Chapter 28, Problem 40P

Just as Fourier's law and the heat balance can be employed to characterize temperature distribution, analogous relationships are available to model field problems in other areas of engineering. For example, electrical engineers use a similar approach when modeling electrostatic fields. Under a number of simplifying assumptions, an analog of Fourier's law can be represented in one-dimensional form as

$D=-\epsilon \frac{dV}{dx}$

whereD is called the electric flux density vector, $\epsilon =\text{permittivity}$ of the material, and $V=\text{electroststic}$ potential. Similarly, a Poisson equation for electrostatic fields can be represented in one dimension as

$\frac{d^{2}V}{d{x}^2}=-\frac{{\rho }_v}{\epsilon }$

where ${\rho }_v=\text{charge}$ density. Use the finite-difference technique with $\Delta x=2$ to determine V for a wire where $V\left(0\right)=1000,V\left(20\right)=0,\epsilon =2,L=20,\text{and}{\rho }_v=30$.