The saturation current from a thoriated-tungsten cathode at `2000 K` is `100 mA`. What will be the saturation current for a pure-tungsten cathode of the same surface area operating at the same temperature? The constant `A` in the Richardson-Dushman equation is `60 xx 10^4 A m^(-2) K^(-2)` for pure tungsten and `3.0 xx 10^(4) A m^(-2) K^(-2)` for thoriated tungsten. The work function of pure tungsten is `4.5 eV` and that of thoriated tungsten is `2.6 eV`.

Explanation

`i=(AST^2) e^(-phi/KT)`
` i_1 = i` i_1 = 100 mA `
` A_1 = 60 xx (10^4) A_2 = 3 xx (10^4)`
` S_1 = S S_1 = SX `
` T_1 = 2000 T_2 = 2000 `
` phi _1 = 4.5 eV phi_2 = 4.6 e V `
` K = 1.38 xx (10^-23) J/K`
` i= (60 xx (10^4))(S) xx (2000)^2 `
` e^((-4.5 xx 1.6 xx (10^-19))/(1.38 xx (10^-23) xx 2000))`
` 100 = (3 xx (10^4)) (S)(2000^2)`
` e^((-2.6 xx 1.6 xx (10^-19))/(1.38 xx (10^-23) xx 2000))`
` Dividing the equation i/100`
` rArr 20 xx e^((-4.5 xx 1.6 xx (10^-19))/(1.38 xx (10^-23) xx 2000) - (-2.6 xx 1.6 xx (10^-19))/(1.38 xx (10^-23) xx 2000))`
` rArr i/100 = 20 xx e^((2.6- 4.5 xx 1.6 xx 10)/(2 xx 1.38))`
` rArr i/100 = 20 xx e ^((16 (2.6 xx 4.5))/ (2 xx 1.38)) `
rArr i/100 = 20 e^((8 xx (-1.9))/1.38)`
` rArr i/100 = 20 e^-11.014 `
`rArr i/100 = 20 xx 0.000016 `
` rArr i = 20 xx 0.0016 `
`= 0.0329A = 33mA.`