Question

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 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.

Answer 1

According to Richardson-Dushman equation, the current of thermions is given by
$i=AS{T}^2{e}^{-\varphi /KT}$
For the thoriated tungsten cathode:
Saturation current, i = 100 mA
Temperature, T = 2000 K
A = 3.0 × 10⁴ Am⁻²K⁻²
ϕ = 2.6 eV.

For the pure tungsten cathode:
Let the saturation current be i.
Temperature, T = 2000 K
A = 60 × 10⁴ Am ⁻² K⁻²
ϕ = 2.6 eV

k = 1.38× 10⁻²³ J/K

Substituting the values of the quantities in the Richard-Dushman equation, we get:
$$\begin{gathered} i=(60\times {10}^4)\left(\mathrm{S}\right)\times (2000{)}^2{e}^{\frac{-4.5\times 1.6\times {10}^{-19}}{1.38\times {10}^{-23}\times 2000}}...(1) \\ 100\times {10}^{-3}=(3\times {10}^4\left)\right(\mathrm{S}\left)\right(2000{)}^2{e}^{\frac{-2.6\times 1.6\times {10}^{-19}}{1.38\times {10}^{-23}\times 2000}}...\left(2\right) \end{gathered}$$

^{Dividing equation (1) by (2)}

$$\begin{gathered} \frac{i}{100\times {10}^{-3}}=20\times e\left({}^{\frac{-4.5\times 1.6\times {10}^{-19}}{1.38\times {10}^{-23}\times 2000}-\frac{-2.6\times 1.6\times {10}^{-19}}{1.38\times {10}^{-23}\times 2000}}\right) \\ \frac{i}{100\times {10}^{-3}}=20\times e^{\frac{(2.6-4.5)\times 1.6\times 10}{2\times 1.38}} \\ \frac{i}{100\times {10}^{-3}}=20\times e^{\frac{8\times (-1.9)}{1.38}} \\ \frac{i}{100\times {10}^{-3}}=20\times 0.000016 \\ \Rightarrow i=32.9\times {10}^{-6}\mathrm{A}\approx 33\mathrm{\mu A} \end{gathered}$$