A Jeweller's Lens and Two Mirrors
Rajinder runs a small jewellery shop in Amritsar. To show fine work on a pendant he holds a convex lens of focal length 8 cm close to the piece and looks through it from the other side. He finds that with the pendant about 5 cm from the lens the image is upright and enlarged, but as he brings it towards 11 cm the image turns upside down and becomes hard to use. A customer wants the design projected on a white card so that three people can examine it together. Rajinder mounts the same lens on a stand, places the pendant 10 cm from it, and hunts for the sharp image with the card. He also has a plane mirror and a small concave mirror on the counter. A supplier offers him a magnifying glass marked 25 D and claims that with it a pendant can be seen at practically infinite magnification.
1. Explain, with lens formula calculations, why the image flips between 5 cm and 11 cm, and work out where the card must go for the pendant at 10 cm.
Read the worked answer
With f = +8 cm, use 1/v − 1/u = 1/f.
At u = −5 cm: 1/v = 1/8 − 1/5 = (5 − 8)/40 = −3/40, so v = −13.3 cm and m = v/u = (−13.3)/(−5) = +2.7. A negative v means a virtual image on the same side as the object - upright and enlarged, which is what he sees.
At u = −11 cm: 1/v = 1/8 − 1/11 = (11 − 8)/88 = 3/88, so v = +29.3 cm and m = v/u = 29.3/(−11) = −2.7. A positive v means a real, inverted image on the far side.
The change happens exactly at the focal point. At u = −8 cm the refracted rays emerge parallel, v is at infinity, and no image can be focused at all - which is why the region around 8 cm feels unusable.
At u = −10 cm: 1/v = 1/8 − 1/10 = (5 − 4)/40 = 1/40, so v = +40 cm, with m = 40/(−10) = −4. He must place the card 40 cm on the far side of the lens, and the design will be real, inverted and four times larger.
2. Assess the supplier's 'infinite magnification' claim for the 25 D magnifier.
Read the worked answer
Power P = 1/f in metres, so 25 D gives f = 1/25 m = 4 cm. Handling the lens as a simple magnifier, the angular magnification is M = 1 + 25/f (with f in cm) when the image is placed at the near point, giving 1 + 25/4 = 7.25×; with the image at infinity the figure is 25/4 = 6.25×. So the practical gain is roughly six to seven times, not infinity.
The claim confuses two different things. The image distance does become infinite when the object sits at the focal point, but an image at infinity is simply one the relaxed eye can view - it is not infinitely large, and nothing is projected anywhere. Real single lenses also lose quality as magnification rises: at f = 4 cm the working distance is tiny, spherical and chromatic aberration blur the edges of a stone, and the field of view shrinks. For inspecting jewellery, a well-corrected 10× loupe or a low-power stereo microscope is a better buy than a very short-focus single lens.
3. Rajinder wants a mirror to keep an eye on the shop behind him while serving at the counter. Which of his two mirrors, or a third type, should he use, and why? What if he instead wants a magnified upright view of a small hallmark?
Read the worked answer
For watching the shop, a convex mirror is the right choice, even though he does not currently own one. It always forms a virtual, erect, diminished image and its field of view is far wider than a plane mirror of the same size, so a single glance covers the whole counter and door.
A plane mirror gives an image the same size, with a field of view limited to about twice the mirror width - too narrow for a shop. The concave mirror is the worst option for this purpose: with the object close (inside the focal length) it magnifies a small area only, and beyond the focal length it produces an inverted real image. It would show him a magnified view of one shelf, or an upside-down shop, not a survey of the room.
The concave mirror is, however, exactly right for the second job - reading a tiny hallmark - because an object held inside its focal length gives an enlarged, upright virtual image. So the mirror should be selected by the image it must produce: wide field and diminished for watching, magnified and upright at close range for inspecting.