One-dimensional solution of Poisson's equation

So, how do we actually solve Poisson's equation,

$$\frac{\partial^2\phi}{\partial x^2} +\frac{\partial^2\phi}{\... ...artial^2\phi}{\partial z^2} = -\frac{\rho(x,y,z)}{\epsilon_0},$$ (703)

in practice? In general, the answer is that we use a computer. However, there are a few situations, possessing a high degree of symmetry, where it is possible to find analytic solutions. Let us discuss some of these solutions.

Suppose, first of all, that there is no variation of quantities in (say) the $y$- and $z$-directions. In this case, Poisson's equation reduces to an ordinary differential equation in $x$, the solution of which is relatively straight-forward. Consider, for instance, a vacuum diode, in which electrons are emitted from a hot cathode and accelerated towards an anode, which is held at a large positive potential $V_0$ with respect to the cathode. We can think of this as an essentially one-dimensional problem. Suppose that the cathode is at $x=0$ and the anode at $x=d$. Poisson's equation takes the form

$$\frac{d^2\phi}{dx^2} = - \frac{\rho(x)}{\epsilon_0},$$ (704)

where $\phi(x)$ satisfies the boundary conditions $\phi(0)=0$ and $\phi(d)=V_0$. By energy conservation, an electron emitted from rest at the cathode has an $x$-velocity $v(x)$ which satisfies

$$\frac{1}{2} m_e v^2(x) - e \phi(x) = 0.$$ (705)

Finally, in a steady-state, the electric current $I$ (between the anode and cathode) is independent of $x$ (otherwise, charge will build up at some points). In fact,

$$I = -\rho(x) v(x) A,$$ (706)

where $A$ is the cross-sectional area of the diode. The previous three equations can be combined to give

$$\frac{d^2\phi}{dx^2} = \frac{I}{\epsilon_0 A}\left(\frac{m_e}{2 e}\right)^{1/2}\phi^{-1/2}.$$ (707)

The solution of the above equation which satisfies the boundary conditions is

$$\phi = V_0 \left(\frac{x}{d}\right)^{4/3},$$ (708)

with

$$I = \frac{4}{9}\frac{\epsilon_0 A}{d^2}\left(\frac{2 e}{m_e}\right)^{1/2} V_0^{3/2}.$$ (709)

This relationship between the current and the voltage in a vacuum diode is called the Child-Langmuir law.

Let us now consider the solution of Poisson's equation in more than one dimension.