Exam practice

Light travels more slowly in glass (2.00 × 10⁸ m/s) than in water (2.25 × 10⁸ m/s); glass has the higher refractive index (1.5 against 1.33). Light going from water into glass bends towards the normal. Of two media, the one in which light is slower is the optically denser one, so glass is denser and water rarer.

Light from the coin passes from water (denser) into air (rarer) and bends away from the normal at the surface. The eye traces the light straight back, so it sees the coin at the point the light seems to come from, which is above the real coin. The apparent depth is less than the real depth (about ¾ of it in water), so the coin seems raised.

Light from the letters passes up through the glass and into the air, bending away from the normal as it leaves the glass. Traced back, it seems to come from a point higher than the letters, so they look raised. The apparent depth of the letters is the thickness of the slab divided by the refractive index of glass: through a 3 cm slab, they look only 2 cm deep (3 ÷ 1.5), raised by 1 cm.

Light from a star passes through many layers of the atmosphere, whose refractive indices are different and keep changing as the air moves. The light is refracted by changing amounts, bent sometimes towards and sometimes away from the normal, so the amount of light reaching the eye and the apparent position of the star keep changing. Because a star is so far away that it is a point of light, this flickering is easy to see: the star twinkles.

Light from the Sun, when it is just below the horizon, enters the atmosphere and passes from rarer into denser layers, bending towards the normal at each. It curves down over the Earth’s surface into the observer’s eye, and seems to come from above the horizon. So the Sun is seen about 2 minutes before it actually rises (and for about 2 minutes after it actually sets).

Diamond’s refractive index is high (2.42), so its critical angle is very small (24°). Light entering a well-cut diamond meets its inner faces at more than 24° again and again, is totally internally reflected many times, and comes back out of the top: the diamond sparkles. Glass has a critical angle of 42°, so much of the light meets the back faces at less than that and passes straight out through them instead of being reflected back to the eye.

Light travels more slowly in glass than in air, so it bends at each face of the prism: it is refracted going in, and again coming out, towards the base. And because the speed of light in glass is different for each color (red fastest, violet slowest), each color bends by a different amount: violet most, red least. So sunlight is not only refracted but dispersed into the seven colors of the spectrum.

A convex lens is like a stack of prisms with their bases towards the middle of the lens. Each prism bends light towards its base, so all of the lens bends light towards the principal axis. The edges are more steeply tilted than the middle, so they bend light more, and rays parallel to the axis all meet at one point, the principal focus.

A concave lens is like a stack of prisms with their bases facing outwards, towards the edges. Each prism bends light towards its base, so every part of the lens bends light away from the principal axis, more near the edges. Parallel rays spread out after the lens as if they came from one point in front of it, the principal focus.

We see in dim light with the rod cells of the retina. They contain a light-sensitive pigment, rhodopsin, made from a protein and vitamin A. Without enough vitamin A, too little rhodopsin is made, the rods can’t respond to dim light, and the person can’t see well at night.

Color is detected by three kinds of cone cells, sensitive to red, green and blue light; the brain works out each color by comparing how strongly they respond. If one kind of cone doesn’t work properly (usually the red or green ones), the brain can’t tell apart colors that differ only in that cone’s response, such as red and green. That is color blindness.

The bending, or change in direction, of light as it passes from one optical medium into another, caused by the change in the speed of light. Light going into a denser medium bends towards the normal; going into a rarer medium, away from it.

1. The incident ray, the normal at the point of incidence, and the refracted ray all lie in the same plane.
2. For a given pair of media, the ratio of the sine of the angle of incidence to the sine of the angle of refraction is constant (Snell’s law): $\dfrac{\sin i}{\sin r} = \mu$, the refractive index.

The pencil looks bent (broken) at the water’s surface, and the part under water looks shorter and raised. Light from the part under water passes from water into air and bends away from the normal, so the eye sees each point of it higher than it is (Figure 10.6). The process is the refraction of light.

(Check: sin 60° ÷ sin 40.5° = 0.866 ÷ 0.649 = 1.33, the refractive index of water.) Light from an object above the water bends towards the normal as it enters the water. An observer under the water traces it straight back, along a line steeper than the light’s real path in the air, so the object seems to be higher, farther from the surface, than it really is. To a fish, a fisherman looks taller and farther away.

The angle of incidence, in the denser medium, for which the angle of refraction in the rarer medium is 90° (the refracted ray just skims along the boundary). It is 42° for glass and 49° for water, against air; sin c = 1/μ.

When light traveling in a denser medium meets the boundary with a rarer medium at an angle of incidence greater than the critical angle, none of it can get out: all of it is reflected back into the denser medium, obeying the laws of reflection. This is total internal reflection.

1. The light must be traveling from a denser medium into a rarer medium.
2. The angle of incidence must be greater than the critical angle for that pair of media.

Data is turned into flashes of light and sent into the glass core of an optical fiber. The core is surrounded by a cladding of lower refractive index, and the light meets the boundary at more than the critical angle, so it is totally reflected again and again, staying in the core even round bends. Because total internal reflection loses no light, the signal travels tens of kilometers before it needs boosting, carrying huge amounts of data at the speed of light in glass.

The instruments (endoscope, colonoscope, laparoscope) contain bundles of optical fibers. Light from a lamp is sent along one bundle by total internal reflection, round all the bends of the tube, to light up the organ. Light reflected by the organ travels back along a second bundle, again by total internal reflection, to a camera or eyepiece, so the doctor sees the organ on a screen. So the inside of the body can be seen through the mouth, the rectum or a very small cut, without opening the body.

The splitting of white light into its seven colors (violet, indigo, blue, green, yellow, orange and red) when it passes through a prism or a similar transparent object.

In a medium such as glass, the speed of light depends on its wavelength (its color): red light, with the longest wavelength, travels fastest, and violet, with the shortest, slowest. So the refractive index is different for each color, and each color is bent by a different amount: violet most, red least. The colors separate.

• i.See Figure 10.2: incident ray AO, refracted ray OB bent towards the normal, emergent ray BC parallel to AO, and the lateral shift CD.
• ii.See Figure 10.22: white light enters one face, and fans out into the seven colors, red bent least (at the top) and violet most (at the bottom).

A rainbow is seen when the Sun is behind the observer and there are water drops in the air in front (after rain). Sunlight entering each drop is refracted and dispersed, reflected from the back of the drop, and refracted again as it comes out. Each color leaves at its own angle: red at about 42° and violet at about 40° to the direction of the sunlight (Figure 10.24).
So the observer receives red light only from drops lying at 42° from the line joining the Sun, the observer’s head and the point directly opposite the Sun (the antisolar point). All such drops lie on a circle around that line, so the rainbow is always a circle of the same angular size, 42° in radius (Figure 10.25). The antisolar point is below the horizon, so the ground hides the lower part of the circle and we see an arc; when the Sun is on the horizon, exactly a semicircle. From an airplane, a full circle can be seen.

• i.Center of curvature: the center of the sphere of which a curved surface of the lens is a part.
• ii.Optical center: the point at the center of the lens, on the principal axis, through which light passes without bending.
• iii.Principal axis: the straight line passing through the optical center and the centers of curvature of the lens.
• iv.Principal focus: the point on the principal axis where rays parallel to the principal axis meet (convex lens), or seem to come from (concave lens), after passing through the lens.

The power of a lens is its ability to converge or diverge light rays. It is the reciprocal of its focal length in meters, $P = \dfrac{1}{f}$, and is measured in diopters (D). Convex lenses have positive power, concave lenses negative.

• i.The +4 D lens is thicker. A greater power means a shorter focal length, which needs more strongly curved surfaces, so (for the same glass) a thicker lens.
• ii.$f = \dfrac{1}{P}$: for +2 D, $f = \dfrac{1}{2} = 0.5\ \text{m} = 50\ \text{cm}$; for +4 D, $f = \dfrac{1}{4} = 0.25\ \text{m} = 25\ \text{cm}$.

When the object is at 2F₁, twice the focal length from the lens. The image is formed at 2F₂ on the other side: real, inverted and the same size.

No: the spear will pass above the fish. Light from the fish bends away from the normal as it leaves the water, so the man sees the fish higher up than it really is (the apparent depth is less than the real depth). He aims at the image, and the spear, which travels straight and doesn’t bend, goes above the real fish. He should aim below where the fish seems to be. (People who fish with spears learn this by experience.)

Total internal reflection of light: the light meets the walls at more than the critical angle and is totally reflected each time, so it follows the curved tube. Devices: optical fiber cables (for internet and telephone) and the endoscope (also the colonoscope and laparoscope; periscopes and binoculars use it in prisms).

Burning paper: a convex lens; the Sun is at infinity, so its parallel rays are brought together at the focus F, forming a real, tiny, very bright and hot image of the Sun that can set paper alight. (The case “at infinity” in the ConvexLensCases widget.)

Magnifying glass: a convex lens with the object placed between F and the lens; it forms a virtual, erect, magnified image on the same side as the object, seen by looking through the lens.

Door peephole: a concave lens (in a real peephole, a combination of lenses that acts like one); whatever is outside the door, it forms a virtual, erect, diminished image (Figure 10.31), so a wide view of the doorstep fits in the small lens.

• a.X is violet (bent most, lowest) and Y is red (bent least, highest).
• b.Red light has a longer wavelength and travels faster in glass than violet, so the refractive index of glass is smaller for red. Red is bent less at each face of the prism, so Y is deviated less than X.

Place a second, identical prism upside down next to the first (Figure 10.23). It bends each color back by the same amount the first bent it, so the colors recombine and come out as white light, parallel to the light that entered. (A spinning Newton’s disc shows the same idea: the seven colors together look white.)

The lens of focal length +20 cm is convex; the lens of −20 cm is concave.
The convex lens forms a virtual, magnified image, because the object (16 cm) is closer than its focus (20 cm): it acts as a magnifying glass. (A concave lens always gives a diminished image.)

Measured from the scale drawing (or found with the lens formula: $\tfrac{1}{v} = \tfrac{1}{20} - \tfrac{1}{16} = -\tfrac{1}{80}$), the image is 80 cm from the lens on the same side as the object, and 80 ÷ 16 = 5 times as tall. (The concave lens would give a virtual, upright image 8.9 cm from the lens, about half the size.)

• i.Ciliary muscle: changes the thickness, and so the focal length, of the eye lens, to focus near and distant objects on the retina (accommodation).
• ii.Cornea: the clear front window of the eye; it lets light in and does most of the bending of light (about two-thirds of the eye’s focusing).
• iii.Lens: fine-tunes the focus, bringing the light to a sharp image on the retina.
• iv.Iris: the colored ring of muscle that controls the size of the pupil, and so how much light enters.
• v.Pupil: the hole in the iris through which light enters the eye.
• vi.Retina: the light-sensitive layer where the image forms; its rods (dim light) and cones (color) turn light into electrical signals sent to the brain by the optic nerve.

Any two: a corneal ulcer (keratitis), an infected sore that can lead to blindness if not treated quickly; corneal edema, fluid between the layers making the cornea cloudy and vision blurred; scarring that leaves the cornea cloudy; a change in its shape (as in keratoconus) causing defects of vision.

Looking at the whiteboard, which is fairly near, the ciliary muscles are partly contracted, so the lens is fairly thick, with a shorter focal length, to focus the spreading light from the board on the retina. When he looks at the distant object, the ciliary muscles relax; the suspensory ligaments pull the lens thinner, its curvature decreases and its focal length increases, so the nearly parallel light from far away is focused on the retina. This automatic adjustment is accommodation.

The defect is long-sightedness (hypermetropia).
Causes:

1. the eyeball is too short, so the retina is too close to the lens;
2. the lens can’t become thick enough (its focal length stays too long), because the ciliary muscles are weak or the lens has stiffened, as in old age.

Correction: a convex lens of suitable power is worn in front of the eye. It converges the light from a near object a little before it enters the eye, so that the light seems to come from the eye’s own (farther) near point, and the eye can focus it on the retina (Figure 10.35).

• i.Short-sightedness (myopia).
• ii.See Figure 10.34a: parallel light from the distant board is focused in front of the retina.
• iii.Two reasons:

1. the eyeball is too long, so the retina is too far from the lens;
2. the eye lens (or the cornea) is too strongly curved, so its focal length is too short even when the ciliary muscles are relaxed.

(iv) A concave lens spreads the parallel rays from the board a little before they enter the eye, so they seem to come from the eye’s far point, and are focused on the retina (Figure 10.34c).

A long-sighted eye can’t focus light from a near object at 25 cm: it would be focused behind the retina. A convex lens placed in front of the eye converges the light a little before it enters, so that it seems to come from the eye’s own near point, farther away, which the eye can focus on the retina (Figure 10.35c). Its power is chosen to match: for a near point of 1 m, +3 D.

Generally yes. A thicker lens is more strongly curved, so it has a shorter focal length and a greater power. A more serious defect needs a more powerful lens to correct it: a very short-sighted eye needs a strong concave lens (thick at the edges), a very long-sighted eye a strong convex lens (thick in the middle). So thick lenses usually mean a large defect. (One caution: lenses made of high-index glass or plastic bend light more, so they can correct the same defect with a thinner lens. It is the power, in diopters, that really measures the defect, not the thickness.)

Laser eye surgery corrects defects of vision by reshaping the cornea, which does most of the eye’s focusing, with an ultraviolet excimer laser. For short sight, the laser removes a very thin layer from the middle of the cornea, making it flatter: it bends light less, and the image moves back onto the retina. For long sight, the laser works on the edges, so the middle bulges a little more and bends light more. In the most common method, LASIK, a thin flap is cut in the front of the cornea and folded back, the laser reshapes the cornea beneath, and the flap is laid back in place to heal (Figure 10.37).

A cataract is a cloudiness of the eye lens, usually in old age, when its proteins clump together. Light can’t pass cleanly to the retina, vision becomes blurred, and in the end the person goes blind; a gray or milky patch is seen in the pupil.
It is treated by surgery: the cloudy lens is removed and replaced by a small artificial intraocular lens (IOL) of acrylic or silicone, which focuses light on the retina again. Imported IOLs used to cost about 100 US dollars, too much for most patients in Nepal. In 1995, Dr Sanduk Ruit’s Tilganga Eye Centre in Kathmandu began making good-quality IOLs for about 4 dollars, and he developed a quick, stitch-free way of operating. This made cataract surgery cheap enough for poor and remote patients, and has given sight back to hundreds of thousands of people in Nepal and other countries.

$$\mu = \frac{c}{v} = \frac{3 \times 10^8\ \text{m/s}}{2 \times 10^8\ \text{m/s}} = 1.5$$
The refractive index of glass is 1.5.

$$v = \frac{c}{\mu} = \frac{3 \times 10^8\ \text{m/s}}{2.42} = 1.24 \times 10^8\ \text{m/s}$$

$$\mu = \frac{\sin i}{\sin r} = \frac{\sin 45^\circ}{\sin 33^\circ} = \frac{0.7071}{0.5446} = 1.30$$

$f = 25\ \text{cm} = 0.25\ \text{m}$
$$P = \frac{1}{f} = \frac{1}{0.25\ \text{m}} = +4\ \text{D}$$
(Taking the lens as convex, the usual meaning when no sign is given.)

$$f = \frac{1}{P} = \frac{1}{-6\ \text{D}} = -0.1667\ \text{m} = -16.67\ \text{cm}$$
The focal length is −16.67 cm: a concave lens, so the student is short-sighted. The minus sign is what tells you the lens is concave.