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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.