Pushing a Lens to Magnification Eight
A student has a converging lens marked 15.0 cm, an illuminated object, a screen and an optical bench 1.50 m long. With the object at the 40.0 cm mark, the lens at the 60.0 cm mark and the screen at the 120.0 cm mark, a sharp image is formed. The student is now asked to produce a real image with magnification exactly 8.0 to fill a display. The lens, object and screen can all be moved along the bench, and the bench scale can be read to ±0.1 cm. The lens is a compound lens, and its true focal length may differ from the marked value by up to 0.5 cm.
1. Using the first arrangement, confirm the focal length. Then determine whether a magnification of 8.0 can be produced on this bench.
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First arrangement: u = 20.0 cm, v = 60.0 cm.
1/f = 1/20 + 1/60 = 3/60 + 1/60 = 4/60, so f = 15.0 cm. This agrees with the marking. Magnification is 60/20 = 3.0.
For a real image of magnification m, v = mu, and 1/f = 1/u + 1/v gives u = f(m + 1)/m and v = f(m + 1). The object–screen separation is therefore u + v = f(m + 1)²/m.
For f = 15.0 cm and m = 8.0: u = 15.0 × 9/8 = 16.9 cm, v = 135 cm, total = 151.9 cm.
The bench is 150.0 cm, so m = 8.0 is just out of reach. The maximum is about 7.9: setting u + v = 150 with uv = 2250 gives u = 16.9 cm, v = 133.1 cm.
2. Describe the adjustment needed to increase the magnification from 3.0 toward the maximum, and explain why the image becomes dimmer.
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Move the object closer to the lens, from 20 cm toward about 17 cm, and move the screen much further away, from 60 cm to beyond 130 cm.
The change is very asymmetric: as u approaches f, v rises steeply. Reducing u by 3 cm moves the screen by more than 70 cm, so a small object adjustment demands a large screen adjustment. That is why the bench length, not the lens, sets the limit.
The image is dimmer for two reasons. The same light is spread over an area proportional to m², so intensity falls roughly as 1/m². Also, the screen is further from the lens, so the light has spread further by the time it arrives (inverse-square spreading). A longer exposure or a brighter object would be needed to compensate.
3. The student concludes firmly that 'magnification 8.0 is impossible with this lens'. Assess how much confidence this conclusion deserves, using the stated uncertainties.
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The margin is 151.9 cm needed against 150.0 cm available - only 1.9 cm, about 1.3%. The bench scale contributes under 1% at these distances, but the lens focal length may be 14.5–15.5 cm, a 3% variation. Since the required separation scales with f, an f of 14.8 cm or less would bring m = 8.0 within the bench.
So the conclusion is not secure: the dominant uncertainty is the unmeasured focal length, not the ruler. A better approach is to measure f directly - for example with a very distant object, or from several u–v pairs plotted as 1/v against 1/u - before rejecting the lens. With the stated data the honest statement is 'probably just impossible, but not established'.