WEBVTT
00:00:00.051 --> 00:00:02.794
This is College Physics
Answers with Shaun Dychko.
00:00:03.377 --> 00:00:07.051
We are told that the first minimum
for this diffraction pattern
00:00:07.411 --> 00:00:09.754
occurs at an angle of
fourteen and a half degrees.
00:00:10.108 --> 00:00:13.577
So if theta subscript one, the
one meaning the first minimum
00:00:14.605 --> 00:00:18.560
and we have this formula that
tells us the diffraction pattern
00:00:18.560 --> 00:00:22.171
based on the width of the slit
and the wavelength of a light
00:00:22.440 --> 00:00:27.051
but neither of these pieces of information
*D* and Lambda are given to us
00:00:27.651 --> 00:00:31.177
but we can actually deal with that because
we can figure out what their ratio is
00:00:32.228 --> 00:00:33.422
because lambda over *D* .
00:00:33.760 --> 00:00:38.285
If you rearrange this formula
we can divide both sides by *mD*.
00:00:40.582 --> 00:00:45.028
You end up with Lambda over *D*
equaling sine theta over *m*
00:00:45.354 --> 00:00:47.200
and both theta and *m* we are given.
00:00:47.382 --> 00:00:50.485
So sine of fourteen and a
half divided by *m* being one
00:00:50.800 --> 00:00:55.811
when theta is fourteen and a half
gives us a ratio of 0.25038.
00:00:56.411 --> 00:00:59.931
So this is something we can
substitute in place of lambda over *D*
00:01:00.348 --> 00:01:01.971
when we're solving for theta.
00:01:03.497 --> 00:01:07.405
So we're going to find the
angle to the second minimum
00:01:08.170 --> 00:01:11.822
and so we're going to rearrange
this to solve for theta
00:01:12.434 --> 00:01:13.977
divide both sides by capital *D*
00:01:14.440 --> 00:01:16.097
and then take the inverse sine both sides
00:01:16.514 --> 00:01:18.360
so theta is the inverse
sine of *m lambda* over *D*
00:01:18.720 --> 00:01:21.348
and we have something to substitute
in place of lambda over *D*.
00:01:22.611 --> 00:01:25.662
So down here we have theta subscript
two for the second minimum
00:01:25.942 --> 00:01:31.657
is the inverse sine of *m* being the
number two now times is ratio 0.25038
00:01:31.908 --> 00:01:35.074
giving an angle of 30.1 degrees
to the second minimum
00:01:35.931 --> 00:01:38.862
and then the third minimum
will occur at inverse sine of
00:01:38.862 --> 00:01:45.354
three times that ratio 48.7 degrees and
part *c* asks us, will a fourth minimum occur
00:01:45.617 --> 00:01:48.594
and so let's try to find the angle
inverse sine of four times this number
00:01:48.777 --> 00:01:51.777
but we find that this
becomes greater than one
00:01:51.960 --> 00:01:54.445
and there is no inverse sine
of a number greater than one
00:01:54.794 --> 00:01:58.480
and so it's undefined and there
is no fourth order minimum.
00:02:00.400 --> 00:02:05.742
Part *d* says compare the angular
width of the first central maximum
00:02:06.988 --> 00:02:10.771
to the angular width of the next maximum
00:02:11.870 --> 00:02:17.434
and so the next maximum occurs between
the minimum here and the minimum here
00:02:17.760 --> 00:02:22.880
and so we have a maximum intensity here
and that's the angular width of that
00:02:23.068 --> 00:02:25.737
which I've called Alpha two
that we're going to compare
00:02:26.051 --> 00:02:30.371
to the angular width this central
maximum which is Alpha one,
00:02:31.771 --> 00:02:38.537
so alpha one is the angle between
the borders of this central maximum
00:02:38.537 --> 00:02:42.891
and those borders are the
first minimum and you decide
00:02:43.777 --> 00:02:47.897
and so that's going to
be a two times theta one
00:02:48.142 --> 00:02:54.720
so this alpha one is gonna be from
here to this central dotted line here
00:02:55.811 --> 00:02:57.645
which is fourteen and a half degrees
00:02:58.200 --> 00:03:03.451
and then it's going to go that many
degrees again so it's twice theta one.
00:03:05.400 --> 00:03:10.480
Now theta two is going to be the difference
between Alpha two I should say
00:03:10.720 --> 00:03:15.714
Alpha two is going to be the difference
between theta two and theta one.
00:03:15.800 --> 00:03:16.645
So it's this here.
00:03:16.925 --> 00:03:18.742
So it's theta two minus theta one
00:03:20.308 --> 00:03:24.422
and then we're gonna compare
the ratio of these two things.
00:03:24.422 --> 00:03:29.097
So Alpha one being two times theta one
we're gonna divide that by Alpha two
00:03:29.251 --> 00:03:31.245
which is theta two minus theta one
00:03:31.525 --> 00:03:37.148
and see if indeed this angular
width of the central maximum is
00:03:37.148 --> 00:03:39.948
twice the angular width of the next maximum.
00:03:40.662 --> 00:03:43.502
So itâ€™s two times fourteen and
a half degrees divided by
00:03:43.554 --> 00:03:46.760
30.1 minus fourteen and a half
which is approximately two.
00:03:47.022 --> 00:03:50.668
And so therefore yes the central
maximum is approximately
00:03:51.017 --> 00:03:53.560
two times the angular width
of the next maximum.