Electrical Engineering ⇒ Topic : Child-Langmuir Law

CHILD-LANGMUIR LAW

It states that

When a vacuum diode is operated in the space charge limited region, the plate current is directly proportional to the three halves power of the plate voltage.Mathematically, it can be written as follows:

                                                  i_p ∝ e_p^3/2                         ...............(1)

Proof

From Figure (a), two plane and parallel plate surfaces have been considered representing the cathode and anode of a vacuum diode. Let the position of cathode be at x = 0 and the anode be at x = d.

The rate of flow of charge through a unit area in the direction of applied field is given by

                                    J = -nev                                     ...............(2)

                                      = -ρv         ( ∴   ρ = ne)             ..............(3)

               figure (a) Two plane and parallel surfaces representing cathode and anode.

where n is the number of electrons per cubic metre, e is the charge of an electron, y is the drift velocity of the electron in m/sec. J is current density, and ρ is total charge of the electrons.

By Poisson's equation, the gradient of electric field at any point can be written as:

                                                               ............(4)

                                                               ............ (5)

Therefore, putting Eq. (5) in Eq. (4), we get

where V is the electric potential at that point.

                                                         [Using Eq. (3)]           .....(6)

If the cathode is at zero potential, the energy acquired by the electron on being accelerated through a potential V is given by

                                                                         ...........(7)

Using Eq. (7), Eq. (6) can be expressed as:

                                     .............. (8)