Engineering & Science
Child-Langmuir Three-Halves (V^3/2) Space-Charge Current Limit & Perveance
Calculate space-charge-limited electron current density J = 0.002334 * (V^1.5) / (d^2) in mA/cm^2 and diode perveance P = I / V^1.5 for vacuum tubes, X-ray anodes, and ion thruster grids.
Why vacuum diodes and ion-thruster accelerator grids obey V^1.5—NOT Ohm's Law!
Even if a hot cathode can emit unlimited electrons, the negative cloud ("space charge") of electrons already in flight between cathode and anode depresses the electric field at the cathode surface down to zero—capping maximum steady-state current! Clement Child (1911) and Irving Langmuir (1913) solved Poisson's equation coupled with energy conservation to prove the universal 3/2-power Child-Langmuir Law: J = ((4 * eps0) / 9) * SQRT((2 * e) / m_e) * (V^1.5 / d^2)!
Evaluating fundamental electron constants yields prefactor K_e = 2.334e-6 A/V^1.5 = 0.002334 mA/V^1.5—and notice a neat dimensional gift: because 1 m^2 = 10^4 cm^2 cancels d_m^2 = 10^-4 d_cm^2 in the denominator, J_mA_cm2 = 0.002334 * (V^1.5) / (d_cm^2) uses the EXACT same constant whether you work in meters or centimeters! Across a d = 0.50 cm gap at V = 1,000 V, J = 295.23 mA/cm^2—and quadrupling voltage to 4,000 V multiplies space-charge current by 4^1.5 = 8x to 2,361.84 mA/cm^2!
| Anode Voltage & Gap (Electron Diode) | Scaling Factor | Space-Charge Density J (mA/cm^2) | Microperveance (uA / V^1.5) |
|---|---|---|---|
| V = 1,000 V, d = 0.50 cm (A = 2 cm^2) | Baseline V^1.5 / d^2 | 295.23 mA/cm^2 (I = 590.46 mA) | 18.67 uP (Geometry invariant!) |
| V = 4,000 V (4x Voltage), d = 0.50 cm | 4^1.5 = 8.0x Current! | 2,361.84 mA/cm^2 | 18.67 uP (Unchanged!) |
| Invalid Zero/Negative Gap d <= 0 | Division by d^2 Guard | Guarded Input Error | "Invalid Diode Geometry" |
Child-Langmuir Electron Space-Charge Current Density J (mA/cm^2)
Evaluates 0.002334 * (V_volts ^ 1.5) / (d_cm ^ 2) for planar non-relativistic electron gaps.
=IF(OR(B2 < 0, B3 <= 0), "Invalid Diode Geometry", ROUND(0.002334 * (B2 ^ 1.5) / (B3 ^ 2), 2))Planar Diode Microperveance (uA / V^1.5 = 2.334 * Area / d^2)
Computes purely geometric microperveance 2.334 * A_cm2 / (d_cm^2), independent of applied voltage.
=IF(OR(B3 <= 0, B4 <= 0), "Invalid Diode Geometry", ROUND((2.334 * B4) / (B3 ^ 2), 2))Try the example
Use the original values or change them to test your understanding.