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     At any cross section of the wire, the area of cross section tells us the number of atoms available at that cross section.  The current flows in the direction perpendicular to the cross section.  So the number of electrons (free conduction band) present in the atoms in a particular cross section result in the strength of the current.  More the number of electrons more the current.

   As the volume density of the metal is constant, the number of electrons available for conduction of current through the cross section is dependent on the area of cross section.

  So less the thickness, less the conductivity and more the resistance.
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