Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 12
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# The Saturation Current from a Thoriated-tungsten Cathode at 2000 K is 100 Ma. - Physics

#### 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.

#### Solution

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)

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

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

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

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

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

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Solution The Saturation Current from a Thoriated-tungsten Cathode at 2000 K is 100 Ma. Concept: Dual Nature of Radiation.
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