01. Each carbon atom bonds to three neighbors
Three of four outer electrons form sigma bonds within the hexagonal layer.
Everyday Science
The same carbon that writes on paper also carries electrical current - and for a surprisingly elegant reason. Carbon is the basis of all life, the core of every pencil, and under most circumstances, a perfectly reasonable electrical insulator. But graphite, which is simply carbon arranged in a specific way, conducts electricity with ease. Change the arrangement, and diamond, also pure carbon, is one of the best insulators known to exist. The material is the same. Only the geometry differs. The answer involves delocalized electrons, quantum chemistry, and a molecular structure that accidentally creates one of the most interesting electronic properties of any common material.
Quick answer
Graphite conducts electricity because each carbon atom in its layered structure forms only three of its four possible bonds with neighboring atoms, leaving one electron per atom free to move throughout the entire layer in a delocalized electron system that can carry electrical current. Graphite conducts electricity much more easily along its layers than across them - the conductivity is dramatically anisotropic, making it directionally selective in a way few everyday materials are.

The mystery
The answer involves delocalized electrons, quantum chemistry, and a molecular structure that accidentally creates one of the most interesting electronic properties of any common material.
The short answer
Graphite conducts electricity because each carbon atom in its layered structure forms only three of its four possible bonds with neighboring atoms, leaving one electron per atom free to move throughout the entire layer in a delocalized electron system that can carry electrical current.
The twist
Graphite conducts electricity much more easily along its layers than across them - the conductivity is dramatically anisotropic, making it directionally selective in a way few everyday materials are.
Common mistake
Many people assume electrical conduction in graphite requires highly pure or specialized graphite.
Everyday Science
Diamond's completely bonded carbon structure makes it the hardest natural material, with all electrons locked into bonds rather than free to move.
The scientists who isolated graphene
The physicists who isolated graphene from graphite using sticky tape in 2004 and won the Nobel Prize in Physics in 2010.
Where graphite's conductive properties are used
Graphite anodes are used in lithium-ion batteries, exploiting both the conductivity and the layered structure's ability to intercalate lithium ions.
Where graphite's conductive properties are used
Graphite's conductivity combined with its lubricating properties make it ideal for the sliding electrical contacts in electric motors.
Doesn't graphite need to be very pure to conduct?
Common pencil graphite conducts electricity reasonably well, though high-purity graphite is used in industrial applications where consistent conductivity is critical.
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