= Solution
<resistive-pulse sensing> detects the change in a conducting aperture's <electric current> as a particle crosses it. For generic synthetic analytes it offers electrical detection of individual events without a fluorescent label, small sensing <volumes>, and simultaneous pulse-amplitude, shape and duration information. Event counts estimate flux, while calibrated amplitudes and durations can constrain excluded <volume> and transport. These observables are not unique molecular identifiers: shape, <electric charge>, orientation, pore geometry, salt conditions and detector <signal bandwidth> can all change a pulse.
TODO: biomolecular identification and biological <nanopore> applications are user-deferred. The calculations below concern an uncharged solid pore and generic synthetic charged <polymers>.
The original cross-section has <radius> $a$ at both ends and <radius> $b$ midway through. In the intended one-dimensional series-resistance approximation, put $s=(b-a)/l$ and
$$
r(z)=\begin{cases}a+sz,&0\leq z\leq l,\\a+s(2l-z),&l\leq z\leq2l.\end{cases}
$$
For constant ionic <electrical resistivity> $\rho$, a slice contributes $d\mathcal R=\rho\,dz/[\pi r(z)^2]$. The <ionic resistance of a conical pore> then gives
$$
\mathcal R=\frac{2\rho}{\pi}\int_0^l\frac{dz}{(a+sz)^2}
=\frac{2\rho}{\pi s}\left(\frac1a-\frac1b\right),\qquad
\boxed{\mathcal R=\frac{2\rho l}{\pi ab}.}
$$
This assumes locally Ohmic transport with negligible surface conduction and approximately uniform axial <current density> over each slice. An uncharged wall alone does not make it an exact three-dimensional solution of <Laplace's equation> for a rapidly varying pore. <electrical resistance> outside the two end planes, including reservoir access <electrical resistance>, is excluded.
Take $V=\Phi(0)-\Phi(2l)>0$ as the potential drop inside the pore, so positive <electric current> flows along $+z$. <current conservation> and <Ohm's law> give
$$
I_0=\frac{V}{\mathcal R}=\frac{\pi abV}{2\rho l},\qquad
\boxed{E(z)=\frac{\rho I_0}{\pi r(z)^2}=\frac{Vab}{2l\,r(z)^2}.}
$$
This is the leading cross-section-averaged axial <electric field>, and the corresponding on-axis estimate in the same approximation. It is largest at the two narrow ends and smallest at the wide centre:
$$
E(0)=E(2l)=\frac{Vb}{2la},\qquad E(l)=\frac{Va}{2lb},\qquad
\frac{E(0)}{E(l)}=\left(\frac ba\right)^2.
$$
The graph is symmetric, falls on the first half and rises on the second, with a minimum at $l$. The <electric potential difference> check is $\int_0^{2l}E(z)\,dz=V$. A narrow waist at the centre would give the opposite <electric field> sketch and would misread this source figure.
Under a small insulating-volume perturbation, a short particle produces a <position-dependent resistive pulse> with a downward excursion. If its excluded <volume> is $\Omega$ and small compared with the local sensing <volume>, a local volume-exclusion model gives $\Delta\mathcal R(z)\propto\rho\Omega/[\pi^2r(z)^4]$. Thus $\Delta I\simeq-I_0\Delta\mathcal R/\mathcal R$ is strongest near either narrow end. A sufficiently resolved event can show two deeper end features separated by a shallower middle depression, then return to baseline. Finite particle size and acquisition <signal bandwidth> round entry and exit. The original illustration below shows this conditional small-particle mechanism, not a fitted amplitude. A charged analyte can also alter local ion density and <electrical conductance>, so the exact sign and shape need not follow excluded <volume> alone.
\Image[/past-exam-of-the-soft-matter-and-biological-physics-course-of-the-university-of-cambridge/2012/paper-1-pore-signals.png]
{title=Pore field and pulse}
For positive drift in this coordinate, require $\mu V>0$. The <electrophoretic transit time in a conical pore> is the integral of inverse local drift speed:
$$
\Delta t=\int_0^{2l}\frac{dz}{\mu E(z)}
=\frac{2l}{\mu Vab}\int_0^{2l}r(z)^2dz
=\frac{4l}{\mu Vab}\frac{b^3-a^3}{3s}.
$$
Therefore
$$
\boxed{\Delta t=\frac{4l^2}{3\mu Vab}\frac{b^3-a^3}{b-a}
=\frac{4l^2}{3\mu V}\left(\frac ba+1+\frac ab\right).}
$$
Reverse the polarity or use $|\mu V|$ when reporting a positive transit duration in the opposite direction. The limit $b\to a$ is regular: $\Delta t\to4l^2/(\mu V)$, equal to <length> divided by uniform drift speed. Also $b/a+a/b\geq2$, so nonuniform <electric field> makes this point-particle transit no faster than the uniform-field value at the same <length> and <electric potential difference>. The wide, weak-field interior dominates residence time.
The final length-dependent observations do not uniquely identify a microscopic mechanism. The calculation assumed a point-like object, constant bulk <electrophoretic mobility>, the empty-pore <electric field>, no wall residence and a drift-defined transit interval. An extended chain can sample entrance and reservoir fields simultaneously, alter its conformation and tension, and change the effective driving-to-friction ratio. These effects can make a bulk-mobility, local-field prediction underestimate the driving. For a small object, wall interactions, reversible trapping, or repeated entry excursions can add residence time that the deterministic formula omits. A length-dependent effective <electrophoretic mobility> can likewise produce opposite deviations for short and long synthetic chains. Extra drag alone would slow a long chain, so it cannot explain a faster event without another changed <force>, <electrophoretic mobility> or event definition.
Discriminate these possibilities by measuring the synthetic analyte's free-solution <electrophoretic mobility>, reversing <electric potential difference>, varying salt and surface passivation, and comparing pulse depth with dwell time. Resolved multiple occupancy or folds can alter amplitude as well as timing. Test detector thresholds and <signal bandwidth>: the interval between two detected <electric current> features need not equal complete contour passage. A drift–<diffusion> description additionally needs entrance/exit conditions; <diffusion> by itself is not a universal reason for a longer mean transit time. No unique anomalous-duration prediction follows from the provided parameters alone.
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