You need to use Ohms law: $J = \sigma E$ which has to be added to Maxwell's equations as a bulk observation, as explained by this answer.

You can then conclude that the electric field is zero in a conductor for:

• perfect conductor where $\rho = 1/\sigma = 0$ and $J$ is finite
• static case where $J = 0$ and $\sigma$ is finite

You need to use Ohms law: $J = \sigma E$ which has to be added to Maxwell's equations as a bulk observation, as explained by this answer.

You can then conclude that the electric field is zero in a conductor for:

• perfect conductor where $\rho = 1/\sigma = 0$ and $J$ is finite
• static case where $J = 0$ and $\sigma$ is finite