^( )law.

03:18.050 --> 03:19.500
So this is a very important
thing.

03:19.500 --> 03:23.530
If you see anything wrong you
should stop me,

03:23.530 --> 03:27.610
because when I go home,
they're going to play the video

03:27.613 --> 03:30.533
for me to watch,
and I'm going to see that

03:30.530 --> 03:31.360
r^(2).

03:31.360 --> 03:34.840
There is nothing I can do until
some voice from the back says,

03:34.840 --> 03:36.510
"Hey, send it
downstairs",

03:36.508 --> 03:38.648
and I appreciate that a lot,
okay?

03:38.650 --> 03:41.960
It's good, so never hesitate to
do that, plus sign,

03:41.961 --> 03:44.561
minus sign, symbols;
anything that goes wrong.

03:44.560 --> 03:47.420
It also tells me that you're
following me.

03:47.419 --> 03:50.909
So for all those reasons you
should not hesitate to correct

03:50.912 --> 03:54.402
anything, and you should not
think that you don't follow it

03:54.404 --> 03:56.094
because it's your fault.

03:56.090 --> 03:57.310
Probably it is.

03:57.310 --> 04:01.370
Sometimes it's my fault as was
demonstrated now.

04:01.370 --> 04:01.850
Okay?

04:01.849 --> 04:04.329
So this is the force law.

04:04.330 --> 04:07.280
You need only one other
ingredient.

04:07.280 --> 04:10.730
That's the superposition
principle.

04:10.729 --> 04:12.969
You need that ingredient
because we're not going to be

04:12.973 --> 04:14.373
talking only about two charges.

04:14.370 --> 04:16.670
We're going to be talking about
many charges.

04:16.670 --> 04:19.570
And the question is:
what will they do when they're

04:19.569 --> 04:22.139
all present,
and it's a great blessing that

04:22.141 --> 04:25.261
we have the principle of
superposition that says that if

04:25.259 --> 04:28.119
you've got,
say, three charges,

04:28.117 --> 04:33.087
say 1,2 and 3,
you want to know the force on 3

04:33.088 --> 04:34.778
due to 2 and 1.

04:34.779 --> 04:37.379
You can find the force that 1
would exert.

04:37.379 --> 04:39.399
You can find the force that 2
would exert.

04:39.399 --> 04:43.199
We have two vectors,
and you can add the two vectors

04:43.204 --> 04:44.924
to get the net force.

04:44.920 --> 04:47.620
In other words,
the interaction between pairs

04:47.620 --> 04:51.610
of charges is insensitive to the
presence of any other charges.

04:51.610 --> 04:55.340
They go about their business
exactly the same way.

04:55.339 --> 05:00.769
That is a principle that is not
deduced by logic.

05:00.769 --> 05:01.959
You cannot say,
"Of course,

05:01.959 --> 05:03.109
it had to be that way."

05:03.110 --> 05:04.510
That's not true.

05:04.509 --> 05:05.669
It doesn't have to be that way.

05:05.670 --> 05:08.380
In fact it is not that way if
you make supreme,

05:08.377 --> 05:10.727
I mean, extremely exquisite
measurements,

05:10.730 --> 05:13.910
but they occur really at the
really quantum level.

05:13.910 --> 05:17.890
For classical electromagnetic
theory it is actually an

05:17.889 --> 05:21.119
experimental fact that you can
superpose.

05:21.120 --> 05:24.840
All right, if you combine those
things you can calculate

05:24.843 --> 05:29.113
anything and everything that we
will deal with for sometime,

05:29.110 --> 05:32.580
but that's summary of--this is
all I said last time,

05:32.579 --> 05:35.609
okay?

05:35.610 --> 05:40.800
Now, I didn't do one thing,
which is to emphasize to you

05:40.800 --> 05:46.560
that the force of gravitation
divided by the electric force is

05:46.557 --> 05:49.857
some number like 10 to the -40.

05:49.860 --> 05:52.680
I put this twiddle here meaning
not exactly.

05:52.680 --> 05:55.550
There are factors of 1 and 2 or
10 missing,

05:55.550 --> 05:59.600
but 10 to the -40 is roughly
the magnitude of this ratio,

05:59.600 --> 06:04.370
and it was done by comparing
the force between electron and a

06:04.372 --> 06:05.092
proton.

06:05.088 --> 06:07.088
It didn't matter what the
distance was because everything

06:07.089 --> 06:09.699
goes like 1 over r^(2),
so when they take the ratio

06:09.696 --> 06:12.866
that cancels out,
but this was really the ratio

06:12.867 --> 06:15.707
of things like mass of the
electron,

06:15.709 --> 06:18.519
mass of the proton,
divided by 1 over

06:18.516 --> 06:21.946
4Πε
_0;

06:21.949 --> 06:25.149
q of the electron and
q of the proton.

06:25.149 --> 06:29.559
If you put all these numbers in
you've got that.

06:29.560 --> 06:31.650
I want to mention one thing.

06:31.649 --> 06:35.659
It may be interesting for you
to think about.

06:35.660 --> 06:38.210
Look at this nucleus.

06:38.209 --> 06:44.939
Nucleus has a lot of protons in
it, and according to Coulomb's

06:44.939 --> 06:48.469
Law they all repel each other.

06:48.470 --> 06:51.170
And the neutrons,
of course, don't do anything

06:51.170 --> 06:53.510
because they have no electric
charge.

06:53.509 --> 06:57.049
So you should ask yourself,
"Why are all these protons

06:57.053 --> 07:00.293
staying together inside the
nucleus if I don't see any

07:00.291 --> 07:02.371
attraction between them?"

07:02.370 --> 07:04.170
I see why the electrons are
hanging around,

07:04.170 --> 07:05.600
because they're attracted to
the nucleus,

07:05.600 --> 07:07.240
because nucleus is positively
charged,

07:07.240 --> 07:10.170
electrons are negatively
charged, and Mr. Coulomb tells

07:10.166 --> 07:11.736
you they'll be attracted.

07:11.740 --> 07:16.510
What are the protons doing
together in that tiny space?

07:16.509 --> 07:21.209
That space is like 10 to the
-13 centimeters.

07:21.209 --> 07:24.619
So have you ever thought of
that, or does anybody know why

07:24.622 --> 07:26.242
the protons are together?

07:26.240 --> 07:27.120
Yes?

07:27.120 --> 07:28.830
Student:  The strong
nuclear force?

07:28.829 --> 07:32.129
Prof: There is a strong
nuclear force.

07:32.129 --> 07:33.909
Okay, that's the answer.

07:33.910 --> 07:36.820
In other words,
but that answer will lead to

07:36.822 --> 07:39.462
more questions,
but I'll first state the

07:39.464 --> 07:40.214
answer.

07:40.209 --> 07:42.599
There's a force even stronger
than the electric force.

07:42.600 --> 07:44.480
You see, this gives the
impression electric force is

07:44.483 --> 07:45.003
very strong.

07:45.000 --> 07:48.690
It is much stronger than
gravity, but there is a force

07:48.694 --> 07:52.814
even stronger than the electric
force that is experienced by

07:52.807 --> 07:54.757
protons and by neutrons.

07:54.759 --> 07:57.939
In other words there is another
charge which is not electric

07:57.940 --> 08:01.070
charge, which the protons are
endowed with and the neutrons

08:01.067 --> 08:02.197
are endowed with.

08:02.199 --> 08:04.549
And the force due to that
charge, if you like,

08:04.548 --> 08:06.998
it's not called charge but it's
a similar thing,

08:07.000 --> 08:08.410
is much, much stronger.

08:08.410 --> 08:11.860
It had to be,
because you have to beat the

08:11.862 --> 08:13.802
electrical repulsion.

08:13.800 --> 08:15.670
So then you can ask yourself,
"Well,

08:15.670 --> 08:18.540
in that case,
how did we ever find the

08:18.543 --> 08:22.503
electrical force,"
because here's another force

08:22.504 --> 08:26.004
even stronger than the
electrical force,

08:26.000 --> 08:28.040
maybe a thousand times stronger.

08:28.040 --> 08:32.180
Then why wasn't it completely
overshadowing the electrical

08:32.178 --> 08:32.758
force?

08:32.759 --> 08:33.729
Yep?

08:33.730 --> 08:35.700
Student:  It's
irrelevant over larger

08:35.700 --> 08:36.250
distances.

08:36.250 --> 08:38.870
Prof: Yes,
so let me repeat what he said.

08:38.870 --> 08:42.360
It has to do with what's called
the range of the force.

08:42.360 --> 08:44.300
In other words,
look at two different

08:44.298 --> 08:45.268
functions, okay?

08:45.269 --> 08:55.549
One function looks like 1 over
137 times 1 over r^(2).

08:55.548 --> 09:01.238
Other function looks like 10
times e to the -r

09:01.241 --> 09:06.261
over r_0
divided by r^(2);

09:06.259 --> 09:12.839
r_0 is some
length, and the length is

09:12.841 --> 09:18.551
roughly 10 to the -13
centimeters, or 10 to -15

09:18.552 --> 09:19.922
meters.

09:19.918 --> 09:24.218
Which force are you more
impressed with is the question,

09:24.221 --> 09:24.771
okay?

09:24.769 --> 09:27.659
If r is much bigger than
r_0,

09:27.663 --> 09:30.453
say 10 times bigger,
you've got e to the -10

09:30.445 --> 09:31.275
on the top.

09:31.278 --> 09:32.728
e to the -10 is a small
number.

09:32.730 --> 09:35.550
e to the -3 is like 1
over 20.

09:35.548 --> 09:38.758
e to the -10 is 1 over
20 cubed.

09:38.759 --> 09:43.139
So then what will happen is,
this force, even though there's

09:43.140 --> 09:47.370
a number 10 in front of it,
will be negligible compared to

09:47.374 --> 09:48.344
this one.

09:48.340 --> 09:49.790
Whereas if r is much
smaller than

09:49.793 --> 09:51.443
r_0,
say, r is .1 of

09:51.442 --> 09:53.572
r_0,
so you can forget this number,

09:53.568 --> 09:55.488
e to the minus point is
roughly 1,

09:55.490 --> 10:00.520
then this 10 will dominate the
1 over 100.

10:00.519 --> 10:03.069
So what happens is,
if you're sitting inside the

10:03.072 --> 10:05.312
nucleus,
the nuclear force is

10:05.312 --> 10:08.872
numerically strong,
not only because of the number

10:08.870 --> 10:11.450
in front of it,
but also because this

10:11.447 --> 10:14.227
exponential factor has not
kicked in.

10:14.230 --> 10:19.240
Therefore, the protons feel an
attraction for each other due to

10:19.241 --> 10:23.771
the nuclear force that is
stronger than the repulsion due

10:23.769 --> 10:25.789
to the coulomb force.

10:25.788 --> 10:29.658
And the neutrons attract the
protons just as much as the

10:29.658 --> 10:34.158
protons attract the protons with
respect to the nuclear force.

10:34.158 --> 10:37.998
So neutrons are actually good,
because when you throw an extra

10:38.001 --> 10:41.971
neutron into an atom you don't
add to the coulomb repulsion,

10:41.970 --> 10:44.900
but you add to the overall
attraction that they all feel

10:44.900 --> 10:47.140
for each other due to the
nuclear force.

10:47.139 --> 10:49.179
So neutrons are like the glue.

10:49.178 --> 10:52.068
As you add more and more
protons, you will find that

10:52.067 --> 10:55.297
you've got to add more and more
neutrons to compensate the

10:55.297 --> 10:56.597
coulomb attraction.

10:56.600 --> 11:01.060
But a time will come when the
nucleus becomes so big,

11:01.058 --> 11:03.648
that even if you add enough
neutrons,

11:03.649 --> 11:06.929
the repulsion between protons
from one end of the nucleus to

11:06.932 --> 11:10.102
the other is now becoming
comparable to the attraction due

11:10.102 --> 11:13.922
to the nuclear force,
because nucleus has become so

11:13.918 --> 11:17.268
big this factor is no longer
negligible.

11:17.269 --> 11:20.219
See, over long distances a
coulomb force will always

11:20.222 --> 11:22.232
triumph,
because no matter what the

11:22.226 --> 11:24.936
prefactors are,
the exponential factor in front

11:24.942 --> 11:27.532
of the nuclear force will always
weaken it.

11:27.528 --> 11:31.018
That's why you cannot have
nuclei beyond some size.

11:31.019 --> 11:33.779
If you make them any bigger,
the nuclear electrical

11:33.779 --> 11:36.759
repulsion between the distant
parts of the protons,

11:36.759 --> 11:39.319
due to the distant protons in
the nucleus,

11:39.320 --> 11:42.800
cannot be compensated by the
short-range attraction.

11:42.798 --> 11:45.798
So it's the range of the
interaction that is a

11:45.803 --> 11:47.543
significant factor here.

11:47.538 --> 11:51.418
So all the strong forces are
strong, but at short distances

11:51.416 --> 11:55.486
the coulomb force is not that
strong, but it falls like 1 over

11:55.493 --> 11:56.633
r^(2).

11:56.629 --> 11:58.449
Now gravity,
on the other hand,

11:58.446 --> 12:01.226
is exactly 1 over
r^(2)^( )with a number

12:01.230 --> 12:04.250
that's much,
much, much smaller than this,

12:04.245 --> 12:08.555
but then we saw the other day
why gravity managed to survive,

12:08.558 --> 12:10.338
because this is
q_1q

12:10.336 --> 12:12.946
_2 and the
q's can be added

12:12.947 --> 12:15.167
algebraically and cancel each
other.

12:15.168 --> 12:16.538
Whereas if you've got
m_1m

12:16.543 --> 12:18.823
_2_ over
r^(2)^( )there is no way

12:18.822 --> 12:19.802
to cancel the m.

12:19.798 --> 12:23.738
This is how the different
forces manage to survive for the

12:23.735 --> 12:25.525
different reasons,
okay?

12:25.528 --> 12:29.488
Nuclear wins in the nuclear
zone, but dies very quickly

12:29.485 --> 12:31.825
outside the size of a nucleus.

12:31.830 --> 12:34.910
Electrical forces falls more
slowly at like 1 over

12:34.907 --> 12:35.847
r^(2).

12:35.850 --> 12:38.570
They dominate atomic physics,
but once you formed an atom

12:38.572 --> 12:41.202
you've got pretty much an
electrical neutral thing,

12:41.200 --> 12:44.070
and once you've got many,
many atoms making up our

12:44.072 --> 12:46.422
planet,
then all that remains is the

12:46.423 --> 12:50.333
gravitational attraction between
a planet and another planet.

12:50.330 --> 12:54.770
Okay, so these are examples of
different forces and why they

12:54.767 --> 12:59.197
were found at various times,
because they all dominate under

12:59.203 --> 13:01.313
different circumstances.

13:01.308 --> 13:06.648
All right, so today I'm going
to start with my new stuff.

13:06.649 --> 13:09.249
So all you need to know,
really, if you want to do your

13:09.245 --> 13:11.895
problem sets and your homework,
is Coulomb's Law,

13:11.899 --> 13:15.319
you know, how to stick the
numbers in the Coulomb's Law.

13:15.320 --> 13:18.280
And the only thing I didn't
mention is this 1 over

13:18.275 --> 13:21.525
4Πε
_0 is 9 times 10

13:21.533 --> 13:22.623
to the 9^(th).

13:22.620 --> 13:27.060
So today we're going to do
something which is a part of a

13:27.062 --> 13:32.142
great abstraction and it goes as
follows: So I'm gong to take two

13:32.138 --> 13:36.428
charges,
a q_1 here and

13:36.432 --> 13:40.882
q_2 here,
and I'm going to give them some

13:40.878 --> 13:41.408
locations.

13:41.408 --> 13:47.588
So let's say this guy is at
vector r_1,

13:47.586 --> 13:52.636
this one is at vector
r_2.

13:52.639 --> 13:55.789
Now, I will really take are of
all the arrows.

13:55.788 --> 14:00.628
This one is
r_2 -

14:00.628 --> 14:06.638
r_1,
that arrow there.

14:06.639 --> 14:09.249
You can check that I didn't
mess up anything,

14:09.246 --> 14:12.206
because r_1
r_2 -

14:12.211 --> 14:15.531
r_1 should be
r_2.

14:15.528 --> 14:19.468
So let's write the coulomb
force now as a vector.

14:19.470 --> 14:24.810
And you've got to say force on
what, so I'm going to say force

14:24.811 --> 14:26.301
on 2, due to 1.

14:26.298 --> 14:28.968
Now, you've got to realize it's
a convention.

14:28.970 --> 14:32.600
So I use a convention:
This is the force on this guy

14:32.601 --> 14:36.521
and the force due to this guy,
with the second later.

14:36.519 --> 14:39.379
Now, let's write it out in
detail.

14:39.379 --> 14:42.389
So that is
q_1 over

14:42.388 --> 14:45.538
4Πε
_0.

14:45.538 --> 14:47.708
I'm going to put the
q_2 here.

14:47.710 --> 14:52.020
Then here I want to write
(r_1 -

14:52.023 --> 14:54.543
r_2)^(2).

14:54.538 --> 14:57.748
That's the 1 over r^(2),
but then I have to make it a

14:57.750 --> 14:58.240
vector.

14:58.240 --> 15:00.850
So for the vector part,
once you have the magnitude of

15:00.850 --> 15:03.740
the vector,
you should multiply by vector

15:03.741 --> 15:08.071
of length unit 1 going in the
direction of this difference

15:08.070 --> 15:08.830
vector.

15:08.830 --> 15:11.070
So you can use any symbol you
like.

15:11.070 --> 15:14.300
One is to say
e_12.

15:14.298 --> 15:17.988
e's always going to be
unit vector going from 1 to 2,

15:17.990 --> 15:21.010
or if you're inclined you can
also write

15:21.014 --> 15:25.134
e_12 as
r_2 -

15:25.126 --> 15:28.536
r_1 divided by
the length of

15:28.539 --> 15:32.419
r_2 -
r_1.

15:32.419 --> 15:34.919
They're all unit vectors.

15:34.919 --> 15:35.719
Now, do you follow that?

15:35.720 --> 15:40.100
I mean, are you having trouble
with unit vectors?

15:40.100 --> 15:42.850
Anytime I have a vector
pointing from here to there,

15:42.850 --> 15:45.170
I want to give a magnitude and
direction.

15:45.168 --> 15:48.458
The magnitude in this case is 1
over the distance squared,

15:48.457 --> 15:51.857
but you have to append to it a
vector of unit length in that

15:51.861 --> 15:52.671
direction.

15:52.668 --> 15:55.488
That's what makes it into a
vector.

15:55.490 --> 16:00.050
For example,
suppose I want to describe that

16:00.053 --> 16:04.623
vector r,
and it is 7 meters long?

16:04.620 --> 16:07.380
I cannot write r = 7,
because that doesn't tell you

16:07.379 --> 16:08.639
which way it's pointing.

16:08.639 --> 16:12.239
So invent the vector called
i, which is a unit vector

16:12.235 --> 16:15.095
in the x direction and I
multiply it by that.

16:15.100 --> 16:19.240
So 7i is a vector
parallel to i and 7 times

16:19.241 --> 16:19.761
long.

16:19.759 --> 16:25.809
7.3i is a vector
parallel to i 7.3 times

16:25.806 --> 16:27.286
long, okay?

16:27.288 --> 16:30.348
So you need to add a vector of
unit length to the magnitude,

16:30.347 --> 16:32.367
multiply it to get the actual
vector.

16:32.370 --> 16:36.110
So that's that formula.

16:36.110 --> 16:38.970
Then if you have many charges,
I'm not going to do that now,

16:38.970 --> 16:41.170
say one more here,
it'll exert a force on

16:41.168 --> 16:43.838
q_2,
but you've got to add to the

16:43.839 --> 16:44.609
force due to 1.

16:44.610 --> 16:45.240
I'm not going to do that.

16:45.240 --> 16:48.500
I'm just taking two guys.

16:48.500 --> 16:56.850
Now, I'm going to formally
write this as equal to the

16:56.847 --> 17:03.747
electric field at
r_2 times

17:03.750 --> 17:12.260
q_2,
and this is called a field.

17:12.259 --> 17:14.219
You see, if you look at this
thing,

17:14.220 --> 17:17.260
all I've done is rewrite the
expression as something that

17:17.258 --> 17:19.808
involves a charge of
q�_2 and

17:19.807 --> 17:23.117
everything else that involves
the q_1 and the

17:23.117 --> 17:25.127
distance from
q_1 and

17:25.125 --> 17:26.695
q_2.

17:26.700 --> 17:30.890
So it looks like there's no
real content to giving this

17:30.890 --> 17:34.050
object a name,
but it's a very profound

17:34.048 --> 17:38.718
notion, so I've got to tell you
the story that goes with it.

17:38.720 --> 17:42.310
In Coulomb's Law you say that
q_1 and

17:42.308 --> 17:45.558
q_2 exert a
force on each other,

17:45.559 --> 17:46.169
okay?

17:46.170 --> 17:49.140
And the force depends on the
charge and the distance between

17:49.136 --> 17:49.486
them.

17:49.490 --> 17:53.600
Now I'm going to say
q_1 produces an

17:53.597 --> 17:58.497
electric field at the location
of q_2 given by

17:58.496 --> 18:01.666
this vector,
and when I multiply by

18:01.674 --> 18:05.354
q_2 it gives me
the force on

18:05.346 --> 18:07.346
q_2.

18:07.348 --> 18:10.888
Now, what's the importance of
the electric field?

18:10.890 --> 18:14.580
Whereas q_1
and q_2 exist

18:14.577 --> 18:17.757
only at these two places,
the electric field can be

18:17.757 --> 18:19.217
defined everywhere.

18:19.220 --> 18:23.210
It doesn't require a second
charge because you see,

18:23.207 --> 18:26.637
this number you can compute for
any value of

18:26.635 --> 18:28.625
r_2.

18:28.630 --> 18:33.530
So the picture we have is that
q_1 produces an

18:33.530 --> 18:37.710
electric field all over space,
and q_2

18:37.710 --> 18:41.310
experiences that field and gets
repelled as a result,

18:41.308 --> 18:46.768
and the force it feels is the
field at that point times

18:46.766 --> 18:49.086
q_2.

18:49.088 --> 18:54.798
In general, we say there's an
electric field at this point in

18:54.798 --> 18:55.558
space.

18:55.558 --> 19:00.578
If you put a charge q
there it should experience a

19:00.576 --> 19:04.336
force q times the
electric field.

19:04.338 --> 19:06.798
So to understand this,
the charges are only in two

19:06.800 --> 19:08.910
places in our example,
but the field due to

19:08.911 --> 19:10.871
q_1 is
everywhere.

19:10.868 --> 19:14.538
At every point in space I can
compute a field due to

19:14.542 --> 19:16.202
q_1.

19:16.200 --> 19:18.240
So the field,
you can see,

19:18.241 --> 19:23.141
will turn into a force if you
multiply it by a charge you put

19:23.144 --> 19:25.844
at the location of the field.

19:25.838 --> 19:28.448
So if you've got one charge
here, I claim there's a field

19:28.452 --> 19:31.562
here, there's field there;
there's a field everywhere due

19:31.558 --> 19:32.338
to this guy.

19:32.338 --> 19:35.108
How do I know,
because if you put a test

19:35.112 --> 19:37.602
charge it begins to feel a
force.

19:37.598 --> 19:41.488
So one way to say it,
is that the field is the force

19:41.493 --> 19:44.933
on a unit charge you put at that
location,

19:44.930 --> 19:48.350
unit charge because if
q_2 is equal to

19:48.345 --> 19:52.135
1 then numerically E is
exactly equal to the force.

19:52.140 --> 19:54.850
If I go to you and say,
"Find out the electric

19:54.845 --> 19:56.355
field at this point."

19:56.358 --> 19:57.038
You say, "Okay,
where?"

19:57.039 --> 19:58.519
I say, "Here."

19:58.519 --> 20:02.769
What do you think you have to
do to measure the field?

20:02.769 --> 20:04.199
Yes?

20:04.200 --> 20:07.120
Student:  Add up all the
forces on that field from the

20:07.117 --> 20:09.447
different charges,
or add up all of the fields at

20:09.452 --> 20:10.282
that point...

20:10.279 --> 20:10.679
Prof: No, okay.

20:10.680 --> 20:13.060
Her answer was,
"If I want the field here,

20:13.063 --> 20:15.603
I should find out where all the
charges are,"

20:15.601 --> 20:16.121
right?

20:16.118 --> 20:18.748
"Find all the forces they
will exert on this guy,

20:18.746 --> 20:20.676
on the unit charge here,"
right?

20:20.680 --> 20:22.780
But I'm telling you,
that's correct.

20:22.778 --> 20:25.438
That's the theoretical way to
calculate the field at that

20:25.439 --> 20:28.139
point, but suppose you're an
experimentalist and you don't

20:28.144 --> 20:29.764
want to know what produced it.

20:29.759 --> 20:33.309
You just want an answer to
what's the field here.

20:33.309 --> 20:34.939
What will you do?

20:34.940 --> 20:35.520
Yes?

20:35.519 --> 20:36.799
Student:  Place a test
charge.

20:36.799 --> 20:37.949
Prof: Okay, and then?

20:37.950 --> 20:41.010
Student:  And then see
what happens to it.

20:41.009 --> 20:44.219
Prof: By `see what
happens', you've got to be

20:44.221 --> 20:46.871
more--so his answer was:
put a test charge,

20:46.865 --> 20:48.435
and see what happens.

20:48.440 --> 20:50.160
Now, you've got to be more
precise.

20:50.160 --> 20:53.850
I think you know what you
meant, but I want you to finish

20:53.854 --> 20:54.914
that sentence.

20:54.910 --> 20:55.670
Student:  Okay.

20:55.666 --> 20:57.836
Well, you can measure the force
by measuring the acceleration on

20:57.836 --> 20:58.316
the charge.

20:58.319 --> 20:58.859
Prof: Very good.

20:58.859 --> 20:59.489
See, that's what I want.

20:59.490 --> 21:01.640
When you said,
"See what happens,"

21:01.640 --> 21:02.740
does it get married?

21:02.740 --> 21:03.900
Does it have children?

21:03.897 --> 21:05.527
That's not what I meant,
okay?

21:05.528 --> 21:07.208
But you did give the right
answer.

21:07.210 --> 21:10.040
The answer was,
by `see what happens',

21:10.042 --> 21:14.182
you put the charge there,
and you see what acceleration

21:14.178 --> 21:15.478
it undergoes.

21:15.480 --> 21:19.630
Then that acceleration is
the--that times the mass is the

21:19.625 --> 21:21.545
force it's experiencing.

21:21.548 --> 21:25.038
That should be q that
you place there times E.

21:25.038 --> 21:27.478
If q was 1 that force
itself is equal to E.

21:27.480 --> 21:33.390
If q was 10 you've got
to divide the force by 10 to get

21:33.394 --> 21:35.144
the field there.

21:35.140 --> 21:40.730
So, the field is like the sound
of one hand clapping.

21:40.730 --> 21:43.720
People say one hand clapping is
Zen concept, but the field is

21:43.720 --> 21:46.410
like that, because you don't
need two charges to have a

21:46.411 --> 21:46.911
field.

21:46.910 --> 21:49.030
You just need one charge.

21:49.029 --> 21:50.879
So here is how we understand
that.

21:50.880 --> 21:54.580
You put a charge q.

21:54.578 --> 21:58.308
If you go here something has
happened there,

21:58.308 --> 21:58.828
see?

21:58.828 --> 22:00.788
You don't need to put a second
charge there to conclude

22:00.790 --> 22:01.700
something has happened.

22:01.700 --> 22:06.160
Something really is different
at this location because the

22:06.160 --> 22:08.430
charge q is present.

22:08.430 --> 22:09.600
What is different?

22:09.598 --> 22:13.288
What is different is that when
this guy was not here and I put

22:13.292 --> 22:15.112
a charge, it just sat there.

22:15.108 --> 22:19.558
Whereas when this guy is here
and I put a charge,

22:19.560 --> 22:21.880
it experiences a force.

22:21.880 --> 22:25.710
So if you put a 1 coulomb here
it experiences a force which is

22:25.711 --> 22:29.711
1 times the field at that point,
therefore, this charge has

22:29.711 --> 22:35.921
distorted the space around it,
in fact, everywhere.

22:35.920 --> 22:39.910
Now, if you've got many,
many, many charges,

22:39.910 --> 22:43.020
then they will all try to
produce a force on a unit charge

22:43.018 --> 22:44.858
and you should,
like you said,

22:44.856 --> 22:47.916
add up the vectors,
which are the forces due to all

22:47.916 --> 22:50.546
the other charges on that one
location,

22:50.548 --> 22:54.308
on a test charge on that
location.

22:54.308 --> 23:01.008
So the field is the sum of the
field due to all the charges at

23:01.010 --> 23:02.440
that point.

23:02.440 --> 23:10.310
Now, Coulomb's Law doesn't work
when charges are moving.

23:10.309 --> 23:11.109
Why is that?

23:11.108 --> 23:15.768
Have you any idea why you
cannot use it?

23:15.769 --> 23:16.449
Yes?

23:16.450 --> 23:17.390
Student:  The radius is
changing?

23:17.390 --> 23:17.880
Prof: Pardon me?

23:17.880 --> 23:19.610
Student:  r is
changing?

23:19.608 --> 23:22.878
Prof: r is
changing so we'll keep changing

23:22.875 --> 23:25.795
it as the charge moves,
but it's only an approximation

23:25.797 --> 23:26.667
when charges move.

23:26.670 --> 23:30.300
Do you know why?

23:30.299 --> 23:31.529
Yep?

23:31.528 --> 23:34.578
Student:  There's a
magnetic field?

23:34.578 --> 23:36.018
Prof: True,
but even the electric

23:36.016 --> 23:37.486
field--his answer was magnetic
field,

23:37.490 --> 23:40.260
but even the electric field is
not properly given.

23:40.259 --> 23:41.309
Yes?

23:41.308 --> 23:44.788
Student:  Is the
electric field affected by

23:44.790 --> 23:47.350
something like the Doppler
Effect?

23:47.348 --> 23:50.438
Prof: Not the Doppler,
something else.

23:50.440 --> 23:53.420
If Coulomb's Law were exact,
okay?

23:53.420 --> 23:55.110
Here is what I can do.

23:55.108 --> 23:59.538
You take a coulomb and you sit
at the other end of the galaxy.

23:59.539 --> 24:01.539
I have a coulomb here.

24:01.538 --> 24:03.848
You know the force my coulomb
exerts on yours because it's

24:03.847 --> 24:04.737
pushing against you.

24:04.740 --> 24:05.630
You hold it.

24:05.630 --> 24:08.470
Now suddenly I move my coulomb
away from you by a tiny amount.

24:08.470 --> 24:10.740
What will you feel?

24:10.740 --> 24:12.910
You will feel the force is
reduced.

24:12.910 --> 24:14.110
Yes?

24:14.108 --> 24:15.308
Student:  Relativity
is...

24:15.309 --> 24:18.019
Prof: Yes,
so the special theory does not

24:18.015 --> 24:20.785
allow that,
because I have managed to

24:20.787 --> 24:24.007
communicate with you,
arbitrarily far,

24:24.008 --> 24:25.408
instantaneously.

24:25.410 --> 24:27.810
The minute I move this charge,
you know about it,

24:27.808 --> 24:29.458
because your charge moved away.

24:29.460 --> 24:31.870
For example,
if your charge was connected to

24:31.866 --> 24:33.666
a spring,
and it had been extended to

24:33.670 --> 24:35.220
some amount because of this
charge,

24:35.220 --> 24:38.320
the minute I move it your
spring will move.

24:38.318 --> 24:42.478
That instantaneous
communication is forbidden by

24:42.477 --> 24:46.367
the special theory,
so it does not happen.

24:46.368 --> 24:50.158
So the correct way to do it,
it'll maybe come later in the

24:50.163 --> 24:52.643
course,
so probably not even at the end

24:52.644 --> 24:55.834
of this course,
but the proper way to do that

24:55.830 --> 25:00.440
is to in the end realize that
the electric field at this point

25:00.440 --> 25:04.520
is not only due to what the
charges are doing now,

25:04.519 --> 25:07.209
but what they were doing in the
past.

25:07.210 --> 25:10.430
Because if some charge at the
other end of the galaxy did

25:10.430 --> 25:13.540
something it takes some time,
namely traveling at the speed

25:13.538 --> 25:16.848
of light,
to carry that information from

25:16.853 --> 25:18.273
there to here.

25:18.269 --> 25:21.439
So it's the delayed response to
all the motion in the charges

25:21.435 --> 25:23.965
that you've got to add to find
the field here.

25:23.970 --> 25:26.390
That's what makes the
computation of the electric

25:26.394 --> 25:27.914
field much more complicated.

25:27.910 --> 25:31.510
But if you promise me charges
never move then the location now

25:31.509 --> 25:34.999
is the location last year,
the location a million years

25:34.999 --> 25:37.349
ago, then you can use Coulomb's
Law.

25:37.348 --> 25:39.468
Coulomb's Law is good for
electrostatics,

25:39.473 --> 25:42.663
but in real life charges are
moving, so you cannot really use

25:42.660 --> 25:43.510
the formula.

25:43.509 --> 25:45.609
Now, in our room,
if you put a charge here and

25:45.612 --> 25:47.392
another charge there,
if you move this,

25:47.386 --> 25:49.766
that guy will move pretty much
instantaneously.

25:49.769 --> 25:52.569
That's because the time it
takes a light signal to go from

25:52.567 --> 25:54.777
here to there is so small,
you may treat it as

25:54.776 --> 25:55.656
instantaneous.

25:55.660 --> 25:58.110
So Coulomb's Law is used in
electrical circuits and so on.

25:58.108 --> 26:01.328
You don't worry about the time
of transit because it's too

26:01.330 --> 26:03.450
small,
but over longer distances,

26:03.449 --> 26:06.539
where the time it takes for
light to travel becomes

26:06.544 --> 26:09.754
non-negligible,
you cannot use Coulomb's Law.

26:09.750 --> 26:10.670
It's not wrong.

26:10.670 --> 26:14.260
It is not appropriate when
charges are moving.

26:14.259 --> 26:20.239
However, it will always be true
that if you go to any one point

26:20.237 --> 26:26.307
the force on any charge q
you put there is q times

26:26.310 --> 26:31.080
electric field at that point,
okay?

26:31.078 --> 26:33.938
So the electric field notion
survives because it doesn't

26:33.938 --> 26:35.028
violate relativity.

26:35.029 --> 26:38.429
It says if the field here is so
much q will experience

26:38.433 --> 26:41.503
the force q times
E, but the complication

26:41.497 --> 26:43.197
is what is the field here?

26:43.200 --> 26:46.710
Well, it's due to everybody
else, and it's not only due to

26:46.707 --> 26:49.567
everybody else right now,
but everybody else from the

26:49.569 --> 26:52.059
dawn of time because things have
been moving and shaking and

26:52.063 --> 26:53.123
sending signals to us.

26:53.118 --> 26:56.748
We collect all that and see
what lands here at this instant.

26:56.750 --> 26:58.920
That decides the field here.

26:58.920 --> 27:01.750
So that's the computation of
the field, but the response to

27:01.750 --> 27:02.970
the field is very easy.

27:02.970 --> 27:06.870
You put a test charge;
q times E is the

27:06.873 --> 27:07.653
answer.

27:07.650 --> 27:10.950
So in modern physics,
in theories that are compatible

27:10.946 --> 27:14.426
with the special theory of
relativity we break the force

27:14.434 --> 27:15.644
into two parts.

27:15.640 --> 27:19.390
Charges don't immediately
interact with other charges.

27:19.390 --> 27:22.350
Charges produce a field,
and the field may even

27:22.346 --> 27:26.456
propagate outwards at the speed
of light if you make motions,

27:26.460 --> 27:29.340
but another charge at the
location of that particular

27:29.339 --> 27:31.999
point will respond to the field
at that point.

27:32.000 --> 27:34.100
So it's not responding to the
charge right now.

27:34.098 --> 27:38.028
It's responding to the field
this charge produced at its

27:38.025 --> 27:38.805
location.

27:38.808 --> 27:41.578
So all of electromagnetic
theory is going to contain two

27:41.577 --> 27:41.977
parts.

27:41.980 --> 27:44.240
The first part is,
find the field due to this

27:44.243 --> 27:46.763
charge configuration,
that charge configuration,

27:46.761 --> 27:49.641
maybe due to various currents,
and the second part is,

27:49.641 --> 27:53.341
given the field,
find the response of charges to

27:53.338 --> 27:54.288
the field.

27:54.288 --> 27:57.038
So you understand charges play
a double role.

27:57.038 --> 28:00.848
They are the producers of the
field.

28:00.848 --> 28:03.828
They are also the ones who
respond to the field.

28:03.828 --> 28:06.018
If you don't have a charge you
cannot produce field.

28:06.019 --> 28:08.079
If you don't have a charge you
cannot experience it,

28:08.078 --> 28:09.208
you cannot play that game.

28:09.210 --> 28:12.450
To gain membership into
electrostatic interactions

28:12.449 --> 28:14.299
you've got to have charge.

28:14.298 --> 28:17.188
So neutrons cannot do that,
but they can do other things.

28:17.190 --> 28:20.560
Like I said,
they take part in nuclear

28:20.556 --> 28:24.646
force, in fact,
just as well as protons do.

28:24.650 --> 28:29.290
All right, so let's go back now
to the simplest problem in the

28:29.289 --> 28:32.789
world: the electric field due to
one charge.

28:32.789 --> 28:34.469
The formula is very simple.

28:34.470 --> 28:36.710
Let's put that charge at the
origin.

28:36.710 --> 28:40.060
And the electric field due to 1
charge is q over

28:40.064 --> 28:42.864
4Πε
_0,

28:42.858 --> 28:47.158
1 over r^(2) if you're
here,

28:47.160 --> 28:48.880
and that's your r vector.

28:48.880 --> 28:52.460
The field at this point is that
magnitude,

28:52.460 --> 28:55.210
and I'm going to write
e_r_

28:55.211 --> 28:58.961
meaning a vector of unit
length in the radial direction.

28:58.960 --> 29:02.490
Once again, I will tell you if
you want, you can write that as

29:02.490 --> 29:05.730
the position vector divided by
the length of the position

29:05.732 --> 29:06.372
vector.

29:06.369 --> 29:09.089
They're equivalent ways.

29:09.088 --> 29:11.438
If I write it the second way,
you've got to be a little

29:11.443 --> 29:11.883
careful.

29:11.880 --> 29:15.860
It'll look like qr over
4Πε

29:15.863 --> 29:18.153
_0r^(3).

29:18.150 --> 29:21.560
Don't get fooled into thinking
the field is falling at 1 over

29:21.557 --> 29:22.407
r^(3).

29:22.410 --> 29:25.820
It's really 1 over
r^(2)^( )because there's

29:25.819 --> 29:27.559
an r on the top.

29:27.558 --> 29:30.108
See, if you write it as 1 over
r^(2) put unit vector.

29:30.108 --> 29:32.678
If you write it as 1 over
r^(3 )put the position

29:32.681 --> 29:33.111
vector.

29:33.108 --> 29:36.348
They're all saying the same
thing.

29:36.349 --> 29:39.049
So here you have this formula.

29:39.048 --> 29:41.678
If you're a person who likes to
work with formulas this is all

29:41.675 --> 29:42.145
you need.

29:42.150 --> 29:45.700
You manipulate the stuff on
paper, and you add different

29:45.700 --> 29:48.540
fields, but people like to
visualize this.

29:48.539 --> 29:50.449
So how do I visualize this?

29:50.450 --> 29:52.200
That's the real question.

29:52.200 --> 29:56.880
So here's a very popular method
for visualizing this formula.

29:56.880 --> 30:00.480
You know, you've got,
for example--suppose someone

30:00.480 --> 30:04.740
asks you, what's the height
above ground level of a certain

30:04.743 --> 30:06.953
part of the United States?

30:06.950 --> 30:07.790
Okay, you've got some mountains.

30:07.789 --> 30:09.219
You've got some valleys.

30:09.220 --> 30:12.120
Well, somebody can give you a
function that gives you the

30:12.115 --> 30:14.385
height at any point in the
United States,

30:14.390 --> 30:17.750
but it's more interesting to
have some kind of a contour map

30:17.751 --> 30:19.831
that looks like this,
right?

30:19.828 --> 30:21.808
And all these contours are
different heights.

30:21.808 --> 30:23.838
If you've gone hiking,
you can see those maps.

30:23.838 --> 30:27.008
They tell you pictorially what
a certain function is trying to

30:27.008 --> 30:27.578
tell you.

30:27.578 --> 30:32.568
So you want a pictorial
representation of this electric

30:32.569 --> 30:33.309
field.

30:33.308 --> 30:37.098
It's very easy to write down
the electric field at one point,

30:37.104 --> 30:40.524
namely you take that point,
you draw an arrow there;

30:40.519 --> 30:41.589
E.

30:41.588 --> 30:45.448
That E is the electric
field at that point.

30:45.450 --> 30:48.670
So we try to do our best by
saying here is my test charge.

30:48.670 --> 30:51.370
I'm going to pick a few points,
four points,

30:51.373 --> 30:54.013
maybe eight,
and I'm going to tell you what

30:54.013 --> 30:56.093
the field is at those points.

30:56.088 --> 30:58.538
At this point it looks like
that.

30:58.538 --> 31:00.208
At this point it looks like
that.

31:00.210 --> 31:05.830
Here it looks like this.

31:05.828 --> 31:09.978
That's already telling you
something.

31:09.980 --> 31:12.620
You've got to be very careful
on the interpretation.

31:12.618 --> 31:16.228
This arrow is not telling you
what's happening throughout the

31:16.233 --> 31:17.563
length of the arrow.

31:17.558 --> 31:19.518
It's telling you what's
happening at the tip.

31:19.519 --> 31:21.799
You understand the arrow is in
your mind.

31:21.799 --> 31:22.479
It's a vector.

31:22.480 --> 31:24.760
It's not really sticking out in
space.

31:24.759 --> 31:26.979
It's a property at a condition
at that point,

31:26.978 --> 31:29.548
but we've got to draw it,
so we draw it that way.

31:29.548 --> 31:33.458
It doesn't tell you the state
of affairs over its length,

31:33.458 --> 31:36.598
but only at its tip,
at the starting point.

31:36.598 --> 31:39.248
Then you can say,
"Okay, what happens when I

31:39.250 --> 31:40.520
go further out?"

31:40.519 --> 31:42.699
When I go further out,
say over here,

31:42.703 --> 31:46.043
it's going to be still--if I
put a test charge here it's

31:46.040 --> 31:49.440
still going to be repelled
radially, but a lot less.

31:49.440 --> 31:53.010
So I do that.

31:53.009 --> 31:58.649
So I draw arrows at other
representative points and make

31:58.651 --> 32:00.191
them shorter.

32:00.190 --> 32:03.610
In fact, the length of the
arrow will be 1 over

32:03.605 --> 32:04.715
r^(2).

32:04.720 --> 32:08.300
So you can do this, okay?

32:08.298 --> 32:12.088
But now where all do you want
to draw these arrows?

32:12.089 --> 32:12.979
It's up to you.

32:12.980 --> 32:13.840
You pick a few points.

32:13.838 --> 32:16.628
You go to another radius,
you draw more arrows.

32:16.630 --> 32:20.240
Someone had this clever idea of
doing the following.

32:20.240 --> 32:22.630
You can probably guess.

32:22.630 --> 32:31.790
Their idea was when I join all
these arrows like that,

32:31.788 --> 32:41.638
now if you go to a point like
this, what have I gained and

32:41.640 --> 32:45.270
what have I lost?

32:45.269 --> 32:48.299
What more information have I
got when I join the arrows?

32:48.299 --> 32:49.639
Yes.

32:49.640 --> 32:51.900
Student:  The more
information is that now you know

32:51.903 --> 32:54.173
the direction that it will
continue on forever if no other

32:54.165 --> 32:56.435
forces act on it,
and what you've lost is the

32:56.442 --> 32:57.122
magnitude...

32:57.118 --> 32:59.178
Prof: Okay,
let me repeat that.

32:59.180 --> 33:02.560
That's what you guys should
have been thinking in your head.

33:02.558 --> 33:04.568
When I joined these lines--by
the way,

33:04.568 --> 33:09.128
I do want you to anticipate
what I'm going to say,

33:09.130 --> 33:13.200
because if I'm struck by
lightning,

33:13.200 --> 33:15.920
another electromagnetic
phenomenon,

33:15.920 --> 33:19.700
can you even complete my...

33:19.700 --> 33:19.900
Student (Chorus):
Sentence!

33:19.900 --> 33:21.180
Prof: Sentence, right.

33:21.180 --> 33:25.200
Okay, now you should be able to
go a little beyond if I'm doing

33:25.204 --> 33:26.184
a derivation.

33:26.180 --> 33:27.960
You've got to be following me,
right?

33:27.960 --> 33:28.690
That's very important.

33:28.690 --> 33:30.120
It's got to be active.

33:30.118 --> 33:33.908
And I sat through a lecture
yesterday for an hour.

33:33.910 --> 33:36.190
I know it's a very long time.

33:36.190 --> 33:38.510
This is what,
an hour and fifteen minutes?

33:38.509 --> 33:42.779
The only way you can survive
this is if you somehow make it

33:42.782 --> 33:44.112
an active event.

33:44.108 --> 33:47.238
You've got to do something that
keeps you awake during the

33:47.240 --> 33:47.790
process.

33:47.788 --> 33:50.598
One of them is to anticipate
what I will do next in a

33:50.602 --> 33:51.362
calculation.

33:51.358 --> 33:54.058
That'll make sure also that
you're on top of it,

33:54.060 --> 33:56.590
that'll make sure that you
catch mistakes.

33:56.589 --> 33:58.459
Okay.

33:58.460 --> 34:00.290
All right, so here are these
lines.

34:00.288 --> 34:03.358
As she said quite correctly,
previously I knew the field

34:03.361 --> 34:05.541
direction only at the chosen
points,

34:05.538 --> 34:08.408
but now I know it throughout
this line,

34:08.409 --> 34:12.199
but I've lost information on
the magnitude of the field,

34:12.199 --> 34:14.809
because the arrows--there are
no lengths of anything.

34:14.809 --> 34:16.309
These arrows don't have any
length.

34:16.309 --> 34:18.829
In fact, you can keep drawing
more lines if you like.

34:18.829 --> 34:22.939
They go like that in all
directions.

34:22.940 --> 34:25.600
It basically tells you,
hey, the charge is pushing

34:25.601 --> 34:28.321
everything out radially no
matter where you are.

34:28.320 --> 34:30.670
That's the thrust of this
picture.

34:30.670 --> 34:36.620
But, due to the miraculous
property of the coulomb force,

34:36.619 --> 34:39.799
namely that it falls like 1
over r^(2),

34:39.800 --> 34:44.990
there is information even on
the strength of the electric

34:44.994 --> 34:47.754
field,
and that information is

34:47.751 --> 34:51.931
contained in the density of
electric field lines.

34:51.929 --> 34:53.549
And I'll tell you precisely
what I mean.

34:53.550 --> 34:57.190
So, here is the charge.

34:57.190 --> 35:03.060
Take a sphere of radius
r and here are all these

35:03.057 --> 35:04.577
lines going.

35:04.579 --> 35:14.549
By density of lines,
I mean the number of lines

35:14.552 --> 35:24.312
crossing a surface perpendicular
to the lines,

35:24.307 --> 35:32.977
divided by the area of that
surface.

35:32.980 --> 35:37.930
Because let's make a convention
that we will draw for every

35:37.925 --> 35:42.615
coulomb a certain number of
lines, 32 lines per coulomb,

35:42.617 --> 35:44.917
32 lines are going out.

35:44.920 --> 35:50.860
I draw a sphere of some radius,
32 lines cross that sphere.

35:50.860 --> 35:55.570
I draw a bigger sphere,
32 lines cross this sphere

35:55.570 --> 35:58.120
also,
but they're less dense,

35:58.117 --> 36:02.597
because the number of lines per
area will be some number of

36:02.599 --> 36:06.849
lines per charge divided by the
area of the sphere,

36:06.849 --> 36:10.199
which is 4Πr^(2).

36:10.199 --> 36:11.679
Do you follow that?

36:11.679 --> 36:14.029
If you take a sphere,
first of all,

36:14.025 --> 36:17.605
every portion of the sphere the
area that you have is

36:17.610 --> 36:19.750
perpendicular to the lines.

36:19.750 --> 36:22.450
So the area intercepts the
lines perpendicularly.

36:22.449 --> 36:24.149
That's the agreement here.

36:24.150 --> 36:27.370
And you see how many are
crossing per unit area.

36:27.369 --> 36:31.819
That is going like 1 over
r^(2).

36:31.820 --> 36:36.050
So these lines naturally
diverge and spread out in space

36:36.048 --> 36:39.428
so that the density falls
precisely as 1 over

36:39.431 --> 36:40.741
r^(2).

36:40.739 --> 36:44.429
That has to do also with the
fact you're living in three

36:44.427 --> 36:45.297
dimensions.

36:45.300 --> 36:48.820
Only in three dimensions where
area goes like r^(2)^(

36:48.822 --> 36:52.232
)does this spreading of the
density of lines coincide with

36:52.228 --> 36:53.958
the decline of the force.

36:53.960 --> 36:58.280
So these lines tell you more
than simply the direction.

36:58.280 --> 37:01.720
They convey to you visually
where the field is strong.

37:01.719 --> 37:04.149
Wherever the lines are dense,
the field is strong.

37:04.150 --> 37:07.430
Wherever the lines are spread
apart, the field is weak,

37:07.425 --> 37:09.605
and it's a very precise
statement.

37:09.610 --> 37:14.090
The only thing not precise is,
how many lines do you want to

37:14.086 --> 37:15.526
draw per coulomb.

37:15.530 --> 37:18.100
That's really up to you,
but you've got to be

37:18.099 --> 37:18.859
consistent.

37:18.860 --> 37:21.290
Once you give 32 lines per
coulomb, then if you've got a

37:21.289 --> 37:23.099
charge of 1 coulomb you should
draw 32.

37:23.099 --> 37:26.249
If you've got two coulombs,
you should draw 64 lines.

37:26.250 --> 37:29.550
As long as you do that,
the number of lines crossing

37:29.552 --> 37:32.792
per unit area will be
proportional to the field.

37:32.789 --> 37:36.549
But I'm going to make a certain
choice that will make the number

37:36.547 --> 37:39.467
of lines per unit area exactly
equal to the field,

37:39.469 --> 37:41.079
and here is the choice.

37:41.079 --> 37:43.219
It's a choice that makes life
simple.

37:43.219 --> 37:47.969
Let us agree that 1 coulomb
gets 1 over

37:47.967 --> 37:52.587
ε_0
lines.

37:52.590 --> 37:54.250
ε_0 is
a number, right?

37:54.250 --> 37:56.800
1 over 4Πε
_0 is 9 times 10

37:56.800 --> 37:57.470
to the 9^(th).

37:57.469 --> 37:58.689
This is some number.

37:58.690 --> 38:02.770
Maybe 40 million,
so one coulomb gets 40 million

38:02.773 --> 38:03.473
lines.

38:03.469 --> 38:04.489
Don't quote the 40 million.

38:04.489 --> 38:05.869
It's whatever this thing is.

38:05.869 --> 38:07.029
I don't know what it is.

38:07.030 --> 38:08.680
It's a definite number.

38:08.679 --> 38:11.419
Then, what's the nice thing?

38:11.420 --> 38:16.750
If you've got q coulombs
you will have q over

38:16.753 --> 38:20.253
ε_0
lines,

38:20.250 --> 38:24.350
and if you take a sphere of
radius r you'll get 1

38:24.353 --> 38:28.013
over 4Πr^(2)^( )as the
line density,

38:28.010 --> 38:32.930
namely lines per unit area,
but that is exactly equal to

38:32.929 --> 38:36.239
the strength of the electric
field.

38:36.239 --> 38:37.939
If you picked a different
number like 2 over

38:37.936 --> 38:40.456
ε_0 you
will always be measuring 2 times

38:40.461 --> 38:41.371
the electric field.

38:41.369 --> 38:43.969
The density will still convey
the electric field,

38:43.971 --> 38:47.331
namely it'll be proportional to
it, but let's make life easy by

38:47.331 --> 38:48.741
making it equal to it.

38:48.739 --> 38:53.689
This is just a convenience.

38:53.690 --> 38:54.590
Now, we are really set.

38:54.590 --> 38:57.210
If you draw pictures this way,
you can go as far as you like

38:57.213 --> 38:58.063
from this charge.

38:58.059 --> 39:00.489
Simply take a unit area with
you.

39:00.489 --> 39:03.529
Take a piece of wood 1 meter by
1 meter, put it there,

39:03.530 --> 39:07.860
see how many lines cross;
that is equal to the electric

39:07.858 --> 39:09.868
field at that point.

39:09.869 --> 39:15.529
Okay, so this is the way one
likes to visualize field lines.

39:15.530 --> 39:18.030
So I'm going to give you some
examples.

39:18.030 --> 39:21.610
For a single charge you just
draw it that way.

39:21.610 --> 39:26.410
For two charges,
let's take two charges,

39:26.405 --> 39:30.705
a minus charge and a plus
charge.

39:30.710 --> 39:35.350
Let's say that one is
-q, the other is

39:35.349 --> 39:36.509
q.

39:36.510 --> 39:39.350
Then you are very near that
charge.

39:39.349 --> 39:41.079
By the way, I'm not going to
draw 1 over

39:41.077 --> 39:43.647
ε_0
lines per coulomb because it's

39:43.648 --> 39:45.288
going to be too many lines,
okay?

39:45.289 --> 39:47.099
I'm just going to draw a few so
you get the picture.

39:47.099 --> 39:54.379
So I'm going to draw four lines
right near the charge.

39:54.380 --> 39:57.930
You can forget about all other
charges in drawing the lines.

39:57.929 --> 40:03.929
Why is that?

40:03.929 --> 40:05.659
Yep?

40:05.659 --> 40:07.619
Student:  The field only
depends on the one charge.

40:07.619 --> 40:08.629
Prof: Pardon me?

40:08.630 --> 40:10.330
Student:  Why does the
field only depends on the one

40:10.326 --> 40:10.586
charge.

40:10.590 --> 40:12.110
Prof: Why does it depend
on the one charge?

40:12.110 --> 40:13.630
In principle it depends on
every charge.

40:13.630 --> 40:14.890
Somebody had an answer back
there?

40:14.889 --> 40:16.199
Yes?

40:16.199 --> 40:17.949
Student:  Since it falls
as 1 over r^(2)^( )when

40:17.949 --> 40:19.469
you're so much closer to one
than the other then...

40:19.469 --> 40:21.989
Prof: Right,
because the field is 1 over

40:21.987 --> 40:25.607
r^(2)^( )and the 1 over
r for this guy is going

40:25.605 --> 40:26.505
to infinity.

40:26.510 --> 40:28.840
1 over r for this guy is
maybe 1 over 1 meter.

40:28.840 --> 40:29.950
It's finite.

40:29.949 --> 40:32.569
So when you come arbitrarily
close to a charge,

40:32.574 --> 40:34.064
it is going to dominate.

40:34.059 --> 40:36.579
Well, if it's the only thing in
the universe,

40:36.579 --> 40:38.869
we know the lines will look
like this.

40:38.869 --> 40:41.309
At least they'll start out this
way, but soon,

40:41.306 --> 40:44.546
of course, it won't go out this
way forever, because you will

40:44.554 --> 40:46.454
realize there's another charge.

40:46.449 --> 40:52.039
Likewise, it's easy to draw the
lines this way.

40:52.039 --> 40:55.309
Remember, the lines are coming
in because if you put a test

40:55.309 --> 40:57.339
charge, it'll be sucked into
this.

40:57.340 --> 41:00.740
Test charge is always assumed
to be unit positive charge,

41:00.739 --> 41:02.529
so the lines will be coming
into a negative charge,

41:02.530 --> 41:05.390
and leaving,
going away from a positive

41:05.391 --> 41:06.071
charge.

41:06.070 --> 41:11.050
Now we've just go to do what
the agencies forgot to do,

41:11.050 --> 41:13.910
which is to connect the dots.

41:13.909 --> 41:15.289
You do this.

41:15.289 --> 41:16.649
You do this.

41:16.650 --> 41:17.860
You do this.

41:17.860 --> 41:20.590
You do this.

41:20.590 --> 41:24.910
Now, at some point you'll have
to think a little harder,

41:24.909 --> 41:27.109
because suppose I go here?

41:27.110 --> 41:29.950
How do I know I should draw the
lines this way?

41:29.949 --> 41:33.889
If I take this guy here,
it will repel it.

41:33.889 --> 41:36.189
This guy will attract it,
and I add the two,

41:36.188 --> 41:39.078
and get a line in that
direction, so you really have to

41:39.077 --> 41:40.197
do a lot of work.

41:40.199 --> 41:42.299
If you really want this picture
to be exact,

41:42.300 --> 41:43.880
you have to compute the vector
everywhere,

41:43.880 --> 41:46.630
but if you want a sketch,
you're allowed to guess,

41:46.630 --> 41:50.930
and things look like this.

41:50.929 --> 42:02.429
So this is called a dipole,
and this is the field of a

42:02.427 --> 42:04.377
dipole.

42:04.380 --> 42:06.120
So here's another example.

42:06.119 --> 42:09.359
Both guys are plus.

42:09.360 --> 42:11.790
Now what do the lines look like?

42:11.789 --> 42:26.159
Again, they will start out this
way near the charges,

42:26.159 --> 42:29.679
but now when you come to the
midpoint here there should be no

42:29.684 --> 42:32.974
electric field right in the
midpoint because it's getting

42:32.974 --> 42:35.034
pushed equally from both sides.

42:35.030 --> 42:41.990
So the lines,
if you think about how they

42:41.989 --> 42:49.819
will add, they will do something
like this.

42:49.820 --> 42:52.630
Okay, look, I'm not going to do
a good job for a variety of

42:52.626 --> 42:54.626
reasons,
but you can look at your

42:54.632 --> 42:58.132
textbook or any other book to
see what the lines will look

42:58.128 --> 43:00.398
like when you've two plus
charges.

43:00.400 --> 43:03.740
If you go a mile away from
these two plus charges,

43:03.742 --> 43:06.882
what do you think the lines
would look like?

43:06.880 --> 43:07.930
Yep?

43:07.929 --> 43:09.899
Student:  Just like a
point charge that's twice as

43:09.902 --> 43:10.222
strong.

43:10.219 --> 43:12.099
Prof: Right,
so that's the intuition you

43:12.097 --> 43:13.197
should keep in your mind.

43:13.199 --> 43:15.819
If you go very far from a
charge distribution where you

43:15.818 --> 43:18.798
cannot look into the details,
all you will see is some little

43:18.795 --> 43:20.585
dot that has the entire charge
in it,

43:20.590 --> 43:22.520
and the lines will be coming
out radially.

43:22.518 --> 43:25.358
So only when you zoom in you
realize, hey,

43:25.360 --> 43:28.970
it's not a single charge
2q, it is two guys of

43:28.965 --> 43:30.485
strength q.

43:30.489 --> 43:35.329
Finally, let's take a case
where this is charge 2q.

43:35.329 --> 43:36.969
This is charge q.

43:36.969 --> 43:41.969
Let's say this is charge
-q.

43:41.969 --> 43:54.999
Then some lines will go like
this and some lines will run off

43:54.998 --> 43:58.038
to infinity.

43:58.039 --> 44:00.809
Here if you go very,
very far away from the two

44:00.813 --> 44:03.953
charges you'll again see
radially outgoing lines,

44:03.949 --> 44:10.599
except there is a charge
q at the center because

44:10.596 --> 44:17.116
2q and -q give you
a net of q.

44:17.119 --> 44:18.429
Okay?

44:18.429 --> 44:21.769
So this is the example of a
dipole.

44:21.768 --> 44:24.718
If you've got more charges it
gets more complicated,

44:24.724 --> 44:26.814
so people don't usually draw
them.

44:26.809 --> 44:29.869
There's one example which is
pretty interesting.

44:29.869 --> 44:35.399
If you've got one plate and
another plate,

44:35.400 --> 44:37.700
this contains all positive
charges,

44:37.699 --> 44:40.719
this contains all negative
charges,

44:40.719 --> 44:47.929
then the field here will look
like this,

44:47.929 --> 44:49.559
will go from the positive to
the negative plate,

44:49.559 --> 44:51.739
because if you put a test
charge between them,

44:51.739 --> 44:54.219
it's getting repelled by the
positive plate and attracted by

44:54.215 --> 44:57.655
the negative plate,
so the lines will go from one

44:57.655 --> 44:58.865
to the other.

44:58.869 --> 45:01.859
Near the edges they may do
something more complicated,

45:01.862 --> 45:04.292
but in the bulk they will look
like this.

45:04.289 --> 45:15.709


45:15.710 --> 45:19.970
Okay, so now I'm going to do
one calculation,

45:19.967 --> 45:25.967
which is: what is the actual
electric field due to a dipole?

45:25.969 --> 45:28.069
In other words,
not just the picture here,

45:28.070 --> 45:28.890
where is that?

45:28.889 --> 45:30.789
That picture on the left is a
dipole.

45:30.789 --> 45:33.969
I'm going to do it
quantitatively.

45:33.969 --> 45:38.639
So here's my goal:
I want to take a minus charge

45:38.637 --> 45:41.317
here, a plus charge here.

45:41.320 --> 45:43.520
This is at x = a.

45:43.519 --> 45:50.159
This is at x = -a.

45:50.159 --> 45:51.719
So I want to find the field
everywhere.

45:51.719 --> 45:53.649
So today I'm not going to do
the field everywhere,

45:53.650 --> 45:56.580
because later on I'll show you
a more effective way to

45:56.576 --> 45:59.716
calculate it,
but I'm going to calculate it

45:59.724 --> 46:02.854
at a couple of interesting easy
places.

46:02.849 --> 46:06.909
In other words,
we all know the lines look like

46:06.911 --> 46:08.061
this, okay?

46:08.059 --> 46:11.659
But I want to go to some
location and find the magnitude

46:11.661 --> 46:13.561
and direction of the field.

46:13.559 --> 46:16.429
But today I will only find it
at two places,

46:16.429 --> 46:19.959
one along the axis at a point
x,

46:19.960 --> 46:26.990
and one on the perpendicular
bisector at a point (x =

46:26.987 --> 46:32.107
0, y),
just going to do those two.

46:32.110 --> 46:38.190
So let's see what's the field
here, the field at this point?

46:38.190 --> 46:42.180
The field at that point,
you agree, is going to be

46:42.184 --> 46:46.594
entirely in the x direction
because this is pushing it,

46:46.588 --> 46:48.788
and that is pulling it.

46:48.789 --> 46:52.969
So E is going to be
i (unit vector) times

46:52.965 --> 46:56.355
q over
4Πε

46:56.358 --> 47:00.848
_0 times that
distance squared,

47:00.849 --> 47:08.139
which happens to be (x -
a)^(2).

47:08.139 --> 47:12.929
That's the repulsion due to the
charge that's nearer to you.

47:12.929 --> 47:14.639
Then there's the attraction due
to the minus q,

47:14.639 --> 47:21.039
but it's a little further away,
so it looks like a minus sign

47:21.043 --> 47:24.143
but it is (x a)^(2).

47:24.139 --> 47:27.919
If a is equal to 0 you
get 0.

47:27.920 --> 47:30.240
If a is equal to 0 the
two guys are sitting exactly on

47:30.237 --> 47:31.007
top of each other.

47:31.010 --> 47:32.920
You will not see them.

47:32.920 --> 47:35.760
So you see them only because
they're not on top of each

47:35.759 --> 47:36.179
other.

47:36.179 --> 47:40.499
This whole thing fails to be 0
because this a is not 0,

47:40.503 --> 47:42.563
and you can understand why.

47:42.559 --> 47:46.029
The minute a is not 0
you're closer to one of the two

47:46.032 --> 47:49.332
charges, so that they cannot
really cancel each other.

47:49.329 --> 47:52.159
So you've got to manipulate
this expression,

47:52.159 --> 47:53.739
so I will do that now.

47:53.739 --> 47:55.879
q over
4Πε

47:55.884 --> 47:58.844
_0 and you find
common denominator.

47:58.840 --> 48:02.480
I remind you (x
a) times (x -

48:02.481 --> 48:06.871
a) is (x^(2 )-
a^(2))^( )and everything

48:06.867 --> 48:08.427
is under squares.

48:08.429 --> 48:15.109
In the numerator you've got
(x a)^(2) -

48:15.105 --> 48:18.695
(x - a)^(2).

48:18.699 --> 48:20.059
So what does that give me?

48:20.059 --> 48:24.149
iq over
4Πε

48:24.150 --> 48:27.060
_0,
(x^(2 )-

48:27.059 --> 48:29.059
a^(2))^(2).

48:29.059 --> 48:33.459
And how about on the top,
can you do that in your head?

48:33.460 --> 48:36.600
This is going to be
x^(2) a^(2)

48:36.601 --> 48:37.531
2xa.

48:37.530 --> 48:43.260
You're going to subtract from
it x^(2) a^(2) -

48:43.255 --> 48:48.585
2xa, and the only thing
that will survive will be

48:48.594 --> 48:50.054
4xa.

48:50.050 --> 48:52.030
Is everything okay?

48:52.030 --> 48:58.480
Student:  Why is it not
x - a squared times x a squared?

48:58.480 --> 49:00.240
Prof: It is.

49:00.239 --> 49:05.029
You are saying why is it not
(x a)^(2) times

49:05.028 --> 49:07.968
(x - a)^(2),
right?

49:07.969 --> 49:13.859
It is, because this would be
x a times x

49:13.858 --> 49:17.588
- a, the whole thing
squared.

49:17.590 --> 49:18.160
Student:  Oh!

49:18.159 --> 49:22.099
Prof: And this guy is
x^(2 )- a^(2).

49:22.099 --> 49:27.419
So this is classified as a nice
try, so.

49:27.420 --> 49:29.110
But I want you to keep doing
this.

49:29.110 --> 49:31.510
This time I am right,
but you never know,

49:31.510 --> 49:31.930
okay?

49:31.929 --> 49:33.839
I don't want you to give up.

49:33.840 --> 49:35.930
There is nothing better than
shooting me down,

49:35.929 --> 49:37.509
but this happened to be
correct.

49:37.510 --> 49:39.170
You satisfied though?

49:39.170 --> 49:40.910
Student:  I knew you
were right, I just couldn't

49:40.911 --> 49:41.461
figure out why.

49:41.460 --> 49:42.010
Prof: No, no, no, no.

49:42.010 --> 49:43.630
I don't want to rush through
this.

49:43.630 --> 49:47.200
Anybody have the same problem
with this?

49:47.199 --> 49:50.959
Look, also I am doing it fast
because this is 958th time I'm

49:50.956 --> 49:54.326
doing this calculation,
so if you're seeing it for the

49:54.329 --> 49:56.749
first time, I've got to slow
down.

49:56.750 --> 49:58.310
So let's see which part.

49:58.309 --> 50:01.609
Everybody okay with this, right?

50:01.610 --> 50:04.710
So I wrote that.

50:04.710 --> 50:07.200
Then this is really an x
a times another x

50:07.197 --> 50:09.067
a,
and an x - a and

50:09.065 --> 50:11.595
another x - a,
but without these guys,

50:11.601 --> 50:14.361
I know it's x^(2 )-
a^(2 )because I've got

50:14.356 --> 50:15.406
two of everything.

50:15.409 --> 50:16.969
I squared everything.

50:16.969 --> 50:21.199
So this answer's actually an
exact formula of the electric

50:21.195 --> 50:23.935
field along the axis of the
dipole.

50:23.940 --> 50:27.300
But normally what one is
interested in is,

50:27.295 --> 50:30.975
when x is much bigger
than a.

50:30.980 --> 50:34.200
When it's much bigger than
a, downstairs you've got

50:34.204 --> 50:35.284
x squared.

50:35.280 --> 50:38.190
This could be 1 kilometer
squared, a squared was

50:38.193 --> 50:39.763
maybe 1 millimeter squared.

50:39.760 --> 50:42.720
So in the first approximation,
it's not an exact formula

50:42.724 --> 50:45.474
anymore, from now on it is
approximate, in the limit

50:45.474 --> 50:47.744
x is much bigger than
a.

50:47.739 --> 50:52.819
You can see it's going to be
i times q over

50:52.822 --> 50:59.282
2Πε
_0 times

50:59.282 --> 51:04.482
2a divided by
x^(3).

51:04.480 --> 51:06.770
So what did I do now?

51:06.769 --> 51:09.289
I took from the 4a.

51:09.289 --> 51:13.139
I borrowed a 2a to write
this here and I canceled the 2

51:13.137 --> 51:15.027
with the 4 here to get that.

51:15.030 --> 51:17.260
Then on the top I had an
x, and the bottom had

51:17.264 --> 51:17.914
x^(4).

51:17.909 --> 51:22.649
I get x^(3).

51:22.650 --> 51:27.030
So let me write--can everybody
see this thing from wherever you

51:27.034 --> 51:31.074
are;
the last formula here?

51:31.070 --> 51:35.350
So I'm going to write it as
i times p divided

51:35.346 --> 51:39.766
by 2Πε
_0 x^(3).

51:39.768 --> 51:48.538
So I will tell you what I'm
doing.

51:48.539 --> 51:53.319
So the final formula I had was
electric field E = i

51:53.322 --> 51:57.552
times p divided by
2Πε

51:57.548 --> 52:01.578
_0x^(3),
where p--I'm sorry,

52:01.579 --> 52:04.379
i times p,
no arrow;

52:04.380 --> 52:08.670
p is equal to 2aq.

52:08.670 --> 52:12.050
So let's delete some extra
arrows I had.

52:12.050 --> 52:13.270
That's right.

52:13.268 --> 52:16.778
You never should settle for
something that looks like that.

52:16.780 --> 52:18.260
So that's p.

52:18.260 --> 52:23.300
So p is called the
dipole moment of this dipole,

52:23.300 --> 52:27.160
and it's given by the product
of the distance between the

52:27.159 --> 52:30.399
charges and the value of one of
the charges,

52:30.400 --> 52:31.560
the plus charge.

52:31.559 --> 52:36.849
So whenever I give you two
charges, call it dipole,

52:36.846 --> 52:40.966
you can associate with them a
vector.

52:40.969 --> 52:43.819
And the vector is,
if you've got a charge

52:43.820 --> 52:47.810
-q here and a charge
q there separated by a

52:47.809 --> 52:52.299
little vector r, then the
dipole moment is q times

52:52.297 --> 52:54.647
the little vector r.

52:54.650 --> 52:58.230
In our example the little
vector r was 2a in

52:58.228 --> 52:59.358
the x direction.

52:59.360 --> 53:04.580
So 2a times i
times q is the dipole

53:04.576 --> 53:05.426
moment.

53:05.429 --> 53:10.109
So this means electric field is
parallel to the dipole moment

53:10.112 --> 53:13.002
and falls like 1 over
x^(3).

53:13.000 --> 53:16.540
That's the most important part
of the dipole.

53:16.539 --> 53:18.739
A single charge,
the field falls like 1 over

53:18.737 --> 53:21.137
x^(2) if you move a
distance x.

53:21.139 --> 53:24.759
A dipole will always fall like
a bigger power of x,

53:24.755 --> 53:28.305
because to go like 1 over
x^(2) you've got to have

53:28.307 --> 53:29.257
net charge.

53:29.260 --> 53:32.510
As long as the net charge is 0
the fact that there are two

53:32.512 --> 53:35.652
opposite charges that don't
quite cancel each other,

53:35.650 --> 53:38.920
it always comes from the fact
that the distance between them

53:38.920 --> 53:39.530
is not 0.

53:39.530 --> 53:41.960
And the distance will appear in
the numerator,

53:41.960 --> 53:43.690
and that must be the
corresponding distance in the

53:43.690 --> 53:45.640
denominator,
because the formula should have

53:45.643 --> 53:46.613
the same dimensions.

53:46.610 --> 53:49.760
That's what turns the 1 over
x^(2) to 1 over

53:49.762 --> 53:50.712
x^(3).

53:50.710 --> 53:51.610
Yep?

53:51.610 --> 53:54.300
Student:  That only
works if x is a lot

53:54.300 --> 53:55.490
bigger than a.

53:55.489 --> 53:56.539
Prof: Yes.

53:56.539 --> 53:59.199
So this formula is good for all
x;

53:59.199 --> 54:01.649
this formula is good only for
x bigger than a.

54:01.650 --> 54:04.280
You'll find whenever you're
working with dipoles,

54:04.277 --> 54:07.617
people will always ask you to
find the field very far from the

54:07.617 --> 54:08.217
dipole.

54:08.219 --> 54:13.219
So here's a second place where
I can go and find the field.

54:13.219 --> 54:18.559
That's going to be here,
-a, a,

54:18.561 --> 54:23.421
and I'm going to find the field
there.

54:23.420 --> 54:26.400
So that's at a distance
y.

54:26.400 --> 54:27.390
So let's look at this.

54:27.389 --> 54:30.959
That's a q here and
-q here.

54:30.960 --> 54:34.040
A q will repel it that
way, and a -q will

54:34.039 --> 54:36.889
attract it this way,
and their sum will be that.

54:36.889 --> 54:43.419
I'm going to compute that sum.

54:43.420 --> 54:45.370
So how do I do this?

54:45.369 --> 54:48.109
Let's look at this guy here.

54:48.110 --> 54:52.860
We know that these arrows have
equal magnitude because this

54:52.862 --> 54:56.142
distance is the same as that
distance.

54:56.139 --> 54:59.619
Therefore, it's the horizontal
part that will remain.

54:59.619 --> 55:04.129
The vertical part will cancel.

55:04.130 --> 55:04.710
You see that?

55:04.710 --> 55:07.960
We've got two arrows of the
same length with this angle and

55:07.963 --> 55:09.033
this angle equal.

55:09.030 --> 55:11.530
The horizontal part will be
additive and the vertical part

55:11.534 --> 55:12.814
will be equal and opposite.

55:12.809 --> 55:16.449
So I'm only going to compute
the horizontal part.

55:16.449 --> 55:22.289
So the electric field now will
be -

55:22.289 --> 55:25.869
i (that's to tell me
it's in the negative x

55:25.873 --> 55:29.823
direction) times q over
4Πε

55:29.822 --> 55:33.772
_0 times 1 over
distance squared which is

55:33.771 --> 55:36.551
(y^(2)^( ) a^(2)).

55:36.550 --> 55:39.070
This is (y^(2)^( )
a^(2)).

55:39.070 --> 55:41.140
That's also (y^(2)^( )
a^(2)),

55:41.139 --> 55:43.169
but I want the horizontal part
of this,

55:43.170 --> 55:45.180
so I want the cosine
θ.

55:45.179 --> 55:47.709
That is the same as this one.

55:47.710 --> 55:53.230
The cosine θ is
a divided by

55:58.510 --> 56:01.430
See, you want to take that
force and find this horizontal

56:01.434 --> 56:01.804
part.

56:01.800 --> 56:06.170
Then I'm going to put on
another 2 because this is going

56:06.170 --> 56:09.430
to contribute an equal
horizontal part.

56:09.429 --> 56:14.929
So the E,
in the end, is equal to

56:14.927 --> 56:22.677
-iq2a over
4Πε

56:22.681 --> 56:28.881
_0 divided by
(y^(2)^( )

56:28.884 --> 56:32.554
a^(2))^(3/2).

56:32.550 --> 56:37.000
That is then -p divided
by 4Πε

56:37.001 --> 56:40.441
_0 divided by
y^(3),

56:40.440 --> 56:43.310
for y much bigger than
a.

56:43.309 --> 56:48.989
I'm sorry, y^(3),
y is really the

56:48.990 --> 56:50.350
distance.

56:50.349 --> 56:54.179
So, right, if y is 1
mile and a is 1

56:54.175 --> 56:57.685
millimeter that's essentially
the distance.

56:57.690 --> 57:01.520
If you like you can call it
-p divided by

57:01.523 --> 57:06.093
4Πε
_0r^(3).

57:06.090 --> 57:10.160
Where r is the distance,
if you like,

57:10.164 --> 57:12.824
from the center of dipole.

57:12.820 --> 57:15.020
Look, the point of this
exercise is twofold.

57:15.018 --> 57:19.948
One is to show you how to add
vectorially the fields due to

57:19.954 --> 57:20.894
two guys.

57:20.889 --> 57:22.639
Another is, to have you
understand,

57:22.639 --> 57:25.659
at least, how to do the
computation at a few simple

57:25.655 --> 57:27.645
places,
where the direction of the

57:27.652 --> 57:29.532
field is not so hard to
calculate.

57:29.530 --> 57:32.070
Actually, one would like to
compute it here,

57:32.074 --> 57:33.794
but it becomes quite nasty.

57:33.789 --> 57:36.409
The magnitude is not so hard,
but the direction is hard to

57:36.411 --> 57:38.161
calculate, so we'll find a
shortcut.

57:38.159 --> 57:41.439
But at these two places,
on the perpendicular bisector

57:41.440 --> 57:43.980
and on the axis,
the mathematics is pretty

57:43.978 --> 57:44.658
simple.

57:44.659 --> 57:45.629
That's the electric field.

57:45.630 --> 57:46.400
Yeah?

57:46.400 --> 57:48.060
Student:  Is p as
a vector different from the

57:48.063 --> 57:49.643
p that you crossed out as
a vector over there?

57:49.639 --> 57:50.499
Prof: No.

57:50.503 --> 57:51.803
It's the same p.

57:51.800 --> 57:56.490
So p as a vector in our
example will be the charge at

57:56.490 --> 58:00.070
either end times the distance
between them.

58:00.070 --> 58:04.220
The vector difference is
2a times i.

58:04.219 --> 58:16.999
So you can write the formula in
terms of the dipole moment.

58:17.000 --> 58:28.190
Okay, so now I'm going to do
the second part of the problem,

58:28.186 --> 58:36.526
which is finding the response
to E.

58:36.530 --> 58:39.280
This was all computing E.

58:39.280 --> 58:41.730
Of course there are more and
more complicated examples,

58:41.731 --> 58:43.141
but we did a few simple ones.

58:43.139 --> 58:45.509
Next is going to be,
if I give you E, can you

58:45.512 --> 58:47.282
find what will happen to the
charge.

58:47.280 --> 58:49.580
So I'm going to do two examples.

58:49.579 --> 58:53.339
One is there are these two
plates I mentioned to you.

58:53.340 --> 58:54.930
This one is all positively
charged.

58:54.929 --> 58:58.509
This one is negatively charged.

58:58.510 --> 59:01.910
And I shoot a particle here,
with some velocity

59:01.914 --> 59:05.324
v�_o in the x
direction,

59:05.320 --> 59:10.670
and the field everywhere is
down, and the electric field is

59:10.670 --> 59:17.410
some constant,
-j times some number

59:17.407 --> 59:21.617
E_0.

59:21.619 --> 59:28.529
-j because i is
this way and j is that

59:28.525 --> 59:29.235
way.

59:29.239 --> 59:31.969
So what will this do is the
question, and where will it end

59:31.974 --> 59:32.214
up?

59:32.210 --> 59:35.920
Well, I think you can all tell
that it'll end up somewhere

59:35.918 --> 59:36.438
there.

59:36.440 --> 59:41.470
What we're trying to find out
is how much does it fall,

59:41.469 --> 59:46.129
and when it comes out,
what's the direction of this

59:46.128 --> 59:48.548
final velocity vector.

59:48.550 --> 59:54.800
Well, the force on this charge
is equal to -q times

59:54.800 --> 59:58.310
E_0 j.

59:58.309 --> 1:00:04.239
The acceleration will be
-qE_0 over

1:00:04.237 --> 1:00:09.467
m times j in the y
direction.

1:00:09.469 --> 1:00:11.829
So what'll be the position?

1:00:11.829 --> 1:00:27.929
Position will be from lecture
number one of your Physics 200.

1:00:27.929 --> 1:00:33.549
So let's say the starting point
r_0 is our

1:00:33.547 --> 1:00:34.417
origin.

1:00:34.420 --> 1:00:38.850
v_0 is
whatever it was projected in

1:00:42.503 --> 1:00:51.823
times qE_0
over m,

1:00:51.818 --> 1:00:57.498
t^(2) j.

1:00:57.500 --> 1:01:01.120
Okay, so as a function of time
this tells you where the

1:01:01.117 --> 1:01:02.387
position will be.

1:01:02.389 --> 1:01:04.929
At t = 0,
you are at the origin.

1:01:04.929 --> 1:01:09.139
As t increases,
it's moving horizontally due to

1:01:09.137 --> 1:01:13.897
v_0 and it's
also dropping vertically due to

1:01:13.902 --> 1:01:15.572
the acceleration.

1:01:15.570 --> 1:01:18.140
Then it's very easy from now on
to compute anything you like.

1:01:18.139 --> 1:01:20.879
For example,
when you want to go to that

1:01:20.875 --> 1:01:23.045
point, what will be the time?

1:01:23.050 --> 1:01:34.100
Anybody tell me what the time
will be when I go to that point?

1:01:34.099 --> 1:01:38.569
How long will it be in the
region between the plates?

1:01:38.570 --> 1:01:40.340
Yep?

1:01:40.340 --> 1:01:43.220
Student:  The distance
of the length of the plate

1:01:43.224 --> 1:01:44.804
divided by initial velocity.

1:01:44.800 --> 1:01:47.290
Prof: Which velocity
should we take?

1:01:47.289 --> 1:01:48.799
Student:  The initial
horizontal velocity.

1:01:48.800 --> 1:01:51.320
Prof: That's correct
because t will be

1:01:51.318 --> 1:01:53.248
L over v_0
where

1:01:53.246 --> 1:01:56.206
v_0�'s
the magnitude of the initial

1:01:56.210 --> 1:01:58.780
velocity because x velocity is
never changing.

1:01:58.780 --> 1:02:00.660
Acceleration is in the y
direction.

1:02:00.659 --> 1:02:03.599
So the time it takes to cross
will be independent of the fact

1:02:03.596 --> 1:02:05.256
it's falling in the y direction.

1:02:05.260 --> 1:02:09.840
So if you put t equal to
all of this you will find out

1:02:09.835 --> 1:02:11.585
where it will end up.

1:02:11.590 --> 1:02:14.570
And that's how you make
pictures on the television.

1:02:14.570 --> 1:02:18.390
You've got a bunch of plates,
and then you drive charges,

1:02:18.389 --> 1:02:22.079
and if you apply the right
electric field the electron will

1:02:22.079 --> 1:02:24.689
land on a screen and make a
little dot.

1:02:24.690 --> 1:02:27.730
The screen will look like this,
and you're looking at it from

1:02:27.728 --> 1:02:28.588
the other side.

1:02:28.590 --> 1:02:29.850
It'll glow.

1:02:29.849 --> 1:02:33.159
Then you want the dot to move
up and down, you can move the

1:02:33.159 --> 1:02:33.729
voltage.

1:02:33.730 --> 1:02:36.480
Then if you want to move it
back and forth you've got to put

1:02:36.478 --> 1:02:38.158
another set of plates,
not like this,

1:02:38.155 --> 1:02:39.875
but coming out of the
blackboard.

1:02:39.880 --> 1:02:43.020
That way you can move the
electron beam in all directions.

1:02:43.018 --> 1:02:47.258
That's how you scan the
television screen.

1:02:47.260 --> 1:02:51.610
You can also use magnetic
fields, but this is one simple

1:02:51.606 --> 1:02:53.736
way using electric field.

1:02:53.739 --> 1:03:07.099
All right, final thing to
discuss is: what is the force of

1:03:07.099 --> 1:03:16.239
a uniform electric field on a
dipole?

1:03:16.239 --> 1:03:23.969
So let's take an electric field
in the x direction like that.

1:03:23.969 --> 1:03:28.889
It's got a magnitude
E_0 and it's in

1:03:28.894 --> 1:03:30.604
the x direction.

1:03:30.599 --> 1:03:36.839
And in this electric field I
stick a dipole in like that.

1:03:36.840 --> 1:03:38.060
Here is the q.

1:03:38.059 --> 1:03:41.309
Here is the -q.

1:03:41.309 --> 1:03:43.459
Let's make that a.

1:03:43.460 --> 1:03:46.760
Let's make that a.

1:03:46.760 --> 1:03:50.130
So the plus charge will
experience a force like that.

1:03:50.130 --> 1:03:52.550
The minus charge will
experience a force like that.

1:03:52.550 --> 1:03:54.780
This will be q times
E_0.

1:03:54.780 --> 1:03:57.930
That'll be -q times
E_0.

1:03:57.929 --> 1:04:00.749
So dipole as a whole will not
feel any net force,

1:04:00.748 --> 1:04:03.738
because the two parts are
getting pulled by opposite

1:04:03.744 --> 1:04:04.394
amount.

1:04:04.389 --> 1:04:06.659
If the electric field were not
uniform,

1:04:06.659 --> 1:04:09.209
namely if it were stronger here
than here,

1:04:09.210 --> 1:04:10.520
then of course it will drift to
the right,

1:04:10.518 --> 1:04:12.368
but I'm taking uniform electric
field,

1:04:12.369 --> 1:04:14.279
and because the charges are
equal and opposite,

1:04:14.280 --> 1:04:18.110
the net force on it is 0.

1:04:18.110 --> 1:04:22.040
But something is not 0.

1:04:22.039 --> 1:04:24.509
You know what something is
that's not 0?

1:04:24.510 --> 1:04:25.340
Yep?

1:04:25.340 --> 1:04:26.040
Student:  The torque.

1:04:26.039 --> 1:04:27.549
Prof: The torque.

1:04:27.550 --> 1:04:30.660
There is a torque because there
is a force here and the force

1:04:30.664 --> 1:04:31.084
there.

1:04:31.079 --> 1:04:35.619
You can imagine they're trying
to straighten out the dipole so

1:04:35.621 --> 1:04:37.931
it ends up looking like this.

1:04:37.929 --> 1:04:40.809
So let's find the magnitude of
that torque.

1:04:40.809 --> 1:04:45.999
Magnitude of the torque is the
force times the perpendicular

1:04:46.001 --> 1:04:46.971
distance.

1:04:46.969 --> 1:04:49.739
So if this angle is
θ here,

1:04:49.737 --> 1:04:52.927
and you want the perpendicular
distance, it is

1:04:52.931 --> 1:04:55.061
asinθ.

1:04:55.059 --> 1:04:57.399
Then there's another
asinθ from

1:04:57.396 --> 1:04:57.906
that one.

1:04:57.909 --> 1:05:02.379
That's the torque.

1:05:02.380 --> 1:05:10.140
But since 2q a is
p, it's pE_0

1:05:10.141 --> 1:05:13.101
sinθ.

1:05:13.099 --> 1:05:14.369
You can see this makes sense.

1:05:14.369 --> 1:05:18.129
If θ was 0,
if the dipole was aligned with

1:05:18.125 --> 1:05:19.925
the field,
the torque vanishes,

1:05:19.932 --> 1:05:22.602
because if the charges are like
this there is no tendency to

1:05:22.597 --> 1:05:23.047
rotate.

1:05:23.050 --> 1:05:26.400
The biggest torque you get,
if the charges are like this,

1:05:26.402 --> 1:05:28.442
then this gets rotated that
way.

1:05:28.440 --> 1:05:29.600
That gets rotated that way.

1:05:29.599 --> 1:05:31.679
You get the maximum torque.

1:05:31.679 --> 1:05:33.809
Put θ equal to
Π by 2,

1:05:33.806 --> 1:05:36.606
you get a torque of p
times E_0.

1:05:36.610 --> 1:05:39.350
That's also the reason it's
called a dipole moment.

1:05:39.349 --> 1:05:44.279
Now, I'm going to write this as
a cross product,

1:05:44.280 --> 1:05:50.680
and I'm assuming you guys are
familiar with the cross product,

1:05:50.679 --> 1:05:51.729
right?

1:05:51.730 --> 1:05:54.820
You take two vectors,
p and E,

1:05:54.820 --> 1:05:57.040
the cross product has a
magnitude, which is p

1:05:57.041 --> 1:05:59.441
times E times sine of the
angle between them,

1:05:59.440 --> 1:06:04.030
and a direction obtained by
turning a screwdriver from

1:06:04.027 --> 1:06:06.017
p to E.

1:06:06.019 --> 1:06:07.959
So p is like this.

1:06:07.960 --> 1:06:09.610
E is like this.

1:06:09.610 --> 1:06:11.330
Turn a screwdriver from
p to E,

1:06:11.329 --> 1:06:15.549
it goes into the board,
and that's the torque,

1:06:15.550 --> 1:06:18.440
and the dipole will then rotate
until it lines up,

1:06:18.440 --> 1:06:22.260
or if you don't want it to
rotate you've got to provide a

1:06:22.255 --> 1:06:24.635
counter-torque of this
magnitude.

1:06:24.639 --> 1:06:28.179
Now, if you take any dipole and
leave it,

1:06:28.179 --> 1:06:32.359
you know it will like to become
horizontal,

1:06:32.360 --> 1:06:36.160
so there is a certain restoring
torque that tries to rotate it

1:06:36.157 --> 1:06:39.487
so it becomes horizontal,
and it's not very different

1:06:39.487 --> 1:06:41.487
from a spring,
where if you pull it from

1:06:41.487 --> 1:06:43.837
equilibrium,
there is a restoring force that

1:06:43.840 --> 1:06:45.830
brings you back to where you
were.

1:06:45.829 --> 1:06:48.569
So this dipole is happiest when
it's horizontal.

1:06:48.570 --> 1:06:53.090
If you go away from horizontal
the torque brings it back.

1:06:53.090 --> 1:06:56.930
So just like for a spring,
if you've got a force which is

1:06:56.925 --> 1:07:00.755
-kx we can assign a
potential energy U,

1:07:05.119 --> 1:07:08.299
is equal to
-dU/dx.

1:07:08.300 --> 1:07:10.840
This is something I am assuming
you guys know;

1:07:10.840 --> 1:07:13.460
the relation between potential
and force.

1:07:13.460 --> 1:07:15.880
Relation between potential and
force,

1:07:15.880 --> 1:07:18.720
the potential at
x_1 minus the

1:07:18.724 --> 1:07:21.144
potential at
x_2 is the

1:07:21.137 --> 1:07:24.167
integral of the force from
x_1 to

1:07:24.168 --> 1:07:25.838
x_2.

1:07:25.840 --> 1:07:29.410
This is how potential is
defined.

1:07:29.409 --> 1:07:32.599
This is the reason why,
if you knew the potential minus

1:07:32.599 --> 1:07:36.869
the derivative gives the force,
and if you knew the force,

1:07:36.869 --> 1:07:41.389
its integral will give you the
change in potential.

1:07:41.389 --> 1:07:44.909
All right, so now when you do
rotations,

1:07:44.909 --> 1:07:48.919
whatever you had for force you
had for torque,

1:07:48.920 --> 1:07:53.660
and whatever played the role of
x is played by

1:07:53.659 --> 1:07:55.209
θ.

1:07:55.210 --> 1:07:57.800
In other words,
in rotational dynamics torque

1:07:57.797 --> 1:08:00.737
is to force--just like one of
the SAT questions.

1:08:00.739 --> 1:08:04.339
Torque is to force as
θ is to x,

1:08:04.344 --> 1:08:06.044
and work is just work.

1:08:06.039 --> 1:08:10.949
If you want to calculate the
work done by the dipole,

1:08:10.949 --> 1:08:15.739
but if you like the potential
energy when it's at an angle

1:08:15.744 --> 1:08:20.964
θ and its potential
energy when it's at angle 0 is

1:08:20.962 --> 1:08:25.252
the integral of the torque
dθ from 0 to

1:08:25.252 --> 1:08:27.022
θ.

1:08:27.020 --> 1:08:29.460
Now, what is the torque?

1:08:29.460 --> 1:08:34.290
The torque is
-pEsinθ

1:08:34.291 --> 1:08:40.331
dθ from 0 to
θ.

1:08:40.328 --> 1:08:44.178
There's a minus sign because if
θ tries to

1:08:44.176 --> 1:08:47.306
increase, the torque tries to
decrease it.

1:08:47.310 --> 1:08:50.930
That's why it has got a minus
sign.

1:08:50.930 --> 1:08:53.180
Now, integral of minus
sinθ is

1:08:53.182 --> 1:08:55.652
cosθ,
so you will get

1:08:55.648 --> 1:09:00.198
pEcosθ -
pEcos 0 which is

1:09:00.201 --> 1:09:03.641
-pE,
and that is supposed to equal

1:09:03.643 --> 1:09:07.273
to U(θ) -
U(0).

1:09:07.270 --> 1:09:12.840
By comparing the two
expressions you can identify

1:09:12.841 --> 1:09:17.901
U(θ) to be--
sorry, this is

1:09:17.899 --> 1:09:22.019
U(0) minus
U(θ),

1:09:22.020 --> 1:09:25.570
therefore
U(θ) is

1:09:25.567 --> 1:09:29.507
equal to
-pEcosθ,

1:09:29.510 --> 1:09:37.430
or just -p dot E.

1:09:37.430 --> 1:09:43.190
So that's the final formula you
have to remember,

1:09:43.194 --> 1:09:47.764
that the--can we bring it down
here?

1:09:47.760 --> 1:09:51.210
When you have a dipole in an
electric field,

1:09:51.210 --> 1:09:54.930
it has a potential energy
associated with the angle which

1:09:54.925 --> 1:09:59.995
is -p dot E,
and if you draw a picture of

1:10:00.002 --> 1:10:05.412
that as a function of
θ it goes like

1:10:05.408 --> 1:10:06.308
this.

1:10:06.310 --> 1:10:10.080
θ = 0 is when the
dipole is parallel to the field.

1:10:10.078 --> 1:10:15.158
That's when it has the minimum
energy, -pE.

1:10:15.159 --> 1:10:17.869
At 90 degrees, energy is 0.

1:10:17.869 --> 1:10:22.009
At 180 degrees, it's maximum.

1:10:22.010 --> 1:10:25.790
And the torque is just
-dU/dθ and

1:10:25.789 --> 1:10:29.789
you can see that the torque here
and there are zero,

1:10:29.788 --> 1:10:32.098
but this is the point of stable
equilibrium.

1:10:32.100 --> 1:10:34.680
That's the point of unstable
equilibrium.

1:10:34.680 --> 1:10:37.880
See, if this was a potential
energy like a shape of the

1:10:37.877 --> 1:10:40.007
ground,
if you left a marble there

1:10:40.006 --> 1:10:42.586
it'll stay there,
but if it moved a little bit,

1:10:42.585 --> 1:10:43.835
it would roll down hill.

1:10:43.840 --> 1:10:46.630
But if you left the marble here
it'll stay there,

1:10:46.626 --> 1:10:49.526
but if you move it,
it'll rattle back and forth.

1:10:49.529 --> 1:10:50.999
That's a stable equilibrium.

1:10:51.000 --> 1:10:52.100
That's unstable.

1:10:52.100 --> 1:10:55.760
So for the dipole when it's
parallel to the field you are

1:10:55.760 --> 1:10:59.420
here, and at anti-parallel to
the field you are there.

1:10:59.420 --> 1:11:02.330
The difference is when you're
parallel and you move it a

1:11:02.328 --> 1:11:05.348
little bit,
it'll have stable oscillations,

1:11:05.354 --> 1:11:09.784
whereas if at anti-parallel if
you move it a little bit it'll

1:11:09.779 --> 1:11:12.949
flip over completely and come
down here.

1:11:12.948 --> 1:11:18.378
All right, so I'm going to
summarize the main points so you

1:11:18.381 --> 1:11:21.381
can carry that with you,
okay?

1:11:21.380 --> 1:11:24.760
We saw today that we should
think in terms of electric field

1:11:24.756 --> 1:11:28.066
from now on, and we no longer
talk about direct interaction

1:11:28.074 --> 1:11:29.224
between charges.

1:11:29.220 --> 1:11:33.660
We say charges produce fields,
and fields act on charges to

1:11:33.661 --> 1:11:34.581
move them.

1:11:34.578 --> 1:11:37.458
The force of a field on a
charge is just q times

1:11:37.462 --> 1:11:38.052
E.

1:11:38.050 --> 1:11:41.490
The field is found by adding
the field due to all the charges

1:11:41.485 --> 1:11:44.055
in the universe,
provided they're all at rest,

1:11:44.063 --> 1:11:46.243
and you just add by Coulomb's
Law.

1:11:46.238 --> 1:11:49.688
So we found the field due to a
dipole allowing the axis and

1:11:49.685 --> 1:11:51.345
perpendicular to the axis.

1:11:51.350 --> 1:11:54.160
We saw the notion of field
lines as a good way to visualize

1:11:54.162 --> 1:11:56.542
what's going on in the vicinity
of the charges.

1:11:56.538 --> 1:11:59.468
The lines tell you the
direction where the field is,

1:11:59.470 --> 1:12:01.440
and they tell you in your line
density,

1:12:01.439 --> 1:12:04.109
lines cutting a unit area
perpendicular to them,

1:12:04.109 --> 1:12:05.869
the strength of the field.

1:12:05.868 --> 1:12:09.748
Then I calculated for you the
field of a dipole along the axis

1:12:09.746 --> 1:12:11.776
and perpendicular to the axis.

1:12:11.779 --> 1:12:14.309
There are a lot of formulas,
but one thing you should carry

1:12:14.314 --> 1:12:14.974
in your head.

1:12:14.970 --> 1:12:18.850
When you've got two equal and
opposite charges and you go very

1:12:18.849 --> 1:12:22.029
far, the field will go like 1
over r^(3).

1:12:22.029 --> 1:12:25.539
The 1 over r^(2) part of
them is canceled,

1:12:25.537 --> 1:12:26.047
okay?

1:12:26.050 --> 1:12:28.350
That's the main point,
and it goes like 1 over

1:12:28.345 --> 1:12:31.245
r^(3) times 2 in one
place, and 1 over r^(3)

1:12:31.253 --> 1:12:32.533
times 1 in one place.

1:12:32.529 --> 1:12:33.489
It doesn't matter.

1:12:33.488 --> 1:12:35.938
The main thing is it's 1 over
r^(3).

1:12:35.939 --> 1:12:39.359
Finally, if you take a dipole
and you put it in an electric

1:12:39.359 --> 1:12:41.409
field,
it tends to line up because

1:12:41.409 --> 1:12:43.509
there's a torque,
p cross E,

1:12:43.506 --> 1:12:44.446
trying to line it up.

1:12:44.448 --> 1:12:48.058
With that torque you can
associate a potential energy by

1:12:48.059 --> 1:12:51.799
the usual formula that the
torque is minus a derivative of

1:12:51.798 --> 1:12:53.438
the potential energy.

1:12:53.439 --> 1:12:56.309
That potential energy is
-p dot E.

1:12:56.310 --> 1:12:59.310
Some of these things may come
in handy later on.

1:12:59.310 --> 1:13:02.240
So you don't have to memorize
them, but they'll be involved

1:13:02.243 --> 1:13:02.803
later on.

1:13:02.800 --> 1:13:08.000