WEBVTT
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SARA DON: Here, I present a
spherical distribution problem
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and how I went about finding
an approximate solution
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in Mathematica.
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If you observe light that passes
through a translucent spherical
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object or radiation
that is transmitted
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through a spherical
attenuator, the intensity
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is not evenly
distributed the intensity
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of transmitted
radiation is described
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by this exponential
function where the intensity
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distribution is governed
by the material thickness x
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and density lambda.
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Let's plot this function.
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You can see that as the
material thickness x increases,
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the intensity of the
transmitted radiation
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decreases exponentially.
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Let's make a function to model
the transmitted radiation
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intensity in 3D.
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In this module, I
defined i, i0, and lambda
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to be local variables.
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c is the function
for a hemisphere.
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We want to look at the
effect on a sphere,
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but it's easier to
work with a hemisphere.
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So I multiplied the
hemisphere by 2.
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And the function
returns intensity
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as a function of
x and y positioned
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where the xy plane is the
projection of the sphere.
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Let's have a look at the
intensity distribution in 3D.
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This plot shows us
that the intensity
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at the edges of
the projection is
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greater than the intensity in
the middle of the projection.
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It's a bit hard to
visualize, so here's
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another way to look at it.
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You can see that the intensity
is highest around the edges
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and lowest in the center
of the projection.
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I did this by making
the arrow length
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a function of the intensity.
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Now let's see what happens when
we replace lambda, which was
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a constant, with a function.
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This means that we are
changing the materials density
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as a function of
the sphere's radius.
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I'm going to alter
the density function
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I made before so
that we can specify
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different functions for lambda.
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Here, I'll show
you three functions
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I tried out earlier for
lambda to try and make
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the intensity constant.
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There's a cosine function,
a nested sine function,
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as suggested by
Professor Carter,
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and a spherical Bessel function.
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Let's check that
our function works
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on the lambda as a constant
case we tried before.
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OK, good, it works.
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Now let's look at our
new density functions.
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The first one is
the cosine function.
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You can see that the shape
is not quite a hemisphere
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and now looks a bit
closer to a cylinder.
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Next is the sine function.
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It looks a lot like
the cosine function,
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more like a cylinder
than a hemisphere.
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And last is the Bessel function.
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It also looks a lot like the
sine and cosine functions.
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Since the sine, cosine,
and Bessel functions
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are very similar, I'll
plot them a different Way
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So you can see
them more clearly.
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Here, the blue line is
the Bessel function.
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The red line is the
cosine function,
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and the yellow line is
the nested sine function.
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OK, since it's hard to see
from the 3D plots which
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of the density functions
is closest to giving me
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constant intensity,
Professor Carter
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suggested that I make a
numerical error analysis.
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If the intensity
were constant, we
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would get a cylinder
in a 3D plot.
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So I found the height of
that theoretical cylinder
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by finding the minima for each
of the intensity functions.
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Then I subtracted the
volume of the intensity
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from the volume of the
theoretical cylinder,
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and the square of this
difference is the error.
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Here's a list of the errors.
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But it will be easier to
compare if I make a bar chart.
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OK, now you can clearly see
which function is the winner.
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When the density is
constant, the intensity
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is the furthest
from being constant.
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The sine and cosine
functions are
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much closer to making
the intensity constant.
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But the spherical
Bessel function
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definitely makes the most
even intensity distribution.
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There is no perfect solution
because at the limit
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of the edge of the sphere,
the material thickness
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goes to zero.
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And therefore, the density
function must go to infinity.
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But it looks like
the Bessel function
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makes a good approximation.