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

\[i = AS T^2  e^{- \phi/KT}\]

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:-

\[i = (60 \times  {10}^4 )(S) \times (2000 )^2  e^\frac{- 4 . 5 \times 1 . 6 \times {10}^{- 19}}{1 . 38 \times {10}^{- 23} \times 2000}  .............(1)\]

\[100 \times  {10}^{- 3}  = (3 \times  {10}^4 )(S)(2000 )^2  e^\frac{- 2 . 6 \times 1 . 6 \times {10}^{- 19}}{1 . 38 \times {10}^{- 23} \times 2000} ...........(2)\]

Dividing equation (1) by (2)

\[\frac{i}{100 \times {10}^{- 3}} = 20 \times e^\left(\frac{- 4 . 5 \times 1 . 6 \times {10}^{- 19}}{1 . 38 \times {10}^{- 23} \times 2000} - \frac{- 2 . 6 \times 1 . 6 \times {10}^{- 19}}{1 . 38 \times {10}^{- 23} \times 2000} \right)\]

\[\frac{i}{100 \times {10}^{- 3}} = 20 \times  e^\frac{(2 . 6 - 4 . 5) \times 1 . 6 \times 10}{2 \times 1 . 38} \]

\[\frac{i}{100 \times {10}^{- 3}} = 20 \times  e^\frac{8 \times ( - 1 . 9)}{1 . 38} \]

\[\frac{i}{100 \times {10}^{- 3}} = 20 \times 0 . 000016\]

\[ \Rightarrow i = 32 . 9 \times  {10}^{- 6}   A   \approx   33  \mu A\]