Child's Law

Definition

Child's Law (or the Child-Langmuir Law or three-halves-power law) gives the maximum space-charge-limited current in a planar diode of infinite radius (i.e. one-dimensional beam) as a function of the length and potential difference between anode and cathode.

J = K V_d^3/2 / d²

where

Note units on left-side of constant: (F m^-1) C^1/2 kg^-1/2 = (C² J^-1) m^-1 C^1/2 kg^-1/2 = C^5/2 J^-1 (J^-1/2 s^-1) = (C s^-1) (J/C)^-3/2 = A V^-3/2.

Use in CPO Charged Particle Optics Software

The CPO software applies a Child's Law relationship near cathode surfaces in the simulation of space-charge limited thermionic Cathode Emissions and can apply a correction (Langmuir relationship) for non-zero temperature.

Derivation

∇² V = - ρ / ε₀

where charge density ρ = J/v, for velocity v, and where v can be expressed in terms of kinetic energy and related to potential energy: (1/2) m v² = -q V, for mass m and charge q. This gives

∇² V = dV'/dx = A V^-1/2
where A = -J ε₀^-1 (-(1/2) m/q)^1/2 is a constant and V' = dV/dx.

Substitute dx = dV/V' to obtain a separable differential equation and solve, while applying boundary conditions V(x=0) = 0, V(x=d) = V_d, and V'(x=0) = 0:

V' dV' = A V^-1/2 dV
V' = dV/dx = 2 A^1/2 V^1/4
V^-1/4 dV = 2 A^1/2 dx
(2/3)V^3/4 = A^1/2 x
where A = (4/9)V_d^3/2 d^-2 (due to V(x=d) boundary condition)

Now knowing A, we can solve for J in the expression for A:

(4/9)V^3/2 d^-2 = A = -J ε₀^-1 (-(1/2) m/q)^1/2 implies
J = K V^3/2 d^-2 where K = (4/9) ε₀ (-2q/m)^1/2

which is the desired result.

Note also that at x=0, v=0 and ρ=∞.

References

• [*1] 2006 CODATA
• [*2] 2006 CODATA
• [*] Wikipedia: Space-Charge has some comments.
• [*] Solving Poisson's Equation: Child-Langmuir Law, Electricity and Magnetism, Professor Grant W. Mason