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QPages 352-354: Q6: HC Verma: Concepts of Physics - Volume 2 Electric Current through Gases Pages 352-354: Q6 Solution: The saturation current from a thoriated-tungsten cathode at…

The saturation current from a thoriated-tungsten cathode at 2000 K is 100 mA. What will be the saturation current for a pure-tungsten cathode of the same surface area operating at the same temperature? 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.

Step-by-Step Verified Solution

**Answer:** i=33μAi = 33 \mu\text{A}. **Step-by-Step Explanation:** Using Richardson-Dushman equation: i=AST2e−ϕ/kTi = AST^2 e^{-\phi/kT}. For thoriated tungsten: i=100i = 100 mA, A=3.0×104A = 3.0 \times 10^4, ϕ=2.6\phi = 2.6 eV, T=2000T = 2000 K. For pure tungsten: A=60×104A = 60 \times 10^4, ϕ=4.5\phi = 4.5 eV, same SS and TT. Dividing the two equations:
i100=60×1043.0×104e−(4.5−2.6)×1.6×10−19/(1.38×10−23×2000)\frac{i}{100} = \frac{60 \times 10^4}{3.0 \times 10^4} e^{-(4.5 - 2.6) \times 1.6 \times 10^{-19}/(1.38 \times 10^{-23} \times 2000)}
i=100×20×e−11.01=2000×1.65×10−5=0.033 mA=33μAi = 100 \times 20 \times e^{-11.01} = 2000 \times 1.65 \times 10^{-5} = 0.033 \text{ mA} = 33 \mu\text{A}

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