Connect an image plane to a separately specified photon readout

Declare the inter-stage premises

A sensing chain can combine correct component formulas into an incorrect overall conclusion if the connections between them are not justified. Here a thin converging lens with focal length10 cm images an object20–40 cm before it. The lens relation determines a real image plane and signed magnification. It does not determine how much optical power reaches the detector: source radiance, aperture, loss and geometry would be needed. We therefore place the virtual detector at the calculated plane and independently stipulate illumination2 nW.

Reconcile state, rate and calibration

The detector surface has work function2 eV and a supplied collected yield0.25 only strictly above threshold. Photon energy controls both the possible per-electron kinetic energy and photon supply at fixed power. Below threshold, or at the exact zero-kinetic-energy boundary in this particular finite-yield prescription, the readout receives zero current. This is not a real spectral-response claim. The current-output element is stipulated independent of readout voltage; a real collection circuit would need its own compliance, bias and source-feedback analysis.

Challenge the inference rather than merely the arithmetic

The ideal current output drives parallel resistance2 Gohm and capacitance0.5 nF from zero voltage. Charge balance is C dV/dt=I−V/R, with solution V=IR[1−exp(−t/RC)] supplied for algebraic use. The time constant is1 s. Current-source work through time t is I²R[t−tau(1−exp(−t/tau))]. Resistor conversion is I²R[t−2tau(1−exp(−t/tau))+tau(1−exp(−2t/tau))/2], and the difference is capacitor energy CV²/2. These formulas are a readout ledger, not a complete optical-to-electrical efficiency calculation. Rejecting an unjustified inter-stage assumption is as important as matching the numerical energy residual.

Worked original case

At object distance30 cm, the image is15 cm after the lens and magnification−0.5. With2.5 eV photons the stipulated current is0.2 nA and asymptotic readout voltage0.4 V. At2 s, voltage is0.4(1−exp(−2))≈0.345865887 V. Source work, resistor conversion and capacitor energy must close separately from the independently supplied2 nW optical input.

Cumulative investigation

Choose two object positions at one photon energy and two photon energies at one object position. Freeze the image/readout predictions before revealing them. Reconstruct charge and energy accounts, then identify one unsupported cross-stage inference and the independent observation that would test it.

Changed-case practice

Try the changed case before feedback

For the taught lens with object distance40 cm, find image distance after the lens in cm.

Numerical result in the requested unitsCompare this result

This practice is public and unsaved. A matching number does not verify understanding. Retain your explanation, assumptions and changed-case evidence in the separate course workspace.

Independent exit evidence

The separate course workspace retains mixed free responses, first report, criterion-linked critique, revision and a newly frozen assignment. A human reviewer must supply a genuinely unfamiliar packet and inspect reasoning, prior exposure and allowed resources. Completed software or matching public answers do not establish practical technique, independent competence or institutional credit.

Sources and boundaries

Original synthesis checked 31 October 2026 against the primary component model, photoelectric energy, RC state equations and decay expectations. Inter-stage inference is derived here, not presented as a source experiment. All observations are supplied synthetic records. No real wiring, optical exposure, detector operation or radioactive-source handling is requested.

Bound an optical parameter rather than only naming uncertainty

In a separate supplied lens measurement, object distance is independently exact at30 cm while the real image-distance bound is19–21 cm. Inverting the thin-lens relation gives f=oi/(o+i). For positive i at fixed o, increasing i increases f: comparing two candidate distances gives a positive difference proportional to o²(i2−i1)/[(o+i1)(o+i2)]. Thus the hard focal-length endpoints are570/49≈11.632653 cm and630/51≈12.352941 cm. Predict then check the changed image interval18–22 cm: endpoints are11.25 and165/13≈12.692308 cm. These are deterministic bounds conditional on exact object distance and the paraxial model, not a confidence level or an estimate of aberration. A common distance-scale error would require an additional independent calibration constraint.