# Energy density of Induced Electric Field

For electrostatic fields, we write the energy density using $$u = \frac12\epsilon_o E^2$$

is this formula also valid for a non-conservative electric field produced by changing magnetic field?

Or, can we say that the energy stored in a non-conservative electric field per unit charge is equal to the rate of change of flux, given by the formula $$\int E.dl = -\partial\phi/\partial t$$ as the line integral of E is equivalent to work done per unit charge?

No. The induced electric field produce magnetic energy but do not produce the electric energy. The electric field is defined as, $$\boldsymbol{E}=-\frac{\partial\boldsymbol{A}}{\partial t}-\nabla\phi$$ Corresponding to $$\boldsymbol{E}_{s}=-\nabla\phi$$, there is the electric energy intensity $$\frac{\epsilon}{2}|\boldsymbol{E}_{s}|^2$$ This energy is usually saved inside the capacity. Corresponding to the induced electric field, $$\boldsymbol{E}_{i}=-\frac{\partial\boldsymbol{A}}{\partial t}$$ This electric field produced the magnetic field, hence has produced magnetic energy intensity. $$\frac{1}{2\mu}|\boldsymbol{B}|^2$$ $$\boldsymbol{E}_{i}$$ is nothing to do with the electric energy. We should discuss this problem just in magnetic quasi-static electromagnetic field theory or electric-magnetic quasi-static electromagnetic field theory instead of radiation magnetic field theory.
Yes, the formula $$E=\frac{1}{2}\epsilon_0 E^2$$ is valid for electric field energy density in vacuum (or other medium such as air that interacts only very weakly with electric field) whether the electric field is purely electrostatic or general (including induced electric field or field of EM waves), but it is necessary that point charges or line charges are not present. Charges have to be distributed with finite density per unit volume. Otherwise total electric energy diverges and validity of the formula is dubious.
In general, the energy density of electromagnetic field in vacuum is given by $$u=\frac12(|E|^2+|B|^2)$$ in suitable unites. The energy can interpolate between electric and magnetic fields. This is what happens in an electromagnetic wave.