Migratory preference in a rearrangement involving carbocations
why is the second path not taken?


$\ce{H}$ can migrate, it is just that phenyl migrates better.  The reason that phenyl migrates better is because it is much better able to stabilize the developing carbocation center.  The phenyl group participates by what is termed neighboring group participation (see the "NGP by an aromatic ring" section).  It forms a phenonium ion.  Note the geometry of this ion, the former aromatic ring is perpendicular to the plane of the screen while the former $\ce{C-C}$ bond is contained in the plane of the screen.



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We can draw several resonance structures to describe the phenonium ion, this leads to it being especially stable.



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Such extra stabilization is not possible with a migrating hydrogen atom. Therefore the pathway involving the phenonium ion is lower energy (faster) than the pathway involving hydrogen migration.