Question 3.10: Consider again Example 3-5, which has been solved analytical...
Consider again Example 3-5, which has been solved analytically after a linearization. Write a MATLAB program to calculate and plot the potential distribution using the function ‘ode45’ for obtaining numerical solutions of differential equations.
We first transform Poisson’s equation into a standard form with the substitution dV / dr = U . This results in two coupled first order differential equations.
\begin{aligned}&\frac{d V}{d r}=U, \\\\&\frac{d U}{d r}=-\frac{2}{r} U+D(\exp (-C . V)-1)\end{aligned}
where C = q / k _{ B } T_{ e } and D = n _{0} q / \varepsilon_{0} . Choosing numerical values of C =0.25 and D=4 , we normalize the Debye length \lambda_{D}=1 / \sqrt{C . D}=1 . Assuming that the value of the potential V at ρ = 0 has a value of 10 and that the electric field U=-\frac{d V}{d r} at the same location has a value of 1 , we are able to numerically evaluate the potential distribution in space. The numerical results are shown below.
