Senior 4 practice book · Unit 2 of 10

Optical Instruments

50 questions that complete the Senior 4 quiz for this unit: 29 core and 21 challenge. Easy and hard questions are mixed. Try each question, then tap “Show answer”.

Common misconceptions

The eye focuses by moving its lens backwards and forwards, like a camera.The eye lens stays in place; the ciliary muscles change its shape (its focal length), while a camera moves its lens.
Glasses for short sight are converging because they help you see further.A short-sighted eye is too strong; it needs a diverging (negative-power) lens to spread the rays from distant objects before they enter the eye.
A bigger objective lens makes a telescope magnify more.Magnifying power depends on the focal lengths (f_o/f_e); a wider objective collects more light, giving a brighter, sharper image, not a larger one.
The total magnification of a microscope is the objective magnification plus the eyepiece magnification.The eyepiece magnifies the already-magnified image of the objective, so the two magnifications multiply.

What this unit covers

Topics marked new are not tested much in the quiz, so this book gives them extra questions.

The lens camera: lens, shutter, diaphragm, focusing and image on the sensor
The slide projector: lamp, concave reflector, condenser and projection lens
Structure of the human eye and accommodation (cornea, iris, lens, ciliary muscles, retina)
Comparing the eye and the camera; persistence of vision
Visual angle and angular magnification (magnifying power)
Short sight and long sight: causes, correction and lens calculations
Other eye defects: presbyopia, astigmatism and cataract
The simple microscope (magnifying glass)
The compound microscope
The astronomical (refracting) telescope in normal and near-point adjustment
Telescopes with an upright image: terrestrial and Galilean telescopes, prism binoculars
Reflecting telescopes and where telescopes are sited

Go to the questions

Questions (1–50)

Easier and harder questions are mixed, just like in a real exam. The 21 harder ones are marked ★ Challenge.

1True or false
The eyepiece of a compound microscope works as a magnifying glass, viewing the real image produced by the objective.
Show answer
Answer: True
The objective’s real image falls just inside F of the eyepiece, which then acts as a simple magnifier to give a large virtual image.
Common mistakeSome learners think the eyepiece looks at the specimen directly; it only ever sees the image made by the objective.
2True or false · ★ Challenge
A telescope with a concave mirror objective shows coloured fringes around the image of a white star, just as a simple lens objective does.
Show answer
Answer: False
Reflection obeys the same law for every colour, so a mirror does not disperse light; coloured fringes (chromatic aberration) come from refraction in lenses.
Common mistakeAnswering "true" assumes all curved optics behave the same; dispersion needs refraction, and a mirror only reflects.
3True or false
A long-sighted person can usually see distant objects clearly but has difficulty focusing on objects close to the eyes.
Show answer
Answer: True
In long sight the near point is further than 25 cm, so close objects (like a book) cannot be focused, while distant objects are still seen clearly.
Common mistakeThe names confuse many learners: "long-sighted" means you see far (long) distances well, and it is near vision that suffers.
4Fill in the blank · ★ Challenge
For the same magnifying glass, the magnifying power with the final image at the near point is exactly ______ more than with the final image at infinity.
Show answer
Answer: 1 (one)
At the near point M = 1 + D/f; at infinity M = D/f; the difference is exactly 1.
Common mistakeAnswering "twice" is a guess; compare the two formulas, M = 1 + D/f and M = D/f, which differ by exactly one.
5True or false
Presbyopia is caused by the eye lens becoming stiffer with age, so the eye can no longer accommodate enough to see near objects.
Show answer
Answer: True
With age the lens loses its flexibility and the ciliary muscles weaken, so the near point moves away from the eye.
Common mistakeSome learners think presbyopia is the same as long sight from birth; it comes with age, and the person may also have other defects.
6True or false
The magnified image of an insect seen through a magnifying glass can be caught on a sheet of paper held on the other side of the lens.
Show answer
Answer: False
A magnifying glass forms a virtual image on the same side as the object; no rays actually meet there, so it cannot appear on a screen.
Common mistakeThe trap is to think every image can be projected; only real images, where rays really meet, can be caught on a screen.
7Fill in the blank · ★ Challenge
A cinema projector shows 24 frames per second, so each frame is on the screen for about ______ s.
Show answer
Answer: 0.042 (1/24)
Time per frame = 1 ÷ 24 = 0.042 s, less than the 1/16 s (0.0625 s) for which an image persists on the retina.
Common mistakeWriting 24 s confuses frequency and period: the time per frame is the reciprocal of the number of frames per second.
8True or false
Prism binoculars use totally reflecting prisms to turn the image the right way up and to fold the light path so the instrument can be short.
Show answer
Answer: True
Each half has two right-angled prisms; total internal reflection inverts the image back to upright and sends the light back and forth, shortening the tubes.
Common mistakeSome think binoculars contain plane mirrors; prisms are used because total internal reflection gives brighter images and never tarnishes.
9True or false · ★ Challenge
A film shown at 24 pictures a second looks like smooth motion because the impression of each picture stays on the retina for about 1/16 s, longer than the gap between pictures.
Show answer
Answer: True
This persistence of vision means a new picture arrives (every 1/24 s ≈ 0.042 s) before the last impression fades (about 0.06 s), so the separate frames blend into motion.
Common mistakeSome learners think the pictures themselves move on the screen; each frame is still, and the eye’s persistence of vision joins them.
10True or false
In normal adjustment, a parallel beam of light from a star enters the objective of an astronomical telescope and a parallel beam leaves the eyepiece.
Show answer
Answer: True
The objective focuses the parallel light at its focus, which is also the focus of the eyepiece, so the eyepiece sends the light out parallel again (final image at infinity).
Common mistakeMany expect the light to leave converging to a point; in normal adjustment it leaves parallel, so the relaxed eye can focus it.
11True or false
Opening the aperture of a camera wider lets in more light, but it also makes the range of distances that are in sharp focus (the depth of field) smaller.
Show answer
Answer: True
A wide aperture uses the whole lens, so rays from points not exactly in focus form larger blur circles; a small aperture keeps more distances sharp.
Common mistakeMany learners think the aperture only changes brightness; it also changes how much of the scene in front and behind is sharp.
12Multiple choice · ★ Challenge
Why are major research telescopes built on high, dry mountains far from towns?
1. AThe telescope is closer to the stars, so they look bigger
2. BLess air above to blur the image, and no city glow
3. CCold mountain air makes the lenses more powerful
4. DStarlight is stronger at high altitude because of gravity
Show answer
Answer: B. Less air above to blur the image, and no city glow
Moving air and water vapour blur and absorb starlight, and town lights brighten the sky; a high, dry, dark site avoids both.
Common mistakeChoosing "closer to the stars" ignores the scale: a few kilometres is nothing compared with the distance to a star.
13Fill in the blank
Because its eyepiece is a diverging lens, a Galilean telescope gives a final image that is ______.
Show answer
Answer: upright (erect)
The diverging eyepiece intercepts the light before the objective’s inverted image forms, so the final virtual image is upright.
Common mistakeAssuming all telescopes invert the image is a slip: only the astronomical type, with two converging lenses, gives an inverted image.
14Multiple choice · ★ Challenge
A learner’s spectacles are labelled −2.5 D. What can you conclude?
1. AShe is long-sighted; her near point is about 40 cm
2. BShe is short-sighted; her far point is about 40 cm
3. CShe is short-sighted; her far point is about 2.5 m
4. DShe is long-sighted; her lenses converge light
Show answer
Answer: B. She is short-sighted; her far point is about 40 cm
Negative power means a diverging lens, used for short sight; f = 1/P = 1 ÷ (−2.5) = −0.40 m, and this lens makes distant objects appear at her far point, 40 cm away.
Common mistakeChoosing 2.5 m reads the dioptre number as a distance; dioptres are 1/metres, so −2.5 D corresponds to 1 ÷ 2.5 = 0.40 m.
15Fill in the blank
In normal use, the real image formed by the objective of a compound microscope lies just inside the ______ of the eyepiece.
Show answer
Answer: principal focus (focal point)
With the intermediate image inside F of the eyepiece, the eyepiece forms a magnified virtual image, as a magnifying glass does.
Common mistakePlacing the intermediate image beyond F of the eyepiece would give a second real, inverted image instead of a virtual one for the eye.
16Fill in the blank
In an astronomical telescope in normal adjustment, the principal focus of the objective is at the same point as the principal focus of the ______.
Show answer
Answer: eyepiece
This is why the lenses are f_o + f_e apart and why the final image is at infinity.
Common mistakeWriting "retina" confuses the instrument with the eye; the two foci that coincide are those of the objective and eyepiece.
17Multiple choice · ★ Challenge
A photographer at a wedding in Huye first takes a photo of the couple from far away, then walks much closer to them. To get a sharp picture, how must the lens move, and how does the image on the sensor change?
1. ACloser to the sensor; the image becomes larger
2. BFurther from the sensor; the image becomes smaller
3. CFurther from the sensor; the image becomes larger
4. DIt stays put; only the aperture must change
Show answer
Answer: C. Further from the sensor; the image becomes larger
For a nearer object the image distance v increases (1/v = 1/f − 1/u), so the lens moves away from the sensor, and m = v/u grows, giving a larger image.
Common mistakeChoosing "closer to the sensor" uses the rule backwards: v is smallest (equal to f) for very distant objects and grows as the object comes nearer.
18Multiple choice
What is the job of the condenser lenses in a slide projector?
1. ATo magnify the picture of the slide on the screen
2. BTo focus a sharp picture of the slide on the screen
3. CTo concentrate the lamp’s light evenly onto the slide
4. DTo absorb heat from the lamp and keep the slide cool
Show answer
Answer: C. To concentrate the lamp’s light evenly onto the slide
The condenser collects light from the lamp (with help from the concave reflector behind it) and directs it evenly through the slide so every part of the picture is bright.
Common mistakeChoosing "focus the picture" gives the job of the projection lens; the condenser only lights the slide, it does not form the image.
19Multiple choice · ★ Challenge
A man 1.8 m tall stands 90 m away across a football pitch. What angle does he subtend at a spectator’s eye?
1. A50 rad
2. B0.200 rad
3. C1.15 rad
4. D0.020 rad
Show answer
Answer: D. 0.020 rad
For a small angle, α ≈ height ÷ distance = 1.8 ÷ 90 = 0.020 rad (about 1.15°).
Common mistakeChoosing 1.15 rad gives the value in degrees but labels it in radians; height ÷ distance gives the angle directly in radians.
20Multiple choice
Which part of a camera does the same job as the retina of the eye?
1. AThe diaphragm (aperture blades)
2. BThe shutter
3. CThe image sensor (or film)
4. DThe lens cap
Show answer
Answer: C. The image sensor (or film)
The retina and the sensor are both the light-sensitive surfaces on which the real image is formed and recorded.
Common mistakeChoosing the diaphragm matches the iris, not the retina; the diaphragm controls the light, the sensor records it.
21Multiple choice · ★ Challenge
A learner swimming in Lake Muhazi opens his eyes underwater and finds everything very blurred. What is the best explanation?
1. AThe eye lens cannot change shape when it is cold
2. BWater absorbs the light before it reaches the retina
3. CThe pupil closes underwater, so too little light enters
4. DThe cornea loses most of its focusing power in water
Show answer
Answer: D. The cornea loses most of its focusing power in water
Most refraction in the eye happens at the air–cornea surface; water has almost the same refractive index as the cornea, so light hardly bends there and the image forms far behind the retina.
Common mistakeChoosing "the lens cannot change shape" ignores where most bending happens: the cornea, not the lens, gives most of the eye’s power.
22Multiple choice
An elderly farmer in Nyagatare has a cataract. Which statement about this defect is correct?
1. AThe eye lens becomes cloudy; it can be replaced by a plastic lens
2. BThe eyeball becomes too long; a diverging lens corrects it
3. CThe cornea is unevenly curved; a cylindrical lens will correct it
4. DThe retina loses its rods; no treatment is possible
Show answer
Answer: A. The eye lens becomes cloudy; it can be replaced by a plastic lens
In a cataract the crystalline lens becomes cloudy and scatters light; surgery removes it and puts in an artificial (plastic) lens.
Common mistakeChoosing "cylindrical lens" describes astigmatism; a cataract is not a focusing error that spectacles can correct, but a cloudy lens.
23Multiple choice
Which part of the eye has no light-sensitive cells, so that an image falling on it cannot be seen?
1. AThe yellow spot, where the cones are packed tightly
2. BThe blind spot, where the optic nerve leaves
3. CThe iris, the coloured ring round the pupil
4. DThe cornea, the clear window at the front
Show answer
Answer: B. The blind spot, where the optic nerve leaves
At the blind spot the nerve fibres leave the eye through the retina, so there are no rods or cones to detect light there.
Common mistakeChoosing the yellow spot (fovea) is the reverse: it is the most sensitive part of the retina, where vision is sharpest.
24Short answer · ★ Challenge
Explain, as you would describe a ray diagram, how a diverging lens allows a short-sighted person to see a distant school sign clearly.
Show answer
Model answer: Without help, parallel rays from the distant sign are focused in front of the retina, because the eye is too powerful (or the eyeball too long). A diverging lens spreads the parallel rays so they seem to come from the person’s far point. The eye can focus objects at its far point, so the rays now meet exactly on the retina and the sign is sharp.
Common mistakeSaying the lens "pushes the image back" without mentioning the far point misses the key idea: the lens forms a virtual image of distant objects AT the far point.
25Multiple choice
In a Newtonian reflecting telescope, what is the job of the small plane mirror near the top of the tube?
1. ATo collect the light from the star before the big mirror
2. BTo magnify the image formed by the large concave mirror
3. CTo reflect the converging light sideways to the eyepiece
4. DTo reflect the light back to the main mirror a second time
Show answer
Answer: C. To reflect the converging light sideways to the eyepiece
The large concave mirror at the bottom focuses the starlight; a small flat mirror at 45° turns the converging beam sideways to an eyepiece in the side of the tube.
Common mistakeChoosing "magnify" gives a plane mirror a job it cannot do: a plane mirror only changes the direction of the light, not the size of the image.
26Short answer · ★ Challenge
An older teacher wears bifocal spectacles. Explain why the upper part of each lens may be diverging while the lower part is converging, and why she looks through the lower part to read.
Show answer
Model answer: She is short-sighted (her far point is closer than infinity), so the upper part is diverging for looking straight ahead at distant things. Her near point has also moved away (presbyopia), so the lower part is converging for reading. People look down to read a book held in the hands, so the reading lens is placed at the bottom.
Common mistakeSaying one eye is short-sighted and the other long-sighted misses the point: the same eye needs two different corrections for two ranges of distance.
27Fill in the blank
To make the picture bigger, a projector is moved further from the screen and the projection lens must then be moved slightly ______ the slide to refocus.
Show answer
Answer: closer to (towards)
A larger image distance v needs a smaller object distance u (1/u = 1/f − 1/v), so the lens comes nearer to the slide, though still further than f.
Common mistakeMoving the lens away from the slide does the opposite: it brings the sharp image closer, which is what you do when the screen is nearer.
28Multiple choice · ★ Challenge
A Galilean telescope (opera glass) has a converging objective of focal length 50 cm and a diverging eyepiece of focal length 5.0 cm. In normal adjustment, what are its magnifying power and its length?
1. AM = 10, length 45 cm
2. BM = 10, length 55 cm
3. CM = 250, length 45 cm
4. DM = 0.1, length 45 cm
Show answer
Answer: A. M = 10, length 45 cm
M = f_o/|f_e| = 50 ÷ 5.0 = 10. The diverging eyepiece is placed inside the focus of the objective: length = f_o − |f_e| = 50 − 5.0 = 45 cm.
Common mistakeChoosing 55 cm uses f_o + f_e as for an astronomical telescope; with a diverging eyepiece the focal lengths subtract, which is why opera glasses are short.
29Multiple choice
Without any equipment, how can an optician quickly tell whether a pair of spectacles is for short sight or long sight?
1. ALook at print through them: magnified print means diverging (short sight)
2. BLook at print through them: magnified print means converging (long sight)
3. CWeigh them: heavy lenses are always for long sight
4. DHold them in sunlight: a diverging lens burns paper
Show answer
Answer: B. Look at print through them: magnified print means converging (long sight)
A converging lens held close to print forms a magnified upright image; a diverging lens always makes the print look smaller.
Common mistakeChoosing "magnified means diverging" reverses the lenses: only a converging lens can act as a magnifying glass.
30Fill in the blank
The magnifying power of an optical instrument is the angle subtended at the eye by the ______ divided by the angle subtended by the object seen with the naked eye.
Show answer
Answer: final image
M = β/α compares the angular size of the final image with the angular size of the object viewed without the instrument.
Common mistakeMany learners define magnifying power as image height ÷ object height; for instruments the eye looks into, it is a ratio of ANGLES.
31Multiple choice · ★ Challenge
A short-sighted driver has a far point of 125 cm. He wears old spectacles of power −0.50 D instead of the correct ones. What is the farthest distance at which he can now see clearly? (real-is-positive)
1. A1.25 m
2. B2.0 m
3. C0.77 m
4. D3.3 m
Show answer
Answer: D. 3.3 m
f = 1/(−0.50) = −200 cm. The lens must form the image at his far point, v = −125 cm: 1/u = 1/f − 1/v = −1/200 + 1/125 = 3/1000, so u = 333 cm ≈ 3.3 m.
Common mistakeChoosing 2.0 m simply reads off the focal length; the farthest clear object is where the lens puts its virtual image at his far point.
32Multiple choice
Which change would increase the magnifying power of a compound microscope?
1. AUsing an objective of longer focal length
2. BUsing an eyepiece of shorter focal length
3. CUsing an eyepiece lens of larger diameter
4. DUsing a brighter lamp under the stage
Show answer
Answer: B. Using an eyepiece of shorter focal length
M = m_o × D/f_e (roughly), so a shorter-focus eyepiece gives a larger angular magnification; a shorter-focus objective would help too.
Common mistakeChoosing a brighter lamp improves how well you see the image, not its size; magnifying power depends on focal lengths.
33Multiple choice · ★ Challenge
An astronomical telescope with f_o = 90 cm and f_e = 6.0 cm is adjusted so that the final image of a distant hill is at the near point (25 cm), not at infinity. What is its magnifying power now?
1. A15.0
2. B16.0
3. C18.6
4. D3.6
Show answer
Answer: C. 18.6
M = (f_o/f_e)(1 + f_e/D) = (90 ÷ 6.0) × (1 + 6.0 ÷ 25) = 15 × 1.24 = 18.6.
Common mistakeChoosing 15.0 gives the normal-adjustment value f_o/f_e; with the image at the near point the eyepiece works harder and M is larger by the factor (1 + f_e/D).
34Multiple choice
A photo of a moto taxi speeding past is blurred, although the camera was held still and focused. Which change would most likely give a sharp photo?
1. AA shorter shutter time
2. BA longer shutter time
3. CA smaller aperture with the same shutter time
4. DMoving the lens closer to the sensor
Show answer
Answer: A. A shorter shutter time
While the shutter is open the image of the moving moto slides across the sensor; a shorter exposure time leaves less time for it to move.
Common mistakeChoosing a smaller aperture improves depth of field, not motion blur; blur from movement depends on how long the shutter is open.
35Short answer
State two ways in which the eye and the lens camera are alike and two ways in which they differ.
Show answer
Model answer: Alike: both use a converging lens system to form a real, inverted, diminished image on a light-sensitive surface (retina / sensor); both control the light with an adjustable opening (iris / diaphragm). Different: the eye focuses by changing the focal length of its lens, the camera by moving the lens; the eye has a fixed lens-to-retina distance; the eye sends a continuous signal to the brain, while the camera records one picture during the shutter time.
Common mistakeWriting "the camera image is upright" is a common slip: both the eye and the camera form inverted real images.
36Short answer · ★ Challenge
Astronomers at a school star party usually set their telescopes in normal adjustment (final image at infinity) rather than with the image at the near point. Explain why, and state what they lose.
Show answer
Model answer: With the final image at infinity the eye is relaxed (ciliary muscles relaxed), so it can observe for a long time without strain. They lose a little magnifying power: at the near point M = (f_o/f_e)(1 + f_e/D) is slightly larger than f_o/f_e.
Common mistakeSome think normal adjustment gives the largest magnification; it is chosen for comfort, and the near-point setting actually magnifies slightly more.
37Short answer · ★ Challenge
For a microscope, the naked-eye angle α is worked out with the object at the near point (25 cm), but for a telescope it is worked out with the object at its real distance. Explain this difference.
Show answer
Model answer: Without a microscope, the best you can do with a small object is bring it to your near point, where it subtends the largest angle you can see clearly, so that is the fair comparison. A star or a distant hill cannot be brought closer, so the telescope is compared with the angle the object subtends at its actual distance.
Common mistakeUsing 25 cm for a telescope object makes no sense: the comparison must always be with the best view possible without the instrument.
38Multiple choice
In a slide projector the slide is put in upside down, a little further than f from the projection lens. Why?
1. AThe lens forms a real, inverted, magnified image, so the picture appears upright
2. BThe lens forms a virtual, upright image, so the slide must be turned the other way round
3. CThe condenser turns the image over before it reaches the lens
4. DThe screen reflects the picture and so inverts it a second time
Show answer
Answer: A. The lens forms a real, inverted, magnified image, so the picture appears upright
With the slide between F and 2F the projection lens forms a real, inverted, magnified image on the distant screen; inverting the slide makes the picture the right way up.
Common mistakeChoosing "virtual image" forgets that only a real image can appear on a screen; a magnifying-glass (virtual) image cannot be projected.
39Short answer
Explain why a magnifying glass must be held so that the object is closer to the lens than its principal focus.
Show answer
Model answer: Only when the object is inside the focal length does a converging lens form a virtual, upright, magnified image that the eye can look at. If the object is beyond F the lens forms a real, inverted image on the other side instead, which is not useful as a magnifier.
Common mistakeHolding the object "as far as possible" from the lens is a common slip; beyond F the image turns upside down and is no longer a magnifier image.
40Multiple choice · ★ Challenge
A watch repairer in Kigali can choose a magnifying lens of power +10 D, +20 D or +40 D. Which gives the greatest magnifying power, and what is it with the image at the near point (D = 25 cm)?
1. A+10 D, about 3.5 times
2. B+40 D, about 10 times
3. C+40 D, about 11 times
4. D+20 D, about 6 times
Show answer
Answer: C. +40 D, about 11 times
The most powerful lens has the shortest focal length: f = 1/40 m = 2.5 cm. M = 1 + D/f = 1 + 25 ÷ 2.5 = 11.
Common mistakeChoosing "about 10 times" uses M = D/f, which is for the image at infinity; with the image at the near point M = 1 + D/f.
41Short answer
Give two reasons why the largest telescopes in the world use a concave mirror rather than a lens as the objective.
Show answer
Model answer: (1) A mirror reflects all colours the same way, so there are no coloured fringes (no chromatic aberration), while a lens disperses light. (2) A large mirror can be supported across its whole back, whereas a large lens can only be held at its edge and sags under its own weight. (3) Only one surface of a mirror has to be ground accurately, and light does not have to pass through thick glass.
Common mistakeSaying "mirrors magnify more than lenses" is not a reason; magnifying power depends on focal lengths, not on whether a mirror or lens is used.
42True or false
The image formed on the retina is upside down, and the brain interprets it so that we see the world the right way up.
Show answer
Answer: True
The eye lens system is converging and forms a real, inverted, diminished image on the retina; the brain learns to read it as upright.
Common mistakeSome learners think the eye lens forms an upright image; a single converging lens system can only form a real image that is inverted.
43Short answer · ★ Challenge
A school projector has a projection lens of focal length 10 cm and makes a sharp picture on a wall 5.0 m from the lens. How far is the slide from the lens, and how many times larger is the picture than the slide?
Show answer
Model answer: 1/u = 1/f − 1/v = 1/10 − 1/500 = 50/500 − 1/500 = 49/500, so u = 500 ÷ 49 = 10.2 cm. m = v/u = 500 ÷ 10.2 = 49, so the picture is 49 times larger.
Common mistakeUsing 5.0 instead of 500 cm mixes units: all distances must be in the same unit before using the lens formula.
44Short answer
A learner reading a book in a dim classroom looks up at a bright window across the room. Describe what the iris and the ciliary muscles do.
Show answer
Model answer: The iris makes the pupil smaller to reduce the amount of light entering the eye in the bright light. The ciliary muscles relax, so the suspensory ligaments pull the lens thinner; the lens becomes less powerful (longer focal length) and focuses the distant window on the retina.
Common mistakeA common mix-up is that the ciliary muscles contract for distant objects; they relax for distant objects and contract for near ones.
45Short answer · ★ Challenge
A phone camera has a lens of focal length 35 mm. It photographs a eucalyptus tree 12 m tall standing 20 m away. Find the lens–sensor distance and the height of the tree on the sensor.
Show answer
Model answer: 1/v = 1/f − 1/u = 1/0.035 − 1/20 = 28.571 − 0.050 = 28.521 m⁻¹, so v = 0.03506 m ≈ 35.1 mm (just beyond f). h_i = h_o × v/u = 12 × 0.03506 ÷ 20 = 0.0210 m = 21 mm.
Common mistakeMixing millimetres and metres (for example 1/35 − 1/20) gives nonsense; convert every length to the same unit first.
46Multiple choice
A terrestrial telescope has an extra converging "erecting lens" of focal length f between the objective and the eyepiece. What does this lens do, and how much does it add to the length of the telescope?
1. AIt turns the image upright and adds 4f to the length
2. BIt increases the magnifying power and adds 2f
3. CIt turns the image upright and adds 2f to the length
4. DIt makes the image brighter and adds 4f to the length
Show answer
Answer: A. It turns the image upright and adds 4f to the length
The erecting lens forms a same-size, inverted copy of the objective’s image with object and image each 2f from it, turning the image upright and adding 2f + 2f = 4f.
Common mistakeChoosing "increases the magnifying power" is wrong: at 2f the erecting lens has magnification 1, so M is still f_o/f_e.
47Short answer · ★ Challenge
A ranger in Akagera uses a terrestrial telescope with objective f_o = 60 cm, eyepiece f_e = 5.0 cm and erecting lens f = 4.0 cm, in normal adjustment. Find its magnifying power and its length.
Show answer
Model answer: Magnifying power M = f_o/f_e = 60 ÷ 5.0 = 12 (the erecting lens has magnification 1). Length = f_o + 4f + f_e = 60 + 4 × 4.0 + 5.0 = 81 cm.
Common mistakeAdding only 2f (or forgetting the erecting lens) gives 73 cm or 65 cm; the erecting lens needs 2f on each side of it.
48Multiple choice
A learner looks at an onion-skin slide through a school compound microscope. Compared with the specimen, the final image is:
1. Areal, upright and magnified
2. Bvirtual, upright and magnified
3. Creal, inverted and diminished
4. Dvirtual, inverted and magnified
Show answer
Answer: D. virtual, inverted and magnified
The objective forms a real, inverted, magnified image; the eyepiece magnifies this without turning it again, so the final image is virtual and still inverted.
Common mistakeChoosing "upright" assumes the eyepiece turns the image back; a magnifying glass keeps the orientation of what it looks at, which is already inverted.
49Multiple choice
A 100 FRW coin held close to the eye looks much bigger than the same coin on a table across the room. Why?
1. AClose up it subtends a larger visual angle, so its retinal image is larger
2. BClose up, the eye lens magnifies it far more strongly than it does from far away
3. CLight from a near coin is brighter, which makes the coin look bigger
4. DA near coin is focused in front of the retina, so its image spreads out
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Answer: A. Close up it subtends a larger visual angle, so its retinal image is larger
The apparent size of an object depends on the angle it subtends at the eye; bringing it closer increases this angle and the size of its image on the retina.
Common mistakeChoosing "the eye lens magnifies more" is wrong: the eye always forms a diminished image; it is the angle, not a magnification by the lens, that changes.
50Short answer · ★ Challenge
In a microscope, the objective (f = 0.50 cm) has the specimen 0.55 cm from it, and the eyepiece (f = 2.5 cm) is adjusted so that the final image is at infinity (D = 25 cm). Find the magnifying power and the distance between the lenses. (real-is-positive)
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Model answer: Objective: 1/v = 1/0.50 − 1/0.55, so v = 5.5 cm and m_o = 5.5 ÷ 0.55 = 10. Eyepiece (image at infinity): M_e = D/f_e = 25 ÷ 2.5 = 10. Total M = 10 × 10 = 100. For a final image at infinity the intermediate image is at F of the eyepiece, so the separation = 5.5 + 2.5 = 8.0 cm.
Common mistakeAdding the two magnifications (10 + 10 = 20) is the classic slip; the eyepiece magnifies an image that is already magnified, so they multiply.