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Zero Electrical Resistance

The hallmark of superconductivity, where current flows without any energy loss due to resistance within the material.
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The statement of the theorem

Let J\mathbf{J} be the current density and E\mathbf{E} be the electric field. The condition of zero electrical resistance (ρ=0\rho = 0) implies that the electric field vanishes for any steady current flow: \nE=0\mathbf{E} = 0 \nThis is consistent with the generalized Ohm's law, E=ρJ\mathbf{E} = \rho \mathbf{J}, requiring the resistivity ρ\rho to be zero, or equivalently, the conductivity σ\sigma to be infinite: \nσ=1ρ=∞\sigma = \frac{1}{\rho} = \infty