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If the temperature of a tungsten filament is raised from 2000 K to 2010 K, by what factor does...

Question:

If the temperature of a tungsten filament is raised from 2000 K to 2010 K, by what factor does the emission current change? The work function of tungsten is 4.5 eV.

Richardson Dushman Equation:

In physics, according to Richardson Dushman's equation, the current density directly varies with the temperature. It described the relationship between the work function and temperature of the material which is emitting.

Answer and Explanation:


  • The temperature increased by: {eq}{T_1} = 2000{\rm{K to }}{{\rm{T}}_2} = 2010{\rm{K}} {/eq}
  • The work function of tungsten is: {eq}w = 4.5{\rm{eV}} {/eq}


Recall the expression for the Richardson Dushman equation.

$$\color{red}{j = A{T^2}{e^{\dfrac{{ - w}}{{kT}}}}} $$

Where,

  • {eq}A {/eq} is the Richardson?s constant
  • {eq}T {/eq} is the temperature
  • {eq}k {/eq} is the Boltzmann constant

The expression for the current emissions relation when temperature of filament is raised is,

$$\begin{align*} \dfrac{{{j_1}}}{{{j_2}}} &= \dfrac{{A{T_1}^2{e^{\dfrac{{ - w}}{{k{T_1}}}}}}}{{A{T_2}^2{e^{\dfrac{{ - w}}{{k{T_2}}}}}}}\\ \dfrac{{{j_1}}}{{{j_2}}} &= {\left( {\dfrac{{{T_1}}}{{{T_2}}}} \right)^2}{e^{\dfrac{w}{k}\left( {\dfrac{1}{{{T_2}}} - \dfrac{1}{{{T_2}}}} \right)}} \end{align*} $$

Substitute the known values in above equation.

$$\begin{align*} \dfrac{{{j_1}}}{{{j_2}}} &= {\left( {\dfrac{{2000}}{{2010}}} \right)^2}{e^{\dfrac{{4.5 \times 1.6 \times {{10}^{ - 19}}}}{{1.38 \times {{10}^{ - 23}}}}\left( {\dfrac{1}{{2010}} - \dfrac{1}{{2000}}} \right)}}\\ \dfrac{{{j_1}}}{{{j_2}}} &\approx 1.15 \end{align*} $$

Thus, the emission current changed with the factor {eq}\boxed{\color{blue}{1.15}}. {/eq}


Learn more about this topic:

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Emission, Dark, and Reflection Nebulae

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