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. The constant A in the Richardson−Dushman equation is 60 × 10⁴ Am ⁻² K⁻² for pure tungsten and 3.0 × 10⁴ Am ⁻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. Find the operating temperature.

Answer 1

For the pure tungsten cathode,

work function, ϕ = 4.5 eV

A = 60 × 10⁴ Am ⁻² K⁻²

For the thoriated-tungsten cathode

work function, ϕ = 2.6 eV

A = 3.0 × 10⁴ Am⁻²K⁻²

Saturation current,

\[i = AS T^2  e^{- \phi/KT}\]

For thoriated tungsten,

\[i_{\text{Thorited Tungsten}}  = 5000 i_{\text{Tungsten}} \]

So,

\[ \Rightarrow S \times 3 \times  {10}^4  \times  T^2  \times  e^\frac{- 2 . 6 \times 1 . 6 \times {10}^{- 19}}{1 . 38 \times {10}^{- 23} \times T}  = 5000 \times 60 \times  {10}^4  \times S \times  T^2  \times  e^\frac{- 4 . 5 \times 1 . 6 \times {10}^{- 19}}{1 . 38 \times T \times {10}^{- 23}} \]

\[ \Rightarrow  e^\frac{- 2 . 5 \times 1 . 6 \times {10}^{- 19}}{1 . 38 \times {10}^{- 23} \times T}  =  {10}^5  \times  e^\frac{- 4 . 5 \times 1 . 6 \times {10}^{- 19}}{1 . 38 \times {10}^{- 23} \times T}\]

Taking 'ln' of both sides, we get:-

\[\frac{- 2 . 89 \times {10}^4}{T} = 11 . 51 + \frac{- 5 . 22 \times {10}^4}{T}\]

\[ \Rightarrow 11 . 51T = 2 . 33 \times  {10}^4 \]

\[ \Rightarrow T = 2024    K\]