The dimension of $\frac{1}{2}\varepsilon_0 E^2$, where $\varepsilon_0$ is permittivity of free space and $E$ is electric field, is

The quantity $\frac{1}{2}\varepsilon_0 E^2$ represents the energy density (energy per unit volume) of an electric field. The dimensional formula of energy is $[M L^2 T^{-2}]$ and that of volume is $[L^3]$. Therefore, the dimensional formula of energy density is $\frac{[M L^2 T^{-2}]}{[L^3]} = [M L^{-1} T^{-2}]$. Alternatively, you can derive it by substituting the individual dimensions: Dimension of permittivity of free space, $\varepsilon_0 = [M^{-1} L^{-3} T^4 A^2]$ Dimension of electric field, $E = \frac{\text{Force}}{\text{Charge}} = \frac{[M L T^{-2}]}{[A T]} = [M L T^{-3} A^{-1}]$ Dimension of $\varepsilon_0 E^2 = [M^{-1} L^{-3} T^4 A^2] \times [M L T^{-3} A^{-1}]^2 = [M^{-1} L^{-3} T^4 A^2] \times [M^2 L^2 T^{-6} A^{-2}] = [M L^{-1} T^{-2}]$.