Include the meter in the circuit being inferred
An ideal voltmeter samples the difference between two nodes with infinite input resistance, drawing no current.
Learning outcome: Compare finite meter resistances at a fixed load, predict the infinite-input-resistance limit, and retain actual branch currents. Invert the indication once with and once without the known meter model. Explain why reading precision alone cannot remove loading bias.
Starting recap
Integrate current or resistance, preserve node topology and derive transient state with a signed energy account.
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Try
For R = 4 kohm and Rm = 8 kohm, Rp = 8/3 kohm. The loaded indication is 24/7 V ≈ 3.42857 V rather than 4 V. Load and meter currents are 6/7 and 3/7 mA. Ignoring the meter would infer 2.66667 kohm rather than the true 4 kohm.
Open the experimentPredict
Ignoring a finite voltmeter in this divider fit identifies which resistance?
Experiment
Compare finite meter resistances at a fixed load, predict the infinite-input-resistance limit, and retain actual branch currents. Invert the indication once with and once without the known meter model. Explain why reading precision alone cannot remove loading bias.
Quantify a finite-resistance measurement branch
Worked teaching model · synthetic values · 6 V ideal source with series 2 kohm feeding the load. Voltmeter is a finite resistance in parallel with load. A separate contrast inserts a 100 ohm ammeter in series without the voltmeter. These are distinct virtual circuits, not a physical wiring procedure.
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Read the node connections before calculating
Two distinct branches connect J to B. Their drawing offsets do not change their shared endpoint nodes. Source positive terminal is S; conventional source-delivered current proceeds S to J. Component orientation and drawing lengths are schematic; source polarities are specified in the caption and node labels.
| First node | Second node | Element |
|---|---|---|
| S | J | 2 kΩ |
| J | B | 4 kΩ load |
| J | B | 8 kΩ meter |
| B | S | 6 V source |
Common schematic symbols and conventions
The symbol key is not a wiring recommendation. An open switch removes a conducting path; an ideal closed switch supplies a zero-drop connection. A current arrow is a chosen conventional-charge reference, not electron drift. A junction dot signifies connection; a crossing without a dot need not connect.
Indication versus meter resistance
Axes rescale independently. Curves join sampled coordinates; a joining segment is not additional measured evidence. Read physical units rather than screen angles or apparent slopes.
Indication versus meter resistance: numerical coordinates
| meter resistance (kohm) | voltage (V) |
|---|---|
| 2 | 2.4 |
| 2.3 | 2.53211 |
| 2.6 | 2.64407 |
| 2.9 | 2.74016 |
| 3.2 | 2.82353 |
| 3.5 | 2.89655 |
| 3.8 | 2.96104 |
| 4.1 | 3.0184 |
| 4.4 | 3.06977 |
| 4.7 | 3.11602 |
| 5 | 3.15789 |
| 5.3 | 3.19598 |
| 5.6 | 3.23077 |
| 5.9 | 3.26267 |
| 6.2 | 3.29204 |
| 6.5 | 3.31915 |
| 6.8 | 3.34426 |
| 7.1 | 3.36759 |
| 7.4 | 3.38931 |
| 7.7 | 3.40959 |
| 8 | 3.42857 |
| 8.3 | 3.44637 |
| 8.6 | 3.46309 |
| 8.9 | 3.47883 |
| 9.2 | 3.49367 |
| 9.5 | 3.50769 |
| 9.8 | 3.52096 |
| 10.1 | 3.53353 |
| 10.4 | 3.54545 |
| 10.7 | 3.55679 |
| 11 | 3.56757 |
| 11.3 | 3.57784 |
| 11.6 | 3.58763 |
| 11.9 | 3.59698 |
| 12.2 | 3.60591 |
| 12.5 | 3.61446 |
| 12.8 | 3.62264 |
| 13.1 | 3.63048 |
| 13.4 | 3.63801 |
| 13.7 | 3.64523 |
| 14 | 3.65217 |
| 14.3 | 3.65885 |
| 14.6 | 3.66527 |
| 14.9 | 3.67146 |
| 15.2 | 3.67742 |
| 15.5 | 3.68317 |
| 15.8 | 3.68872 |
| 16.1 | 3.69407 |
| 16.4 | 3.69925 |
| 16.7 | 3.70425 |
| 17 | 3.70909 |
| 17.3 | 3.71377 |
| 17.6 | 3.71831 |
| 17.9 | 3.7227 |
| 18.2 | 3.72696 |
| 18.5 | 3.73109 |
| 18.8 | 3.7351 |
| 19.1 | 3.73899 |
| 19.4 | 3.74277 |
| 19.7 | 3.74643 |
| 20 | 3.75 |
| Quantity | Value | Unit |
|---|---|---|
| Ideal unloaded voltage | 4 | V |
| Loaded voltmeter indication | 3.42857 | V |
| Actual load current | 0.857143 | mA |
| Voltmeter current | 0.428571 | mA |
| Naively inferred load resistance | 2.66667 | kohm |
| True load resistance | 4 | kohm |
| Separate nonideal-ammeter current | 0.983607 | mA |
| Current without that ammeter | 1 | mA |
The finite meter is part of the measured circuit, not an invisible observer.
A voltmeter belongs across two nodes and idealizes to infinite resistance; a current meter belongs in the selected branch and idealizes to zero resistance. Here the loaded voltage identifies Rload parallel Rmeter if the meter is ignored. Recovering Rload requires the known meter conductance and a positive inferred remaining conductance.
Derive before checking. Retain contrasting captures and explain one assumption that could fail; calculated evidence is not actual laboratory work.
The finite meter is part of the measured circuit, not an invisible observer.
A voltmeter belongs across two nodes and idealizes to infinite resistance; a current meter belongs in the selected branch and idealizes to zero resistance. Here the loaded voltage identifies Rload parallel Rmeter if the meter is ignored. Recovering Rload requires the known meter conductance and a positive inferred remaining conductance.
Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.
Understand
Apply
Public practice checks a specific claim. It does not establish independent mastery.
Which source, topology, initial condition or instrument assumption would invalidate this inference?
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