Townsend’s Current Growth Equation

Townsend’s Current Growth Equation 

• At any distance x, let number of electrons be n_x. Since current is the rate of flow of electrons, so we will start with the electron equation.

dn_x = n_x^.α. dx/

where,

α = Townsends first ionization coefficient which represents the number of electrons liberated by a free electron due to its collisions with gas molecule per unit distance in the direction of field. It depends on gas pressure, p, and E/P.

dn_x / dx = α. n_x

Integrating both sides,

∫1/n_x dn_x
= ∫α. dx

L_nn_x =
α x + A

At x = 0,n_x = n_o

L_nn_o
= 0 + A

A = L_nn_o

L_nn_x =
α x + L_nn_o

L_n(n_x –
n_o) = α x

L_n(n_x /
n_o) =  e^α.x

n_x = n_o e^αx

Number of Electrons reaching the anode will be i.e.,at x = d,

n_d = n_o e^αd

• Since average current in the gap is equal to number of electrons produced by one electron travelling per second.

n_d / t = n_o / t  e^αd

I_o = I_o e^αd… (1)

Where I_o =  initial current at cathode.

 e^αd = electron avalanche.

• It represents number of electrons produced by one electron travelling from cathode to anode.
• Equation (1) is called as Townsends current growth equation due to primary ionization coefficient α.

Drawbacks of Equation

• Townsend’s observed that current increases more rapidly as compared to Equation (1). To explain this, he explained secondary ionization coefficient y which influences current growth. The increase in the current is due to positive ion impact, metastable or photons on the cathode which liberate secondary electrons.