What it shows
If an object and a screen are further apart than four focal lengths, there are two lens positions that give a sharp image. The distance between them gives f from f = (D² − d²) / 4D.
The science
For a lens at distance u from the object, the screen at distance D from it, the image distance is v = D − u and 1/u + 1/v = 1/f, which gives u² − D u + D f = 0. When D > 4f this has two roots, so the lens gives a sharp image at two positions: one with a magnified image and one with a diminished image. If the two positions differ by d, then f = (D² − d²) / (4D). The method needs no measurement from the lens centre, so it avoids the usual error in finding the optical centre.
Materials
Cost: under Rs 200. Many items come from home or from waste; spare parts such as LEDs, motors and wire are in the store.
- A convex lens of 10 to 15 cm focal length (reading glass)
- A metre scale or a taped tape measure on a table
- A bright torch with a cross-wire in card as the object
- A white card screen
- Clay holders for the lens, object and screen
- A calculator
Safety
For this project: Do not look at the torch beam directly and never aim the lens at the Sun.
Never use mains electricity: power models with cells, batteries or a low-voltage adapter. Sharp and hot tools, flames and lab chemicals need an adult or your teacher beside you. Keep magnets, button cells and small parts away from young children, and wash your hands after making.
All the safety rulesStep by step
7 steps. Do them in order, and test each part before you go on to the next. Write what you notice in a notebook as you go.
Fix the object at one end and the screen at the other so that D is at least 4.5 times the rough focal length, say 70 cm for a lens of f about 15 cm.
Measure D between the object and the screen carefully and note it.
Slide the lens along from the object to the screen and stop at the first position giving a sharp, enlarged image; note the lens position x1.
Continue sliding and find the second position giving a sharp, small image; note x2.
Find d = x2 − x1, and compute f = (D² − d²) / 4D.
Repeat for D = 80, 90 and 100 cm, taking the average of three trials for each position.
Compare the average f with the value found from a distant object, and calculate the percentage difference.
What you should see
All values of D give almost the same f, within a few millimetres, and agree with the distant-object value. The enlarged image occurs close to the object and the diminished image close to the screen.
Viva questions
A teacher or a judge will ask how it works. Try to answer each question before you open it, then compare.
1What is the condition for two positions to exist?
The object-to-screen distance D must be more than 4f.
2What happens if D = 4f?
The two positions come together at the middle and d = 0, so only one sharp image appears.
3Why is this method better than the u-v method?
It uses only the shift of the lens, so the error in finding the lens centre cancels.
4What are the two images like?
One is magnified and one is diminished, both real and inverted.
Take it further
Measure the sizes of both images, h1 and h2, and show that the object size is √(h1 × h2).
Change one thing, measure what happens, and add a table or chart of your readings: that is what turns a model into your own investigation.
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