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bobred
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Homework Statement
A slab of radioactive material of thickness L lies in the x-y plane surrounded by a material that can be thought of as extending to +/- infinity from -L/2 and +L/2. The system is in steady. Find the general solution for the diffusion equations in each region(ok). Find the form of the general solution in Region 1 that is an even function and in 2 that is physically meaningful(ok). Find the general solution that satisifes the boundary conditions(?)
Homework Equations
Region 1
[tex]0=D_1 \frac{\partial^2 c}{\partial x^2}+H[/tex]
for [itex]\left| z \right|<L/2[/itex]
Region 2
[tex]0=D_2 \frac{\partial^2 c}{\partial x^2}-Rc[/tex]
for [itex]\left| z \right|>L/2[/itex]
The Attempt at a Solution
The general steady state solutions for each region are
[tex]c=-\frac{H}{2D_1}z^2+Az+B[/tex]
and
[tex]c=Ce^{\lambda z}+De^{-\lambda z}[/tex]
[itex]\lambda=\pm \sqrt{\frac{R}{D_2}} [/itex]
In region 1 to be even A=0, for region 2 the concentration must tend to zero as z approaches +/- infinity, so
[tex]c=-\frac{H}{2D_1}z^2+B[/tex]
[tex]c(z)=\begin{cases}
De^{-\lambda z} & z>\tfrac{L}{2}\\
Ce^{\lambda z} & z<-\tfrac{L}{2}
\end{cases}[/tex]
The flux is the same on both sides and concentration continuous. I am having difficultly picturing these, what is happening at z=0 and z=+/-L/2.
Any pointers would be greatly appreciated.
James