An image is 4.0 mm in front of a converging lens with focal length 5.0 mm. Where is the object?

Question

BlackTom AI · Accepted Answer

Thin-lens formula: 1/f = 1/do + 1/di. Use the sign convention that f > 0 for a converging lens and di < 0 for a virtual image (image on the same side of the lens as the object, i.e., "in front" of the lens).

Given: f = +5.0 mm, image is 4.0 mm in front of the lens → di = −4.0 mm (virtual image on object side).

Step-by-step algebra:
1. Start with 1/f = 1/do + 1/di.
2. Solve for 1/do: 1/do = 1/f − 1/di.
3. Substitute numbers: 1/do = 1/(5.0 mm) − 1/(−4.0 mm) = 1/5 + 1/4.
4. Compute common denominator: 1/5 + 1/4 = (4 + 5)/20 = 9/20 (mm^-1).
5. Invert to find do: do = 20/9 mm ≈ 2.222... mm.

Thus the object is approximately 2.2 mm in front of the lens.

Why this is correct: using the correct sign for a virtual image (di negative) gives a positive, small object distance (do ≈ 2.22 mm) located on the same side of the lens as the image. The calculation directly yields the provided answer.

Why the other choices are incorrect (briefly):
- "2.2 mm behind the lens": would correspond to do = −2.2 mm or a different sign; it does not satisfy the lens equation with di = −4 mm and f = +5 mm.
- "5.0 mm behind the lens": if di = +5 mm then 1/do = 1/5 − 1/5 = 0 implying an object at infinity, not 5 mm.
- "20 mm in front of the lens": if do = 20 mm then 1/di = 1/5 − 1/20 = 3/20 → di ≈ +6.67 mm (image behind lens), which contradicts the given image location.

Conclusion: The correct object position is 2.2 mm in front of the lens, matching the provided answer.

An image is 4.0 mm in front of a converging lens with focal length 5.0 mm. Where is the object?单项选择题

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