Q.What happens if a bar magnet is cut into two pieces

[!TLDR] Either way, each piece becomes its own complete, weaker bar magnet with a fresh N and S pole -- isolated magnetic poles are never produced. [!ANSWER] Both pieces become independent (weaker) magnets, each with its own N and S pole.

By fact (ii) of section 12.1, breaking a magnet into pieces never isolates a single pole -- every fragment reorganises itself into a complete, though somewhat weaker, magnetic dipole. Cutting TRANSVERSE to the length (across the bar, perpendicular to its axis) produces two shorter magnets, each of which develops a brand-new pole at its freshly cut face (opposite in sign to whichever original pole is still at its other, uncut end), so each half ends up with its own full N-S pair, just over a shorter magnetic length. Cutting ALONG the length (splitting the bar lengthwise into two thinner strips) instead produces two thinner magnets, each retaining the FULL original length, but with roughly half the original pole strength (since the pole strength scales with the magnet's cross-sectional area), each strip still having its N pole and S pole at the same ends as the original bar. In both cases, no isolated N or S pole is ever produced -- the search for a genuine magnetic monopole, mentioned in section 12.1, remains unsuccessful even by this simple mechanical route. [!ANSWER] Both pieces become independent (weaker) magnets, each with its own N and S pole.

Apply the textbook fact that breaking a magnet into pieces never isolates a single pole -- each fragment reorganises into its own complete, weaker dipole, whether the cut is transverse or longitudinal.

Expecting a transverse cut to separate the N pole from the S pole into two single-pole fragments, when in fact each fragment develops its OWN brand-new pair of N-S poles at the cut.