 Insertion barrier of a charged polymer

If only the inserted length $x$ feels a uniform inward electric field, its charge experiences force $qEx$, where $q$ is charge per contour length. Integrating gives electric energy $-qE{x}^2/2$. Adding a constant entropic confinement force gives the displayed free-energy profile. Its interior maximum is at $x=f/(qE)$ and has height $f^2/(2qE)$; entry waiting and subsequent directed insertion are distinct times.