What Is The Focal Length Of A Makeup Mirror That Has A Power Of 1.50 D?

25.1 The Ray Aspect of Light

one.

Suppose a human stands in front of a mirror as shown in Effigy 25.49. His eyes are 1.65 m above the flooring, and the top of his caput is 0.13 k higher. Find the pinnacle above the floor of the height and lesser of the smallest mirror in which he can run into both the top of his head and his feet. How is this distance related to the man'south height?

Figure 25.49 A full-length mirror is 1 in which you tin run across all of yourself. It need not be as big as y'all, and its size is independent of your distance from it.

25.2 The Law of Reflection

ii .

Show that when light reflects from two mirrors that meet each other at a right angle, the outgoing ray is parallel to the incoming ray, equally illustrated in the following figure.

Figure 25.l A corner reflector sends the reflected ray dorsum in a direction parallel to the incident ray, contained of incoming direction.

three .

Low-cal shows staged with lasers apply moving mirrors to swing beams and create colorful effects. Show that a light ray reflected from a mirror changes direction past $2\theta$ when the mirror is rotated by an angle $\theta$ .

4 .

A flat mirror is neither converging nor diverging. To prove this, consider two rays originating from the same betoken and diverging at an angle $\theta$ . Bear witness that subsequently hitting a plane mirror, the bending between their directions remains $\theta$ .

Figure 25.51 A flat mirror neither converges nor diverges calorie-free rays. 2 rays continue to diverge at the same angle after reflection.

25.three The Law of Refraction

5.

What is the speed of lite in h2o? In glycerine?

6 .

What is the speed of light in air? In crown glass?

7.

Calculate the index of refraction for a medium in which the speed of light is $2\text{.}\text{012}\times {\text{10}}^8\text{thou/s}$ , and place the near likely substance based on Tabular array 25.one.

eight .

In what substance in Table 25.1 is the speed of light $\mathrm{ii}\text{.}\text{290}\times {\text{ten}}^8\text{thou/s}$ ?

nine.

There was a major collision of an asteroid with the Moon in medieval times. It was described by monks at Canterbury Cathedral in England as a red glow on and effectually the Moon. How long after the asteroid striking the Moon, which is $3\text{.}\text{84}\times {\text{10}}^5\text{km}$ away, would the lite first arrive on Earth?

ten .

A scuba diver training in a pool looks at his instructor as shown in Effigy 25.52. What angle does the ray from the instructor's face make with the perpendicular to the h2o at the point where the ray enters? The angle between the ray in the water and the perpendicular to the water is $\text{25}\text{.}\mathrm{0º}$ .

Effigy 25.52 A scuba diver in a pool and his trainer expect at each other.

11.

Components of some computers communicate with each other through optical fibers having an index of refraction $n=1\text{.}\text{55}$ . What time in nanoseconds is required for a betoken to travel 0.200 k through such a fiber?

12 .

(a) Using information in Figure 25.52, find the height of the teacher's head above the water, noting that you lot volition first have to summate the angle of refraction. (b) Find the credible depth of the diver's head below water as seen by the teacher. Assume the diver and the diver'south image are the aforementioned horizontal distance from the normal.

13.

Suppose you accept an unknown clear substance immersed in water, and you wish to identify it by finding its index of refraction. You arrange to have a beam of calorie-free enter information technology at an bending of $\text{45}\text{.}\mathrm{0º}$ , and y'all observe the angle of refraction to be $\text{xl}\text{.}\mathrm{3º}$ . What is the alphabetize of refraction of the substance and its probable identity?

14 .

On the Moon'south surface, lunar astronauts placed a corner reflector, off which a laser beam is periodically reflected. The distance to the Moon is calculated from the round-trip time. What percent correction is needed to account for the filibuster in time due to the slowing of light in Earth's atmosphere? Assume the distance to the Moon is precisely $3\text{.}\text{84}\times {\text{10}}^{\mathrm{eight}}\text{g}$ , and Earth's atmosphere (which varies in density with altitude) is equivalent to a layer 30.0 km thick with a abiding index of refraction $n=\mathrm{i}\text{.}\text{000293}$ .

fifteen .

Suppose Figure 25.53 represents a ray of low-cal going from air through crown glass into water, such as going into a fish tank. Calculate the corporeality the ray is displaced past the glass ( $\mathrm{\Delta }x$ ), given that the incident angle is $\text{twoscore}\text{.}\mathrm{0º}$ and the glass is ane.00 cm thick.

sixteen .

Figure 25.53 shows a ray of light passing from one medium into a second and and so a 3rd. Show that ${\theta }_{\mathrm{three}}$ is the same every bit it would be if the second medium were not nowadays (provided total internal reflection does not occur).

Figure 25.53 A ray of calorie-free passes from i medium to a third by traveling through a second. The final direction is the same as if the second medium were not present, but the ray is displaced by $\mathrm{\Delta }\mathrm{ten}$ (shown exaggerated).

17.

Unreasonable Results

Suppose lite travels from water to some other substance, with an bending of incidence of $\text{x}\text{.}\mathrm{0º}$ and an angle of refraction of $\text{xiv}\text{.}\mathrm{9º}$ . (a) What is the alphabetize of refraction of the other substance? (b) What is unreasonable about this consequence? (c) Which assumptions are unreasonable or inconsistent?

18 .

Construct Your Own Problem

Consider sunlight entering the Earth's atmosphere at sunrise and dusk—that is, at a $\text{90º}$ incident angle. Taking the boundary between nearly empty space and the atmosphere to be sudden, calculate the bending of refraction for sunlight. This lengthens the time the Sun appears to be above the horizon, both at sunrise and sunset. At present construct a problem in which yous decide the angle of refraction for different models of the atmosphere, such as various layers of varying density. Your teacher may wish to guide y'all on the level of complexity to consider and on how the index of refraction varies with air density.

xix.

Unreasonable Results

Light traveling from water to a gemstone strikes the surface at an angle of $\text{eighty}\text{.}\mathrm{0º}$ and has an angle of refraction of $\text{15}\text{.}\mathrm{2º}$ . (a) What is the speed of light in the gemstone? (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

25.4 Full Internal Reflection

xx .

Verify that the disquisitional angle for lite going from water to air is $\text{48.6º}$ , equally discussed at the end of Case 25.four, regarding the critical angle for lite traveling in a polystyrene (a type of plastic) pipe surrounded past air.

21 .

(a) At the terminate of Instance 25.iv, it was stated that the critical angle for light going from diamond to air is $\text{24}\text{.}\mathrm{4º}$ . Verify this. (b) What is the critical angle for calorie-free going from zircon to air?

22.

An optical fiber uses flint glass clad with crown glass. What is the critical angle?

23 .

At what minimum angle will y'all get full internal reflection of light traveling in water and reflected from water ice?

24.

Suppose you are using full internal reflection to make an efficient corner reflector. If there is air outside and the incident bending is $\text{45}\text{.}\mathrm{0º}$ , what must be the minimum index of refraction of the cloth from which the reflector is fabricated?

25 .

You can determine the alphabetize of refraction of a substance by determining its critical angle. (a) What is the alphabetize of refraction of a substance that has a critical angle of $\text{68}\text{.}\mathrm{4º}$ when submerged in h2o? What is the substance, based on Table 25.ane? (b) What would the critical bending be for this substance in air?

26.

A ray of light, emitted beneath the surface of an unknown liquid with air to a higher place it, undergoes total internal reflection as shown in Figure 25.54. What is the index of refraction for the liquid and its likely identification?

Effigy 25.54 A light ray inside a liquid strikes the surface at the critical angle and undergoes total internal reflection.

27 .

A light ray entering an optical fiber surrounded by air is first refracted and then reflected as shown in Figure 25.55. Bear witness that if the cobweb is made from crown glass, any incident ray volition be totally internally reflected.

Figure 25.55 A calorie-free ray enters the cease of a fiber, the surface of which is perpendicular to its sides. Examine the conditions under which it may be totally internally reflected.

25.v Dispersion: The Rainbow and Prisms

28 .

(a) What is the ratio of the speed of crimson low-cal to violet lite in diamond, based on Table 25.ii? (b) What is this ratio in polystyrene? (c) Which is more than dispersive?

29.

A beam of white calorie-free goes from air into h2o at an incident angle of $\text{75}\text{.}\mathrm{0º}$ . At what angles are the blood-red (660 nm) and violet (410 nm) parts of the light refracted?

30 .

By how much exercise the critical angles for cherry (660 nm) and violet (410 nm) light differ in a diamond surrounded past air?

31.

(a) A narrow axle of light containing yellowish (580 nm) and green (550 nm) wavelengths goes from polystyrene to air, striking the surface at a $\text{xxx}\text{.}\mathrm{0º}$ incident bending. What is the bending betwixt the colors when they emerge? (b) How far would they have to travel to be separated by 1.00 mm?

32 .

A parallel beam of light containing orange (610 nm) and violet (410 nm) wavelengths goes from fused quartz to water, striking the surface between them at a $\text{60}\text{.}\mathrm{0º}$ incident angle. What is the angle between the two colors in h2o?

33.

A ray of 610 nm calorie-free goes from air into fused quartz at an incident bending of $\text{55}\text{.}\mathrm{0º}$ . At what incident angle must 470 nm light enter flint glass to have the same angle of refraction?

34 .

A narrow beam of lite containing red (660 nm) and blue (470 nm) wavelengths travels from air through a 1.00 cm thick apartment piece of crown glass and back to air once again. The axle strikes at a $\text{30}\text{.}\mathrm{0º}$ incident bending. (a) At what angles do the two colors emerge? (b) By what distance are the red and bluish separated when they emerge?

35.

A narrow beam of white light enters a prism made of crown glass at a $\text{45}\text{.}\mathrm{0º}$ incident angle, as shown in Effigy 25.56. At what angles, ${\theta }_{\text{R}}$ and ${\theta }_{\text{V}}$ , do the red (660 nm) and violet (410 nm) components of the light sally from the prism?

Figure 25.56 This prism will disperse the white light into a rainbow of colors. The incident angle is $\text{45}\text{.}\mathrm{0º}$, and the angles at which the scarlet and violet calorie-free emerge are ${\theta }_{\text{R}}$ and ${\theta }_{\text{V}}$.

25.6 Image Germination past Lenses

36 .

What is the ability in diopters of a camera lens that has a l.0 mm focal length?

37.

Your photographic camera's zoom lens has an adjustable focal length ranging from 80.0 to 200 mm. What is its range of powers?

38 .

What is the focal length of ane.75 D reading glasses found on the rack in a pharmacy?

39.

You note that your prescription for new eyeglasses is –4.50 D. What will their focal length exist?

40 .

How far from the lens must the motion-picture show in a camera be, if the lens has a 35.0 mm focal length and is being used to photo a flower 75.0 cm abroad? Explicitly show how you follow the steps in the Trouble-Solving Strategy for lenses.

41.

A certain slide projector has a 100 mm focal length lens. (a) How far away is the screen, if a slide is placed 103 mm from the lens and produces a precipitous image? (b) If the slide is 24.0 by 36.0 mm, what are the dimensions of the image? Explicitly show how you follow the steps in the Problem-Solving Strategy for lenses.

42.

A doc examines a mole with a xv.0 cm focal length magnifying drinking glass held 13.5 cm from the mole (a) Where is the image? (b) What is its magnification? (c) How big is the image of a 5.00 mm diameter mole?

43.

How far from a piece of newspaper must yous hold your male parent'south two.25 D reading spectacles to try to burn down a hole in the paper with sunlight?

44 .

A camera with a 50.0 mm focal length lens is existence used to photograph a person standing 3.00 m away. (a) How far from the lens must the picture show exist? (b) If the film is 36.0 mm high, what fraction of a 1.75 m tall person will fit on it? (c) Discuss how reasonable this seems, based on your experience in taking or posing for photographs.

45.

A camera lens used for taking close-upwards photographs has a focal length of 22.0 mm. The uttermost it tin can be placed from the film is 33.0 mm. (a) What is the closest object that can be photographed? (b) What is the magnification of this closest object?

46 .

Suppose your 50.0 mm focal length camera lens is 51.0 mm away from the film in the photographic camera. (a) How far away is an object that is in focus? (b) What is the peak of the object if its image is ii.00 cm high?

47.

(a) What is the focal length of a magnifying glass that produces a magnification of three.00 when held five.00 cm from an object, such as a rare coin? (b) Summate the power of the magnifier in diopters. (c) Discuss how this ability compares to those for store-bought reading glasses (typically 1.0 to iv.0 D). Is the magnifier's power greater, and should it exist?

48 .

What magnification volition exist produced by a lens of power –4.00 D (such as might exist used to correct myopia) if an object is held 25.0 cm away?

49.

In Instance 25.7, the magnification of a book held 7.50 cm from a 10.0 cm focal length lens was found to be 4.00. (a) Detect the magnification for the volume when it is held 8.l cm from the magnifier. (b) Do the same for when it is held nine.50 cm from the magnifier. (c) Comment on the trend in m as the object distance increases as in these ii calculations.

50 .

Suppose a 200 mm focal length telephoto lens is being used to photo mountains 10.0 km away. (a) Where is the paradigm? (b) What is the height of the image of a 1000 g high cliff on one of the mountains?

51.

A photographic camera with a 100 mm focal length lens is used to photograph the sun and moon. What is the meridian of the paradigm of the sun on the film, given the sunday is $\mathrm{one}\text{.}\text{40}\times {\text{10}}^6\text{km}$ in diameter and is $\mathrm{ane}\text{.}\text{50}\times {\text{10}}^8\text{km}$ away?

52 .

Combine thin lens equations to show that the magnification for a thin lens is determined by its focal length and the object distance and is given past $m=f/\left(f-d_{\text{o}}\right)$ .

25.7 Image Formation by Mirrors

53.

What is the focal length of a makeup mirror that has a power of i.50 D?

54 .

Some telephoto cameras use a mirror rather than a lens. What radius of curvature mirror is needed to supplant a 800 mm focal length telephoto lens?

55.

(a) Calculate the focal length of the mirror formed by the shiny dorsum of a spoon that has a three.00 cm radius of curvature. (b) What is its power in diopters?

56 .

Observe the magnification of the heater chemical element in Example 25.9. Notation that its big magnitude helps spread out the reflected energy.

57.

What is the focal length of a makeup mirror that produces a magnification of 1.l when a person'south face is 12.0 cm away? Explicitly evidence how you follow the steps in the Problem-Solving Strategy for Mirrors.

58 .

A shopper standing 3.00 k from a convex security mirror sees his image with a magnification of 0.250. (a) Where is his prototype? (b) What is the focal length of the mirror? (c) What is its radius of curvature? Explicitly show how y'all follow the steps in the Problem-Solving Strategy for Mirrors.

59.

An object one.50 cm high is held iii.00 cm from a person's cornea, and its reflected image is measured to exist 0.167 cm high. (a) What is the magnification? (b) Where is the image? (c) Find the radius of curvature of the convex mirror formed past the cornea. (Note that this technique is used by optometrists to measure the curvature of the cornea for contact lens plumbing fixtures. The instrument used is called a keratometer, or curve measurer.)

60 .

Ray tracing for a flat mirror shows that the paradigm is located a distance behind the mirror equal to the distance of the object from the mirror. This is stated $d_{\text{i}}={\mathrm{–d}}_{\text{o}}$ , since this is a negative image distance (information technology is a virtual image). (a) What is the focal length of a flat mirror? (b) What is its power?

61.

Prove that for a flat mirror $h_{\text{i}}=h_{\text{o}}$ , knowing that the image is a altitude behind the mirror equal in magnitude to the distance of the object from the mirror.

62 .

Apply the police force of reflection to prove that the focal length of a mirror is half its radius of curvature. That is, prove that $f=R/2$ . Note this is true for a spherical mirror only if its diameter is small compared with its radius of curvature.

63.

Referring to the electric room heater considered in the commencement example in this section, summate the intensity of IR radiations in ${\text{Due west/thou}}^2$ projected by the concave mirror on a person three.00 g abroad. Assume that the heating element radiates 1500 W and has an expanse of $\text{100}{\text{cm}}^2$ , and that half of the radiated power is reflected and focused by the mirror.

64 .

Consider a 250-West heat lamp fixed to the ceiling in a bathroom. If the filament in 1 light burns out and so the remaining three even so work. Construct a trouble in which you determine the resistance of each filament in club to obtain a certain intensity projected on the bath floor. The ceiling is three.0 m high. The problem will demand to involve concave mirrors behind the filaments. Your instructor may wish to guide you lot on the level of complexity to consider in the electrical components.

Source: https://openstax.org/books/college-physics-ap-courses/pages/25-problems-exercises

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