## 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}/

*d*

^{2}

where

*J*= current density (mA mm

^{-2})

*V*= potential difference between anode and cathode (V)

_{d}*d*= distance between anode and cathode (mm)

*K*is a constant given by

*K*= (4/9)

*ε*

_{0}(-2

*q/m*)

^{1/2}with units mA V

^{-3/2}, where

*ε*= permittivity of free-space ~ 8.854187817*10

_{0}^{-12}F m

^{-1}[*1]

*q/m*= charge (C) to mass (kg) ratio of particle (absolute value). For an electron,

*q/m*= 1.758820150*10

^{11}C kg

^{-1}[*2]

For an electron,

*K*~ 0.0023340 mA V

^{-3/2}.

Note units on left-side of constant: (F m^{-1}) C^{1/2} kg^{-1/2} = (C^{2} 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

Child's Law is a fairly direct result of the Poisson equation:
*∇*^{2} *V* = - *ρ* / *ε _{0}*

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* *v*^{2} = -*q*
*V*, for mass *m* and charge *q*. This gives

*∇*^{2} *V* = dV'/dx = A *V*^{-1/2}

where *A* = -*J* *ε _{0}*

^{-1}(-(1/2)

*m/q*)

^{1/2}is a constant and

*V*' = d

*V*/d

*x*.

Substitute d*x* = d*V*/*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*' d*V*' = *A* *V*^{-1/2} d*V*

*V*' = d*V*/d*x* = 2 *A*^{1/2} *V*^{1/4}

*V*^{-1/4} d*V* = 2 *A*^{1/2} d*x*

(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* *ε _{0}*

^{-1}(-(1/2)

*m/q*)

^{1/2}implies

*J*=

*K*

*V*

^{3/2}

*d*

^{-2}where

*K*= (4/9)

*ε*

_{0}(-2

*q/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