Question

The saturation current from a thoriated –tungsten cathode at $2000k$ is $100mA$ . 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 $60\times {{10}^{4}}A{{m}^{-2}}{{k}^{-2}}$ for pure tungsten and $3.0\times {{10}^{4}}A{{m}^{-2}}{{k}^{-2}}$ for thoriated tungsten. The work function of pure tungsten is $4.5eV$ and that of thoriated tungsten is $2.6eV$ .

Answer 1

Hint: Tungsten is a metal which is used as an electron emitter and the symbol used to represent tungsten is W. Atomic number of tungsten is 74 and it is a rare metal. Tungsten has a very high melting point of 3655k and a moderate work function. Tungsten metal which belongs to group 6 in the periodic table and it is a d-block element.

Complete step-by-step answer:
For thermionic emission tungsten is the suitable metal and it starts emission at 2500k and boiling point of tungsten metal is 6203k.tungsten metal has a density of \[19.3gc{{m}^{-3}}\].
The Richardson-Dushman equation which relates the current density of a thermionic emission to the work function ($W$ ) and temperature(T) of the emitting material.
The current is given by:
$i=A{{S}^{2}}{{e}^{-\dfrac{\phi }{KT}}}$
Where $i$ = current density of the emission
             $T$ =Temperature
              $\phi $ =Work function of the cathode material
From the data
 $\begin{align}
  & {{A}_{1}}=60\times {{10}^{4}} \\
 & {{A}_{2}}=3\times {{10}^{4}} \\
\end{align}$
 $\begin{align}
  & {{S}_{1}}=S \\
 & {{S}_{2}}=S \\
\end{align}$
${{T}_{1}}={{T}_{2}}=2000k$
$K=1.38\times {{10}^{-23}}J/k$
 $\begin{align}
  & {{i}_{1}}=i \\
 & {{i}_{2}}=100mA \\
\end{align}$
${{\phi }_{1}}=4.5eV$ and ${{\phi }_{2}}=2.6eV$
After substituting these values in the above equation
$i=(60\times{{10}^{4}})(S)\times{{(2000)}^{2}}\times{{e}^{\dfrac{-4.5\times1.6\times{{10}^{-19}}}
{1.38\times {{10}^{-23}}\times 2000}}}$
$100=(3\times{{10}^{4}})(S)\times{{(2000)}^{2}}{{e}^{\left(\dfrac{-2.6\times1.6\times{{10}^{-19}}}
{1.38\times {{10}^{-23}}\times 2000} \right)}}$
After dividing the equation
$\begin{align}
  & \dfrac{i}{100}=20\times {{e}^{\left[ \left( \dfrac{-4.5\times 1.6\times 10}{1.38\times 2} \right)\times \left( \dfrac{-2.6\times 1.6\times 10}{1.38\times 20} \right) \right]}} \\
 & \dfrac{i}{100}=20\times {{e}^{-11.014}} \\
 & \dfrac{i}{100}=20\times 0.000016 \\
 & i=20\times 0.0016=33\mu A \\
\end{align}$

Note: Tungsten is used in heavy metal alloys and tungsten which is solid at room temperature .Tungsten metal which has excellent corrosion resistance and also has high tensile strength. The thermal conductivity of tungsten metal is $174W/(m\cdot K)$ and the molar heat capacity of tungsten metal is$24.27J/(mol\cdot K)$.