Calculate n(T)/n(1000 K) for tungsten emitter at T = 300 K, 2000 K and 3000 K, where n(T) represents the number of thermions emitted per second by the surface at temperature T. Work function of tungsten is 4.52 eV.

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Solution

According to Richardson-Dushman equation, the number of thermions (n) emitted by a surface, in a given time (t), is given by
$i = n e = A S T^{2} e^{- ϕ / k T} A' = \frac{A}{e} \Rightarrow n = A' S T^{2} e^{- ϕ / k T}$
Here,
$ϕ = 4.52 e . V = 4.52 \times (1.6 \times 10^{- 19}) J k = 1.38 \times 10^{- 23} J / K n (1000) = A' S \times (1000)^{2} \times e^{(- 4.52 \times 1.6 \times 10^{- 19}) / (1.38 \times 10^{- 23} \times 1000)} n (1000) = A' S \times 10^{6} \times 1.7396 \times 10^{- 23} n (1000) = A' S \times 1.7396 \times 10^{- 17}$
$\frac{n (300 K)}{n (1000 K)} = \frac{A' S \times (300)^{2} \times e^{(- 4.52 \times 1.6 \times 10^{- 19}) / (1.38 \times 10^{- 23} \times 300)}}{A' S \times 1.7396 \times 10^{- 17}} \frac{n (300 K)}{n (1000 K)} = \frac{9 \times 10^{4} \times 1.364 \times 10^{- 76}}{1.7396 \times 10^{- 17}} \frac{n (300 K)}{n (1000 K)} = 7.056 \times 10^{- 55}$
$\frac{n (2000 K)}{n (1000 K)} = \frac{A' S \times (2000)^{2} \times e^{(- 4.52 \times 1.6 \times 10^{- 19}) / (1.38 \times 10^{- 23} \times 2000)}}{A' S \times 1.7396 \times 10^{- 17}} \frac{n (2000 K)}{n (1000 K)} = \frac{4 \times 10^{6} \times (4.1712 \times 10^{- 12})}{(1.7396 \times 10^{- 17})} \frac{n (2000 K)}{n (1000 K)} = 9.73 \times 10^{11}$

$\frac{n (3000 K)}{n (1000 K)} = \frac{A' S \times (3000)^{2} \times e^{(- 4.52 \times 1.6 \times 10^{- 19}) / (1.38 \times 10^{- 23} \times 3000)}}{A' S \times 1.7396 \times 10^{- 17}} \frac{n (3000 K)}{n (1000 K)} = \frac{(9 \times 10^{6}) \times (2.5913 \times 10^{- 8})}{(1.7396 \times 10^{- 17})} \frac{n (3000 K)}{n (1000 K)} = 1.34 \times 10^{16}$

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