A Tungsten cathode and a thoriated-tungsten cathode have the same geometrical dimensions and are operated at the same temperature. The thoriated-tungsten cathode gives 5000 times more current than the other one. Find the operating temperature. Take relevant data from the previous problem.

Explanation

Pure tungsten
` phi = 2.6e V`
` A = 60 xx (10^4) A/m^2 =k^2 `
` I = AST^2 e^(-phi/KT) `
` Thoriated tungsten `
` phi = 2.6 eV `
` A = 3 xx (10^4) A/m^2 -k^2 `
` i_(Thoriated Tungsten) = 5000 i_(Tungsten) `
` So, 5000 xx S xx 60 xx (10^4) xx `
` T^2 + e^((-4.5 xx 1.6 xx (10^-19))/ (1.38 xx T xx (10^-23)))`
` = S^3 xx (10^4) xx (T^2) xx e^((-2.6 xx 1.6 xx (10^-19))/(1.38 xx (10^-23) xx T)`
` rArr 1 + (10^4) xx e^((-4.5 xx 1.6 xx (10^-19))/(1.38 xx (10^-23) xx T)) = e^ ((-2.5 xx 1.6 xx (10^-19))/(1.38 xx (10^-23) xx T)) `
` Taking 'In the both sides . `
` rArr 9.21 T = 22029 `
` T = 22029/9.21 = 2391.856K ` .