Luke.A.Johnson

Child-Langmuir Law

Graph showing Child-Langmuir Law. S and d are constant and equal to 1.

Also known as Child's Law or the Three-Halves Power Law, Child's Law states that the space charge-limited current (SCLC) in a plane-parallel diode varies directly as the three-halves power of the anode voltage ${V_{a}}$ and inversely as the square of the distance $d$ separating the cathode and the anode. That is,

$I_{a}=JS=2.33\times 10^{-6}{\frac {S\nu {V_{a}}^{3/2}}{d^{2}}}$.

Where $I_{a}$ is the anode current, $J$ the current density, and $S$ the area. This assumes the following:

1. The electrodes are planar, parallel, equipotential surfaces of infinite dimensions.
2. The electrons have zero velocity at the cathode surface.
3. In the interelectrode region, only electrons are present.
4. The current is space-charge limited.
5. The anode voltage remains constant for a sufficiently long time so that the anode current is steady.

Extension of Child's Law

By dropping the assumption that the electrode surface is infinite, ...