# What is the constant of proportionality?

Oct 23, 2014

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

$x = k y$

If I know the value of $y$, I can calculate the value of $x$. If the value of $y$ doubles, then I know that the value of $x$ 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 (${j}^{\cdot}$) and the temperature ($T$). 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.

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

The constant of proportionality $\sigma$ 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 ($c$), Boltzmann's Constant ($k$), Planck's Constant ($h$), and $\pi$.
$\sigma = \frac{2 {\pi}^{5} {k}^{4}}{15 {c}^{2} {h}^{3}} = 5.670 \cdot {10}^{-} 8 \frac{J}{s {m}^{2} {K}^{4}}$