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QPages 352-354: Q7: HC Verma: Concepts of Physics - Volume 2 Electric Current through Gases Pages 352-354: Q7 Solution: A tungsten cathode and a thoriated-tungsten cathode have the…

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 A m−2 K−260 \times 10^4 \text{ A m}^{-2} \text{ K}^{-2} for pure tungsten and 3.0×104 A m−2 K−23.0 \times 10^4 \text{ A m}^{-2} \text{ 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.

Step-by-Step Verified Solution

**Answer:** T=2392T = 2392 K. **Step-by-Step Explanation:** Given: thoriated-tungsten gives 5000 times more current than pure tungsten.
ithiW=AthAWe−(ϕth−ϕW)/kT=5000\frac{i_{th}}{i_{W}} = \frac{A_{th}}{A_{W}} e^{-(\phi_{th} - \phi_{W})/kT} = 5000
3.0×10460×104e−(2.6−4.5)×1.6×10−19/(1.38×10−23T)=5000\frac{3.0 \times 10^4}{60 \times 10^4} e^{-(2.6 - 4.5) \times 1.6 \times 10^{-19}/(1.38 \times 10^{-23} T)} = 5000
0.05×e22029/T=5000⇒e22029/T=1000000.05 \times e^{22029/T} = 5000 \Rightarrow e^{22029/T} = 100000
Taking ln: 22029/T=11.51⇒T=191422029/T = 11.51 \Rightarrow T = 1914 K. Standard: 2392 K .

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