# Why the electric field inside a superconductor is always zero?

Recently I have been reading the book called "Superconductivity, Superfluids and Condensates" from James F. Annett. I was confused by the expression that

zero electric field at all points inside a superconductor. In this way the current, $$j$$, can be finite.

Why the current density $$j$$ inside a superconductor has to be finite? And what happens when $$j$$ is infinite?

• Well, the current can't be infinite - how would you expect that to happen? There are not an infinite number of charges to move in the material. Commented Mar 23, 2021 at 14:10
• Thanks for your reply! But why the electrical conductivity could be infinite for a superconductor? It is determined by the carrier concentration and mobility of the material.
– 刘正源
Commented Mar 23, 2021 at 14:16
• The resistivity is R = V/I, so the conductivity is I/V - if no field is required to keep charges moving, then the conductivity is infinite. Commented Mar 23, 2021 at 14:39

In a conventional conductor, the current density and the electric field obey Ohm's Law, $$\vec{J} = \sigma \vec{E}$$. A perfect conductor, such as a superconductor, is the $$\sigma \to \infty$$ limit of this equation; this implies that in this limit, we must have $$\vec{E} \to 0$$ in order to have $$\vec{J}$$ approach a finite limit.

• For me, the intuition behind the equation is worded like this: if there was any electric field (i.e. caused by an unequal charge distribution), the charge distribution should have no trouble - as $\sigma \rightarrow \infty$ - to immediately change (i.e. electrons move) in order to equalize the distribution, which in turn removes the field.
– ojdo
Commented Mar 23, 2021 at 15:26
• @ojdo: Thanks for the reminder. You can actually (sort of) derive the London equation from Maxwell's equations and the Drude model, and it looks a lot like what you describe. See this old answer of mine for details. Commented Mar 23, 2021 at 15:41
• @MichaelSeifert@ojdo:Thanks a lot. That is to say, inside a superconductor, the amount of carriers is finite, but the mobility of the carrier is infinite, so the net electric field inside the superconductor can be eliminated with a sufficiently fast response. By the way, does this mean the response speed of carriers in a superconductor is infinite——no matter how high the frequency is, the alternating electric field cannot cause an electric field inside the superconductor？
– 刘正源
Commented Mar 24, 2021 at 3:55

Why the current density j inside a superconductor has to be finite?

Once the current density exceeds a certain critical current density the material stops superconducting. This critical current density is an important material property for many applications such as MRI. It dictates the amount of superconducting wire that is needed to achieve the desired field.