Answer
The equation $~~I = ~\lt u \gt c~~$ is dimensionally consistent.
Work Step by Step
$I$ is intensity in units of $W~m^{-2}$
$\lt u \gt$ is energy density in units of $J~m^{-3}$
$c$ is the speed of light in units of $m~s^{-1}$
We can verify the units of the term $\lt u\gt ~c$:
$(J~m^{-3})~(m~s^{-1}) = J~s^{-1}~m^{-2} = W~m^{-2}$
We can see that both sides of the equation $~~I = \lt u \gt c~~$ have units of $\frac{W}{m^2}$
Therefore the equation $~~I = ~\lt u \gt c~~$ is dimensionally consistent.
