Re: Does the Child - Langmuir law apply to separate electrical charges too ?

The Child-Langmuir law is valid for space-charge-limited currents.
It says that under these conditions the current is proportional to the
3/2 power of the voltage difference between the anode and the cathode.
The range of voltage for which the Child-Langmuir law is valid depends
on the temperature of the cathode.  It is larger for higher temperatures,
since the density of emitted electrons near the surface of the cathode
rises with temperature.  For more details, see, for example, the article
on "thermionic emission" in the Encyclopaedia Britannica Online.

The current from a cold cathode would not be space-charge-limited.  If
the current is limited by the quantum mechanical tunneling of electrons
through the surface of the cathode, then the current should be an
exponential function of the voltage.