Definition:Stefan-Boltzmann Constant/Dimension

Definition

The Stefan-Boltzmann constant has the dimension $\mathsf {M T^{-3} \Theta^{-4} }$.

This arises from its definition as:

$\sigma := \dfrac {2 \pi^5 k^4} {15 c^2 h^3}$

where:

$k$ is Boltzmann's constant, with dimension $\mathsf {M L^2 T^{-2} \Theta^{-1} }$

$c$ is the speed of light, with dimension $\mathsf {L T^{-1} }$

$h$ is Planck's constant, with dimension $\mathsf {M L^2 T^{-1} }$

Hence by dimensional analysis:

$\dfrac {\paren {\mathsf {M L^2 T^{-2} \Theta^{-1} } }^4} {\paren {\mathsf {L T^{-1} } }^2 \paren {\mathsf {M L^2 T^{-1} } }^3} = \dfrac {\mathsf {M^4 L^8 T^{-8} \Theta^{-4} } } {\paren {\mathsf {L^2 T^{-2} } } \paren {\mathsf {M^3 L^6 T^{-3} } } } = \mathsf {M T^{-3} \Theta^{-4} }$