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The Possion equation in its simplest
form is
e0erd2V/dx2 = -r(x). |
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Differentiating the potential including the
discontinuities thus will gives us the charge distribution. We can do that very
easily in a qualitative way a shown on the left hand side below. |
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However, an infinitely sharp discontinuity will
not be noticed in the dV/dx curves, The curves we get
are identical to the old
curves without a discontinuity. |
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Infinitely sharp discontinuities, or singularities
in general, often 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 at the minimum in the order of the atom size) |
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Differentiating graphically in a qualitative way
now is easy, this is shown on the right hand side. |
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We now get a sharp "wiggle" in the
charge distribution, corresponding to a dipole layer right at the
interface. |