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 th esame surface area operating at the same temperature ? The constant A in the Rihardson- Dushman equation is $60\times {10}^{4}A{m}^{-2}{K}^{-2}$ for pure tungsten and $3.0\times {10}^{4}A{m}^{-2}{K}^{-2}$ for thoriated tungsten. The work function of pure tungsten is 4.5 eV and that of thoriated tungsten is 2.6 eV.

Answer 1

$$\begin{array}{cc}i=AS{T}^2{e}^{-\varphi \sqrt{KT}}& \\ i_1=i& i_1=100mA\\ A_1=60\times {10}^4& A_2=3\times {10}^4\\ S_1=S& S_1=SX\\ T_1=2000& T_2=2000\\ {\varphi }_1=4.5eV& {\varphi }_2=4.6eV\\ K=1.98\times {10}^{-23}J/K& \end{array}$$
$i=(60\times {10}^4)\left(S\right)\times (2000{)}^2{e}^{\frac{-4.5\times 1.6\times {10}^{-19}}{1.38\times {10}^{-23}\times 2000}}100=(3\times {10}^4\left)\right(S\left)\right(2000{)}^2{e}^{\frac{-2.6\times 1.6\times {10}^{-19}}{1.38\times {10}^{-23}\times 2000}}$
Dividing the equation $\frac{i}{100}$
$=20\times e^{\frac{-4.5\times 1.6\times {10}^{-19}}{1.38\times {10}^{-23}\times 2000}}-\frac{-2.6\times 1.6\times {10}^{-10}}{1.38\times {10}^{-23}\times 2000}\Rightarrow \frac{i}{100}=20\times e^{\frac{(2.6-4.5)\times 1.6\times 10}{2\times 1.38}}\Rightarrow \frac{i}{100}=20\times e^{\frac{16(2.6-4.5)}{2\times 1.38}}\Rightarrow \frac{i}{100}=20e^{\frac{8\times (-1.9)}{1.38}}\Rightarrow \frac{i}{100}=20e^{-11.014}\Rightarrow \frac{i}{100}=20\times 0.000016\Rightarrow i=20\times 0.0016=0.0329A=33mA.$