What is the constant of proportionality?

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What is the constant of proportionality?

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Answer 1 · 2014-10-23T14:33:21

The ratio between two quantities is called the constant of proportionality. If it is true that some quantity $x$ $x$ changes as you change another quantity $y$ $y$ then there is some constant of proportionality $k$ $k$ which can be used to mathematically relate the two.

$x = k y$ $x = ky$

If I know the value of $y$ $y$ , I can calculate the value of $x$ $x$ . If the value of $y$ $y$ doubles, then I know that the value of $x$ $x$ will also double.

This question is asked in the context of Stefan's Law where the two quantities being related are the total energy radiated per unit area ( $j^{\cdot}$ $j^*$ ) and the temperature ( $T$ $T$ ). They don't relate directly the way the mathematical example above does. Instead, the total energy radiated varies as the fourth power of the temperature.

$j^{\cdot} = σ \cdot T^{4}$ $j^* =sigma*T^4$

The constant of proportionality $σ$ $sigma$ is the value which relates the two. The value can be shown to derive from several other fundamental constants. It is related to the speed of light ( $c$ $c$ ), Boltzmann's Constant ( $k$ $k$ ), Planck's Constant ( $h$ $h$ ), and $π$ $pi$ .
$σ = \frac{2 π^{5} k^{4}}{15 c^{2} h^{3}} = 5.670 \cdot 10^{- 8} \frac{J}{s m^{2} K^{4}}$ $sigma = (2pi^5k^4)/(15c^2h^3) = 5.670 * 10^-8 (J)/(sm^2K^4)$

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What is the constant of proportionality?

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