Use a signed intermediate object in a two-lens system
Two lenses cannot generally be replaced by adding their separated focal lengths or image distances. First solve what the first lens alone would do. Here lens1 at global x=0 has f1=10 cm and object x=−30 cm, so its predicted image is at x=15 cm with magnification−1/2. A second lens at x=d then acts on the rays reaching that plane, not on the original object distance thirty centimetres away.
Learning outcome: Classify the second object as real, virtual or focal before solving each allowed separation. Retain both local distances and global positions. Explain why the d=10 case needs a negative second object distance even though the original object was real.
Starting recap
Use normal-based angles, signed conjugate distances and explicit optical-path/reflection phases. Separate geometric direction, amplitude and energy; preserve focal and critical boundaries.
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At d=10 cm, o2=−5 cm and i2=10/3 cm, so the final image lies at x=40/3 cm. m2=2/3 and total magnification−1/3. At d=20 cm, o2=5 cm gives i2=−10 cm and global x=10 cm: a virtual final image for lens2. At d=30 cm, i2=30 cm and the final image is at x=60 cm.
Open the experimentPredict
At lens separation10 cm in the stated system, lens2 receives:
Experiment
Classify the second object as real, virtual or focal before solving each allowed separation. Retain both local distances and global positions. Explain why the d=10 case needs a negative second object distance even though the original object was real.
Treat an intermediate image as the next signed object
Worked teaching model · synthetic values · First thin lens at x=0 has f1=10 cm and object x=−30 cm; its image without the second lens would lie at x=15 cm. Second lens f2=10 cm lies at selected x=d. Both paraxial, no apertures/aberrations. d=10 gives a virtual object for lens2; d=25 gives its exact focal branch.
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Two lenses and signed conjugate coordinates
Spatial axes use the same pixels per physical unit, preserving geometry. 1: original object (solid).2: first lens (dashed).3: second lens (dotted).4: intermediate image if lens2 absent (dotted).5: final image location (dotted). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Two lenses and signed conjugate coordinates
original object
| global axis coordinate (cm) | height (cm) |
|---|---|
| -30 | 0 |
| -30 | 1 |
At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.
first lens
| global axis coordinate (cm) | height (cm) |
|---|---|
| 0 | -2 |
| 0 | 2 |
At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.
second lens
| global axis coordinate (cm) | height (cm) |
|---|---|
| 20 | -2 |
| 20 | 2 |
At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.
intermediate image if lens2 absent
| global axis coordinate (cm) | height (cm) |
|---|---|
| 15 | 0 |
| 15 | -0.5 |
At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.
final image location
| global axis coordinate (cm) | height (cm) |
|---|---|
| 10 | 0 |
| 10 | -1 |
At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.
| Quantity | Value | Unit |
|---|---|---|
| Lens separation | 20 | cm |
| First intermediate image coordinate | 15 | cm |
| First magnification | -0.5 | ratio |
| Second signed object distance | 5 | cm |
| Second virtual object branch | 0 | flag |
| Second focal-object branch | 0 | flag |
| Second image distance from second lens | -10 | cm |
| Final global image coordinate | 10 | cm |
| Second magnification | 2 | ratio |
| Overall magnification | -1 | ratio |
| Final image height | -1 | cm |
Use the first image as the second signed object, then translate the final coordinate back to the common axis.
Lens1 alone gives i1=10×30/(30−10)=15 cm and m1=−1/2. The second signed object distance is d−15, not d or15. Apply the same reciprocal relation with this signed object, multiply m1m2 and add lens2 position to its image distance. A predicted intermediate focus beyond an intercepting lens is not an actual completed first image; it defines the converging input rays for the next element.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
Use the first image as the second signed object, then translate the final coordinate back to the common axis.
Lens1 alone gives i1=10×30/(30−10)=15 cm and m1=−1/2. The second signed object distance is d−15, not d or15. Apply the same reciprocal relation with this signed object, multiply m1m2 and add lens2 position to its image distance. A predicted intermediate focus beyond an intercepting lens is not an actual completed first image; it defines the converging input rays for the next element.
Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.
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Public practice checks a specific claim. It does not establish independent mastery.
Which independent observation would reject the approximation, sign convention or calibration premise rather than merely refit it?
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