Q.Show that the quantity $1/\sqrt{\mu_0\epsilon_0}$ has the dimensions of speed, and state what this quantity represents physically for an electromagnetic wave in vacuum.

Dimensional check. From $\vec{F}=q\vec{v}\times\vec{B}$ and the Biot-Savart law, $\mu_0$ has SI unit $\text{T m A}^{-1}$, equivalent to $\text{kg m A}^{-2}\text{s}^{-2}$; from Coulomb's law, $\epsilon_0$ has SI unit $\text{C}^2\text{N}^{-1}\text{m}^{-2}$, equivalent to $\text{A}^2\text{s}^4\text{kg}^{-1}\text{m}^{-3}$. Multiplying these together, the kg, A, and one factor of m cancel appropriately, leaving $[\mu_0\epsilon_0]=\text{s}^2\text{m}^{-2}=\text{T}^2\text{L}^{-2}$ (time-squared per length-squared). Taking the reciprocal square root,

which is exactly the dimension of a SPEED (length divided by time) -- so the combination $1/\sqrt{\mu_0\epsilon_0}$ is dimensionally consistent with being a velocity, even before any numbers are substituted. …

Find the SI units of $\mu_0$ and $\epsilon_0$ from the laws that define them, combine to check the dimension is a spee …

• Substituting numbers first and only checking the final unit comes out as m/s, without actually showing the DIMENSIONAL combination reduces to $\text{L}\,\text{T}^{-1}$ in general. …