# 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}

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