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At a perfect conductor in electrostatics, the electric field inside is zero, so the surface is an equipotential. If any tangential component of the field existed at the surface, it would drive charges to move along the surface until that tangential field vanished. Therefore the tangential component of the electric field at the surface is zero.

The field just outside the conductor is perpendicular to the surface, and its normal component is related to the surface charge density by E_normal = σ/ε0. This means the normal part can be nonzero, while the tangential part remains zero. The other statements misidentify which component is tied to the surface charge or imply the field is undefined or nonzero in the tangential direction.