Slide Projector: Lens Equation, Image Orientation and Overheating
A student builds a slide projector for the science fair using a thin convex lens of focal length 15.0 cm and a transparent slide measuring 3.0 cm × 4.0 cm. The screen is a whiteboard 1.2 m × 0.9 m placed 3.00 m from the lens. He places the slide 16.0 cm from the lens, with the picture the right way up as the audience sees it, and finds that the image on the board is upside down and blurred, with dull edges. The 50 W halogen lamp sits 5 cm behind the slide with no condenser lens, and after ten minutes the slide has begun to warp and the lens is hot to the touch. The teacher says two things are wrong: the lens aperture is too small, and the slide is the wrong way up. She asks him to work out where the slide should actually be placed.
1. Calculate the image distance for the current 16.0 cm object distance, and determine where the slide must be moved to focus sharply on the 3.00 m screen.
Read the worked answer
Use (\frac{1}{f}=\frac{1}{u}+\frac{1}{v}), with f = 15.0 cm and u = 16.0 cm.
(\frac{1}{v}=\frac{1}{15.0}-\frac{1}{16.0}=\frac{16-15}{240}=\frac{1}{240})
(v = 240\text{ cm} = 2.40\text{ m})
The image forms 2.40 m from the lens, i.e. 60 cm in front of the whiteboard - that is why it is blurred.
For the screen at v = 300 cm:
(\frac{1}{u}=\frac{1}{15.0}-\frac{1}{300}=\frac{20-1}{300}=\frac{19}{300})
(u = 15.79\text{ cm} \approx 15.8\text{ cm})
So the slide must be moved about 0.2 cm closer to the lens, not further away - a small and slightly counter-intuitive shift, because near (u \approx f) the image distance changes extremely fast with object distance. This sensitivity is why a projector needs a fine focusing rack, and it also warns that the 15.8 cm setting must be accurate to a fraction of a millimetre.
2. Explain the orientation of the image, and check whether the projected picture will fit on the whiteboard.
Read the worked answer
Orientation. A single converging lens with (u > f) forms a real, inverted image. Inversion happens in both directions - up-down and left-right - so the audience sees a picture rotated by 180°. To make the final image upright the slide itself must be inserted upside down and back-to-front relative to the audience's view. That is exactly how slides and film strips were traditionally loaded; the student's observation confirms the theory rather than indicating a fault.
Size check. Magnification:
(m = \frac{v}{u} = \frac{300}{15.79} = 19)
(h_{image} = m \times h_{object} = 19\times3.0 = 57\text{ cm}) (height) (w_{image} = 19\times4.0 = 76\text{ cm}) (width)
The image is 57 cm × 76 cm on a 90 cm × 120 cm board, so it fits comfortably with a margin. Note that m ≈ 19 also means the area is magnified about 360 times, so the light originally spread over 12 cm² is now spread over roughly 4300 cm² - this is the real reason the picture looks dim, and it cannot be fixed by geometry alone.
3. Explain why the picture has dull edges and why the slide warps, and suggest two practical modifications.
Read the worked answer
Dull edges. The slide is an extended, diffusely lit object, not a point source. Each point on the slide sends a cone of light towards the lens, but only the rays that actually strike the lens can be focused. With a small aperture, the cone is clipped, so only the centre of the slide is well transmitted and the edges lose intensity - this is vignetting. A larger-diameter lens collects more of each cone and brightens the whole field. At the same time, however, the lens must be stopped down for sharpness, so brightness and image quality trade off; a condenser lens placed between lamp and slide is the standard fix, because it bends the diverging beam towards the lens so the aperture is filled without making the lens huge.
Warping. The halogen lamp radiates strongly in the infrared. With no filter, the slide absorbs this radiation and heats up; the plastic softens and warps, and the lens also heats and its focal length can shift slightly, blurring the picture.
Suggested modifications:
- Insert a heat-absorbing (IR-blocking) glass or a dichroic filter between the lamp and the slide.
- Add a pair of condenser lenses (or a concave mirror behind the lamp) to collect and direct light through the slide, and use a fan for forced-air cooling.
The judgement to state clearly is the priority order: fix the geometry (position and orientation) first, because no optical add-on can sharpen an image formed 60 cm in front of the screen.