The constant `A` in the Richardson-Dushman equation for tungsten is `60 xx 10^(4) A m^(-2)`. The work function of tungsten is `4.5 eV`. A tungsten cathode having a surface area `2.0 xx 10^(-5) m^2` is heated by a `24 W` electric heater. In steady state, the heat radiated by the cathode equals the energy input by the heater and the temperature becomes constant. Assuming that the cathode radiates like a blackbody, calculate the saturation current due to thermions. Take Stefan constant `= 6 xx 10^(-8) W m^(-2) K^(-4)`. Assume that the thermions take only a small fraction of the heat supplied.

Explanation

Given, `A = 60 xx (10^4)A/m^2 - k^2`
` phi = 4.5 eV,`
` delta = 6 xx (10^-8)omega/ m^2 -k^4 `
` S = 2 xx (10^-5)m^2, `
` k = 1.38 xx (10^-23) J/K `
` H= 24 omega `
The cathode acts as a black body ,
i.e. emissivity = 1
` :. E = delta AT^4 (A is area) `
` rArr T^4 = E/ sigma A `
` = (24/(6 xx (10^-8) xx 2 xx (10^-5)))`
` = 2 xx (10^13)K = 20 xx (10^12)K `
` rArr T= 2.1147 xx (10^3) = 2.114.7 K. `
` Now, i = AST^2 e^(phi/ KT) `
` = 6 xx (10^5) xx 2 xx (10^-5) xx `
` (2114.7)^2 xx e^((-4.5 xx 1.6 xx (10^-19))/(1.38 xx (10^-23) xx 2114.7)) `
` = 1.03456 xx (10^-3)A = 1mA. `