


The
Poisson equation
in its simplest form reads 


e_{0}e_{r} · 
d^{2}V
dx^{2} 
= – 
r(x) 




Differentiating the potential V
including possible discontinuities thus will gives us the charge distribution
r(x). We can do that very easily in a
qualitative way as shown on the left hand side below. 





Note that an infinitely sharp discontinuity will
not be noticed in the
dV/dx curves. The curves we get are identical to the
old curves that did not
contain a discontinuity. 

But infinitely
sharp discontinuities, or singularities in general, mostly do not make sense in
physics. All we have to do therefore, is to redraw the potential with the
discontinuity spread over a small distance (obviously in the order of the atom
size at the very minimum) 


Differentiating graphically in a qualitative way
now is easy, this is shown on the right hand side. 


We now get a sharp "wiggle" in the
charge distribution, corresponding to a dipole
layer of charge right at the interface. 