If the conductor is net neutral then Gauss's law tells us that the divergence of the electric field is zero. This means there can be no electric field in the $R$ direction (using the $R$, $\phi$, $z$ cylindrical coordinates).

If the current is in a steady state then there will be no changing electric or magnetic fields. From Faraday's law we also know then that the curl of the electric field is zero.

We know that in an ideal conductor that ${\bf J} = \sigma {\bf E}$ and so if the current flows along the wire, there must be an $E_z$ component. But if the curl is zero then there cannot be a $\partial E_z/\partial R$ and so this component is uniform across the wire and hence the current density is also uniform across the wire.

Maxwell's equations do permit an $E_{\phi}$, as long as it depends on $R^{-1}$, which would produce a circulating current density in the same direction. But that cannot be a steady-state solution because the current would ohmically dissipate without a source of EMF to keep it going.

By symmetry there can be no gradient of the electric field with respect to $z$ or $\phi$ (using the $R$, $\phi$, $z$ cylindrical coordinates). If the conductor is net neutral then Gauss's law tells us that the divergence of the electric field is zero. This means there can also be no electric field in the $R$ direction.

If the current is in a steady state then there will be no changing electric or magnetic fields. From Faraday's law we also know then that the curl of the electric field is zero.

We know that in an ideal conductor that ${\bf J} = \sigma {\bf E}$ and so if the current flows along the wire, there must be an $E_z$ component. But if the curl is zero then there cannot be a $\partial E_z/\partial R$ and so this component is uniform across the wire and hence the current density is also uniform across the wire.

Maxwell's equations do permit an $E_{\phi}$, as long as it depends on $R^{-1}$, which would produce a circulating current density in the same direction. But that cannot be a steady-state solution because the current would ohmically dissipate without a source of EMF to keep it going.

If the conductor is net neutral then Gauss's law tells us that the divergence of the electric field is zero. This means there can be no electric field in the $R$ direction (using the $R$, $\phi$, $z$ cylindrical coordinates).

If the current i in a steady state then there will be no changing electric of magnetic fields. From Faraday's law we know then that the curl of the elctric field is zero.

We know that in an ideal conductor that ${\bf J} = \sigma {\bf E}$ and so if the current flows along the wre, there must be an $E_z$ component. But if the curl is zero then ther cannot be a $\partial E_z/\partial R$ and so this component is uniform across the wire and hence the current density is also uniform across the wire.

Maxwell's equations do permit an $E_{\phi}$, as long as it depends on $R^{-1}$, which would produce a circulating current density in the sme direction. But that cannot be a steady-state solution because the current would ohmically dissipate without a source of EMF to keep it going.

If the conductor is net neutral then Gauss's law tells us that the divergence of the electric field is zero. This means there can be no electric field in the $R$ direction (using the $R$, $\phi$, $z$ cylindrical coordinates).

If the current is in a steady state then there will be no changing electric or magnetic fields. From Faraday's law we also know then that the curl of the electric field is zero.

We know that in an ideal conductor that ${\bf J} = \sigma {\bf E}$ and so if the current flows along the wire, there must be an $E_z$ component. But if the curl is zero then there cannot be a $\partial E_z/\partial R$ and so this component is uniform across the wire and hence the current density is also uniform across the wire.

Maxwell's equations do permit an $E_{\phi}$, as long as it depends on $R^{-1}$, which would produce a circulating current density in the same direction. But that cannot be a steady-state solution because the current would ohmically dissipate without a source of EMF to keep it going.