When an object is placed 60 cm from a converging lens, it forms a real image. When the object is moved to 40 cm from the lens, the image moves 10 cm farther from the lens. What is the focal length of the lens?

Question

BlackTom AI · Accepted Answer

The thin-lens equation 1/f = 1/do + 1/di relates focal length f, object distance do, and image distance di (sign convention for a converging lens with real images: do > 0, di > 0).

1. Relevant concepts:
- Thin lens equation (above).
- Real image distances are positive; moving the object changes di according to the lens equation.
- Use algebra to eliminate f or di between two situations.

2. Step-by-step reasoning:
Let the image distance when the object is at do1 = 60 cm be di1 = d (unknown, positive). When the object is moved to do2 = 40 cm the image moves 10 cm farther so di2 = d + 10. Apply the lens equation to both placements:
(1) 1/f = 1/60 + 1/d
(2) 1/f = 1/40 + 1/(d+10)
Set the right-hand sides equal (both equal 1/f).

3. Intermediate algebra and calculations:
1/60 + 1/d = 1/40 + 1/(d+10)
Rearrange: 1/d - 1/(d+10) = 1/40 - 1/60
Compute right side: 1/40 - 1/60 = (3 - 2)/120 = 1/120
Left side becomes (d+10 - d)/(d(d+10)) = 10/[d(d+10)]
So 10/[d(d+10)] = 1/120 → d(d+10) = 10 × 120 = 1200
Solve quadratic: d^2 + 10d - 1200 = 0. Discriminant = 10^2 + 4800 = 4900, sqrt = 70.
d = (-10 ± 70)/2 → positive root d = (60)/2 = 30 cm (negative root -40 cm is unphysical).

Now find f from (1): 1/f = 1/60 + 1/30 = 1/60 + 2/60 = 3/60 = 1/20 → f = 20 cm.

4. Why this is correct:
The algebra yields a physically valid positive image distance (30 cm) and a focal length consistent with both object positions and the stated 10 cm image shift.

5. Why other options are incorrect:
- f = 30 cm would give di1 = 60 cm and a much larger shift when the object moves; it does not produce a 10 cm change.
- f = 40 cm yields di1 = 120 cm and di2 → ∞ for do2 = 40 cm (so not a 10 cm change).
- f = 42 cm (or other values) do not satisfy the pair of lens equations simultaneously.

6. Conclusion:
The focal length is 20 cm, which matches the provided answer.

When an object is placed 60 cm from a converging lens, it forms a real image. When the object is moved to 40 cm from the lens, the image moves 10 cm farther from the lens. What is the focal length of the lens?单项选择题

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