A potential barrier of 250 meV exists across a p - n junction. A...

The potential barrier across a $p$-$n$ junction opposes the motion of majority charge carriers. The electric field in the depletion region is directed from the $n$-side to the $p$-side.

When a hole (which acts as a positive charge) moves from the $p$-side to the $n$-side, it moves against the electric field and loses kinetic energy equal to the potential barrier energy.

By conservation of energy, the final kinetic energy $K_{f}$ of the hole after crossing the junction is given by:

$K_{f} = K_{i} - e V_{0}$

where $K_{i}$ is the initial kinetic energy and $e V_{0}$ is the potential barrier energy.

Substituting the given values:

$K_{f} = 300 \text{ meV} - 250 \text{ meV} = 50 \text{ meV}$