I have found the 1D differential equation relating voltage and position for a vacuum diode in the space charge region, which is(adsbygoogle = window.adsbygoogle || []).push({});

[tex] \frac{d^2V}{dx^2} = constant * V^{-1/2} [/tex]

and I know the solution to be

[tex] V(x) = V_0 \left(\frac{x}{d}\right)^{4/3} [/tex]

which is found by multiplying both sides by [itex] V' = \frac{dV}{dx} [/itex] and then integrating the following expression with homogeneous boundary conditions:

[tex] \int V' dV' = constant*\int V^{-1/2} dV [/tex]

What I don't understand is why this trick is even necessary. As far as I can tell the differential equation can be solved by a simple separation of variables, which gives an answer of

[tex] V(x) = V_0 \left(\frac{x}{d}\right)^{4/5} [/tex]

The two answers are different so obviously it's a mistake to separate variables, but for the life of me I can't tell where it is. Could anyone enlighten me? Thanks.

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# Homework Help: Derivation of voltage for vacuum diode space charge region

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