= Solution
Let the channel <length> be $\ell_c\gg R$ and assume uniform solution conductivity. Its Ohmic resistance scales as $\ell_c/(\sigma R^2)$, whereas access resistance of a reservoir mouth scales as $1/(\sigma R)$. The fraction of voltage in the entrance regions is therefore of order $R/\ell_c$. Most of the potential drop lies inside the long channel; the small entrance field region is neglected in the model, not asserted identically zero.
Write $x$ for inserted contour <length> and choose $qE>0$ to pull the chain inward. The charge already inside feels <force> $qEx$, giving
$$
\boxed{U_{\rm el}(x)=-\tfrac12qEx^2.}
$$
Confinement removes <polymer> configurations. In the stiff-chain regime, the <confinement deflection length> obeys $R^2\sim\lambda^3/\ell_p$, hence $\lambda\sim(R^2\ell_p)^{1/3}$. A free-energy cost of order $k_BT$ per deflection gives a roughly constant resisting <entropic force> $f\sim k_BT/\lambda$. The <insertion barrier of a charged polymer> follows from $F(x)\simeq fx-qEx^2/2$, with an entry maximum at $x_*=f/(qE)$ when that point lies within the insertion interval. Its height is $f^2/(2qE)$. The field supplies work against this confinement <entropy>.
Put $\zeta_\ell=2\pi\eta/\ln(R/a)$, so $\gamma(x)=\zeta_\ell x$. After capture, neglect the entry barrier, exterior-chain drag and thermal corrections to strong directed motion. Then
$$
\gamma(x)\dot x=qEx,\quad \dot x=\frac{qE}{\zeta_\ell},\qquad
\boxed{\tau_{\rm in}=\frac{\zeta_\ell L_0}{qE}=\frac{\gamma(L_0)}{qE}.}
$$
The <confined-polymer entry and recoil times> have different length dependence; this entry graph is linear through the origin. Charge and drag both grow with inserted <length>, leaving a constant <velocity>. The same cancellation explains length-independent mobility in an ideal free-draining <electrophoresis> model. It does not include a capture waiting time or claim that every free-solution chain has identical hydrodynamic drag.
A measured superlinear entry time can arise when exterior-coil drag, <tension> propagation, wall adsorption or nonuniform friction adds <length> dependence absent from this model. For example $\gamma(x)\propto x^p$ with the same $qEx$ drive gives $\tau\propto L_0^p$ for $p>0$. Under the assumptions of <uniform wall charge and polymer entry scaling>, a homogeneous thin-double-layer electro-osmotic model adds a length-independent fluid <velocity> $u_{\rm eo}=-\epsilon\zeta_wE/\eta$. It changes the drift coefficient, not its <length> exponent. This does not rule out charge heterogeneity, adsorption or near-cancellation followed by <diffusion> in a real channel; those are additional mechanisms.
With the field off, the confinement <entropy> drives outward motion. Assuming the constant <force> $f$ and the same local friction,
$$
\zeta_\ell x\dot x=-f,\qquad
\boxed{\tau_{\rm out}=\int_0^{L_0}\frac{\zeta_\ell x}{f}\,dx
=\frac{\gamma(L_0)L_0}{2f}.}
$$
The small final end region lies outside the long-confined-segment approximation but gives only a small correction. Thus ideal insertion scales as $L_0$, while recoil scales as $L_0^2$.
TODO: user-deferred DNA-specific numerical stopping-voltage estimate and its biological parameter-based inference. No biological quantities are evaluated.
For <entropic-trap length separation>, a reservoir/channel array can repeat length-dependent capture, confinement and relaxation delays, turning them into different mean residence times. The uniform inside-channel drift <speed> derived above is itself <length> independent, so that drift alone cannot separate <lengths>. Separation must retain <length> dependence of the barrier or of the capture/recoil dynamics and can be tuned with a forward-biased time-dependent field. For large bulk throughput, pressure-driven <advection> is preferable to <diffusion> alone; a <pressure> gradient can be combined with electric capture control. In the unobstructed continuum limit $Q=\pi R^4\Delta P/(8\eta\ell_c)$. Excessive driving erases trapping differences, so throughput and resolution compete; the question supplies no geometry or adsorption law selecting a unique optimal drive.
\Image[/past-exam-of-the-soft-matter-and-biological-physics-course-of-the-university-of-cambridge/2014/paper-1-polymer-channel.png]
{title=Original dimensionless insertion barriers and the distinct insertion and recoil length scalings}
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