I have started studying QFT, and I am currently reviewing briefly on the classical field theory. I have come across the Maxwell Lagrangian given by $$ \mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}. $$ I have seen how the spacetime translation symmetry leads to the conserved current called the energy momentum tensor, given by Noether's recipe $$ T^{\mu\nu}=\frac{\partial\mathcal{L}}{\partial(\partial_{\mu}\phi)}\partial^{\nu}\phi-\eta^{\mu\nu}\mathcal{L}. $$ From the electromagnetic theory, I know that the energy density of the electromagnetic field is given by $$ u=\frac{1}{2}\left(\epsilon_0E^2+\frac{1}{\mu_0}B^2\right), $$ which is also given by the $T^{00}$ component of the above energy momentum tensor. However, I am unable to reproduce the expression of energy density starting from the above Lagrangian and doing spacetime translation and applying Noether's recipe. Can somebody please show a detailed calculation?
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