Question

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

Answer 1

For the pure tungsten cathode,
work function, ϕ = 4.5 eV
A = 60×10⁴ A/m²-K²
For the thoriated-tungsten cathode
work function, ϕ = 2.6 eV
A = 3×10⁴ A/m²-K²

Saturation current,
$i=AS{T}^2{e}^{-\varphi /KT}$
For thoriated tungsten,
$$\begin{gathered} i_{\mathrm{Thorited}\mathrm{Tungsten}}=5000i_{\mathrm{Tungsten}} \\ \mathrm{So}, \\ \Rightarrow S\times 3\times {10}^4\times T^2\times e^{\frac{-2.6\times 1.6\times {10}^{-19}}{1.38\times {10}^{-23}\times T}}=5000\times 60\times {10}^4\times S\times T^2\times e^{\frac{-4.5\times 1.6\times {10}^{-19}}{1.38\times T\times {10}^{-23}}} \\ \Rightarrow e^{\frac{-2.5\times 1.6\times {10}^{-19}}{1.38\times {10}^{-23}\times T}}={10}^5\times e^{\frac{-4.5\times 1.6\times {10}^{-19}}{1.38\times {10}^{-23}\times T}} \end{gathered}$$

Taking 'ln' of both sides, we get:

$$\begin{gathered} \frac{-2.89\times {10}^4}{T}=11.51+\frac{-5.22\times {10}^4}{T} \\ \Rightarrow 11.51T=2.33\times {10}^4 \\ \Rightarrow T=2024K \end{gathered}$$