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How can I prove the relation: $$\epsilon=\frac{3}{4}\frac{Q}{c}\tau$$ where: $\epsilon$ is the energy density of radiation, $Q$ is the flux and $\tau$ is the optical thickness.

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I think a bit more information is needed about your specific case and dimensions of your symbols. Are you considering a photon gas? Are you just trying to count the amount of energy?

For example, let’s says I know $Q$, the radiant heat flux per unit area per time $Jm^{-2}s^{-1}$ (we will ignore solid angle and direction, let’s say this is normal to some surface).

Then the energy density $ Jm^{-3 }$, is $Q$ divided by the group velocity of the light, which in a vacuum is $c$,

$$ U = Q / c $$

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