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