Question

Calculate n(T)/n(1000 K) for tungsten emitter at T = 300 K, 2000 K and 3000 K, where n(T) represents the number of thermions emitted per second by the surface at temperature T. Work function of tungsten is 4.52 eV.

Answer 1

According to Richardson-Dushman equation, the number of thermions (n) emitted by a surface, in a given time (t), is given by

`i= n e =AST^2e^(-phi"/"kT)`

`A'=A/e`

`rArr n = A'ST^2e^(-phi"/"kT)`

Here,

\[\phi = 4 . 52\text{ }e . V   = 4 . 52 \times (1 . 6 \times  {10}^{- 19} )  J\]

\[k = 1 . 38 \times  {10}^{- 23} J/K\]

\[n(1000) = A'S \times (1000 )^2  \times  e^{( - 4 . 52 \times 1 . 6 \times {10}^{- 19} )/(1 . 38 \times {10}^{- 23} \times 1000)} \]

\[n(1000) = A'S \times  {10}^6  \times 1 . 7396 \times  {10}^{- 23} \]

\[n(1000) = A'S \times 1 . 7396 \times  {10}^{- 17}\]

\[\frac{n(300K)}{n(1000K)} = \frac{A'S \times (300 )^2 \times e^{( - 4 . 52 \times 1 . 6 \times {10}^{- 19} )/(1 . 38 \times {10}^{- 23} \times 300)}}{A'S \times 1 . 7396 \times {10}^{- 17}}\]

\[\frac{n(300K)}{n(1000K)} = \frac{9 \times {10}^4 \times 1 . 364 \times {10}^{- 76}}{1 . 7396 \times {10}^{- 17}}\]

\[  \frac{n(300K)}{n(1000K)} = 7 . 056 \times  {10}^{- 55}\]

\[\frac{n(2000K)}{n(1000K)} = \frac{A'S \times (2000 )^2 \times e^{( - 4 . 52 \times 1 . 6 \times {10}^{- 19} )/(1 . 38 \times {10}^{- 23} \times 2000)}}{A'S \times 1 . 7396 \times {10}^{- 17}}\]

\[\frac{n(2000K)}{n(1000K)} = \frac{4 \times {10}^6 \times (4 . 1712 \times {10}^{- 12} )}{(1 . 7396 \times {10}^{- 17} )}\]

\[ \frac{n(2000K)}{n(1000K)} = 9 . 73 \times  {10}^{11}\]

\[\frac{n(3000K)}{n(1000K)} = \frac{A'S \times (3000 )^2 \times e^{( - 4 . 52 \times 1 . 6 \times {10}^{- 19} )/(1 . 38 \times {10}^{- 23} \times 3000)}}{A'S \times 1 . 7396 \times {10}^{- 17}}\]

\[\frac{n(3000K)}{n(1000K)} = \frac{(9 \times {10}^6 ) \times (2 . 5913 \times {10}^{- 8} )}{(1 . 7396 \times {10}^{- 17} )}\]

\[\frac{n(3000K)}{n(1000K)} = 1 . 34 \times  {10}^{16}\]