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A Tungsten Cathode and a Thoriated-tungsten Cathode Have the Same Geometric Dimensions and Are Operated at the Same Temperature. - Physics

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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.

Solution

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:-

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

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

\[ \Rightarrow T = 2024    K\]

  Is there an error in this question or solution?

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Solution A Tungsten Cathode and a Thoriated-tungsten Cathode Have the Same Geometric Dimensions and Are Operated at the Same Temperature. Concept: Davisson-Germer Experiment.
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