#set document(title: "2.1 Images Formed by Plane Mirrors", author: "OpenStax / XYZ Homework") #set page(width: 8.5in, height: auto, margin: 1in) #import "@preview/cetz:0.5.2" #set text(font: ("STIX Two Text", "Libertinus Serif", "New Computer Modern"), size: 10.5pt, lang: "en") #show math.equation: set text(font: ("STIX Two Math", "New Computer Modern Math")) #set par(justify: true, leading: 0.62em, spacing: 0.9em) #set enum(spacing: 1.1em) // room between list items so tall inline fractions don't collide #set list(spacing: 1.1em) #set table(stroke: 0.5pt + rgb("#c7ccd3")) #let BLUE = rgb("#183B6F") // brand navy — section bars + example/solution labels (white on navy 11.09:1) #let ORANGE = rgb("#A94509") // brand primary-700 — AA-safe deep orange for TEXT (5.93:1 on white; raw brand #F37021 is 2.94:1 and must never carry text) #let RED = rgb("#DC2626") // brand error-600 #let GREEN = rgb("#059669") // brand success-600 (decoration only; small green text uses green-text #007942) #show heading.where(level: 1): it => block(width: 100%, above: 0pt, below: 16pt, fill: gradient.linear(BLUE, rgb("#2C5AA0")), inset: (x: 14pt, y: 12pt), radius: 3pt, text(fill: white, weight: "bold", size: 19pt, it.body)) #show heading.where(level: 2): it => block(width: 100%, above: 18pt, below: 10pt, fill: BLUE, inset: (x: 10pt, y: 6pt), radius: 2pt, text(fill: white, weight: "bold", size: 12pt, it.body)) #show heading.where(level: 3): it => text(fill: ORANGE, weight: "bold", size: 12.5pt, it.body) #show heading.where(level: 4): it => text(fill: BLUE, weight: "bold", size: 10.5pt, it.body) #let examplebox(label, title, body) = block(width: 100%, breakable: true, fill: rgb("#EFF1F5"), stroke: 0.5pt + rgb("#CFDDF0"), radius: 4pt, inset: 10pt, above: 12pt, below: 12pt)[ #block(below: 6pt)[#box(fill: BLUE, inset: (x: 6pt, y: 2pt), radius: 2pt, text(fill: white, weight: "bold", size: 8.5pt, label)) #h(0.4em) #strong[#title]] #body] // rail = decorative left rule (raw brand token); labelcolor = AA-safe label text shade #let notebox(label, rail, labelcolor, tint, body) = block(width: 100%, breakable: true, fill: tint, stroke: (left: 3pt + rail), inset: (left: 10pt, rest: 8pt), radius: (right: 4pt), above: 11pt, below: 11pt)[ #text(fill: labelcolor, weight: "bold", size: 7.5pt, tracking: 0.5pt)[#upper(label)] #linebreak() #body] #let solutionbox(body) = block(above: 4pt, below: 8pt)[ #text(fill: BLUE, weight: "bold", size: 8.5pt)[Solution] #linebreak() #body] #let figph(msg) = block(width: 100%, height: 60pt, fill: rgb("#f6f7f9"), stroke: (paint: rgb("#c7ccd3"), dash: "dashed"), radius: 4pt, inset: 10pt)[ #align(center + horizon, text(fill: rgb("#889"), style: "italic", size: 9pt, msg))] // Standardize inlined figure sizes: measure the natural CeTZ canvas, then scale to a // consistent envelope (aspect-aware; see build_typst.py FIG_* constants). Unlike the // print preamble, dimensions are FLOORED: in an editor a user can trim a figure to a // degenerate 1-D shape (a bare line), and w/h or tw/w would then divide by zero. #let _STD_W = 3.5 #let _WIDE_W = 5.6 #let _MAX_H = 3.4 #let _ASPECT_WIDE = 2.2 #let _UPSCALE_MAX = 1.15 #let stdfig(body) = context { let m = measure(body) let w = calc.max(m.width / 1in, 0.01) let h = calc.max(m.height / 1in, 0.01) let tw = if w / h > _ASPECT_WIDE { _WIDE_W } else { _STD_W } let s = calc.min(tw / w, _MAX_H / h, _UPSCALE_MAX) align(center, box(scale(x: s * 100%, y: s * 100%, reflow: true, body))) } #show figure: set block(breakable: false) #set figure(gap: 8pt) #show figure.caption: set text(size: 8.5pt, fill: rgb("#555")) == 2.1#h(0.6em)Images Formed by Plane Mirrors You only have to look as far as the nearest bathroom to find an example of an image formed by a mirror. Images in a #strong[plane mirror] are the same size as the object, are located behind the mirror, and are oriented in the same direction as the object (i.e., “upright”). To understand how this happens, consider . Two rays emerge from point #emph[P], strike the mirror, and reflect into the observer’s eye. Note that we use the law of reflection to construct the reflected rays. If the reflected rays are extended backward behind the mirror (see dashed lines in ), they seem to originate from point #emph[Q]. This is where the image of point #emph[P] is located. If we repeat this process for point #math.equation(block: false, alt: "P prime")[$P^(′)$], we obtain its image at point #math.equation(block: false, alt: "Q prime")[$Q^(′)$]. You should convince yourself by using basic geometry that the image height (the distance from #emph[Q] to #math.equation(block: false, alt: "Q prime")[$Q^(′)$]) is the same as the object height (the distance from #emph[P] to #math.equation(block: false, alt: "P prime")[$P^(′)$]). By forming images of all points of the object, we obtain an upright image of the object behind the mirror. #figure(figph[Figure shows cross section of a flat mirror in the center, a bottle to its left and a faded bottle (indicating that it is an image) to its right. The distances of the object and the image from the base of the mirror are labeled d subscript o and d subscript i respectively. Two rays originating from point P, at the base of the object hit the mirror at two separate points. The reflected rays from these points reach the eye of the observer, shown at the top left. The rays are extended to the right by dotted lines, such that they seem to originate from point Q, at the base of the image. Similarly, two rays, starting from point P prime, at the top of the object hit the mirror and are reflected to the eye of the observer. When extended at the back, these reflected rays seem to originate from point Q prime, at the top of the image.], alt: "Figure shows cross section of a flat mirror in the center, a bottle to its left and a faded bottle (indicating that it is an image) to its right. The distances of the object and the image from the base of the mirror are labeled d subscript o and d subscript i respectively. Two rays originating from point P, at the base of the object hit the mirror at two separate points. The reflected rays from these points reach the eye of the observer, shown at the top left. The rays are extended to the right by dotted lines, such that they seem to originate from point Q, at the base of the image. Similarly, two rays, starting from point P prime, at the top of the object hit the mirror and are reflected to the eye of the observer. When extended at the back, these reflected rays seem to originate from point Q prime, at the top of the image.", caption: [Two light rays originating from point #emph[P] on an object are reflected by a flat mirror into the eye of an observer. The reflected rays are obtained by using the law of reflection. Extending these reflected rays backward, they seem to come from point #emph[Q] behind the mirror, which is where the virtual image is located. Repeating this process for point #math.equation(block: false, alt: "P prime")[$P^(′)$] gives the image point #math.equation(block: false, alt: "Q prime")[$Q^(′)$]. The image height is thus the same as the object height, the image is upright, and the object distance #math.equation(block: false, alt: "d sub o")[$d_("o")$] is the same as the image distance #math.equation(block: false, alt: "d sub i")[$d_("i")$].]) Notice that the reflected rays appear to the observer to come directly from the image behind the mirror. In reality, these rays come from the points on the mirror where they are reflected. The image behind the mirror is called a #strong[virtual image] because it cannot be projected onto a screen—the rays only appear to originate from a common point behind the mirror. If you walk behind the mirror, you cannot see the image, because the rays do not go there. However, in front of the mirror, the rays behave exactly as if they come from behind the mirror, so that is where the virtual image is located. Later in this chapter, we discuss real images; a #strong[real image] can be projected onto a screen because the rays physically go through the image. You can certainly see both real and virtual images. The difference is that a virtual image cannot be projected onto a screen, whereas a real image can. === Locating an Image in a Plane Mirror The law of reflection tells us that the angle of incidence is the same as the angle of reflection. Applying this to triangles #emph[PAB] and #emph[QAB] in and using basic geometry shows that they are congruent triangles. This means that the distance #emph[PB] from the object to the mirror is the same as the distance #emph[BQ] from the mirror to the image. The #strong[object distance] (denoted #math.equation(block: false, alt: "d sub o")[$d_("o")$]) is the distance from the mirror to the object (or, more generally, from the center of the optical element that creates its image). Similarly, the #strong[image distance] (denoted #math.equation(block: false, alt: "d sub i")[$d_("i")$]) is the distance from the mirror to the image (or, more generally, from the center of the optical element that creates it). If we measure distances from the mirror, then the object and image are in opposite directions, so for a plane mirror, the object and image distances should have the opposite signs: #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #math.equation(block: true, alt: "d sub o equals − d sub i .")[$d_("o") = "−" d_("i") .$] ] An extended object such as the container in can be treated as a collection of points, and we can apply the method above to locate the image of each point on the extended object, thus forming the extended image. === Multiple Images If an object is situated in front of two mirrors, you may see images in both mirrors. In addition, the image in the first mirror may act as an object for the second mirror, so the second mirror may form an image of the image. If the mirrors are placed parallel to each other and the object is placed at a point other than the midpoint between them, then this process of image-of-an-image continues without end, as you may have noticed when standing in a hallway with mirrors on each side. This is shown in , which shows three images produced by the blue object. Notice that each reflection reverses front and back, just like pulling a right-hand glove inside out produces a left-hand glove (this is why a reflection of your right hand is a left hand). Thus, the fronts and backs of images 1 and 2 are both inverted with respect to the object, and the front and back of image 3 is inverted with respect to image 2, which is the object for image 3. #figure(figph[Figure shows cross sections of two mirrors placed parallel to each other, mirror 1 being on the left and mirror 2 on the right. Four human faces are shown, labeled object, image 1, image 2 and image 3. The object is in between the two mirrors, facing left towards mirror 1. Image 1 is to the left of mirror 1, facing right. Image 2 is to the right of mirror 2, facing right. Image 3 is to the far left, facing left. It is smaller than the other three faces.], alt: "Figure shows cross sections of two mirrors placed parallel to each other, mirror 1 being on the left and mirror 2 on the right. Four human faces are shown, labeled object, image 1, image 2 and image 3. The object is in between the two mirrors, facing left towards mirror 1. Image 1 is to the left of mirror 1, facing right. Image 2 is to the right of mirror 2, facing right. Image 3 is to the far left, facing left. It is smaller than the other three faces.", caption: [Two parallel mirrors can produce, in theory, an infinite number of images of an object placed off center between the mirrors. Three of these images are shown here. The front and back of each image is inverted with respect to its object. Note that the colors are only to identify the images. For normal mirrors, the color of an image is essentially the same as that of its object.]) Infinite reflections may terminate. For instance, two mirrors at right angles form three images, as shown in part (a) of . Images 1 and 2 result from rays that reflect from only a single mirror, but image 1,2 is formed by rays that reflect from both mirrors. This is shown in the ray-tracing diagram in part (b) of . To find image 1,2, you have to look behind the corner of the two mirrors. #figure(figph[Figure a shows mirror 1 and mirror 2 placed at right angles to each other and a lego man in front of them. Mirror 1 shows image 1, mirror 2 shows image 2 and the image of image 1, labeled image 1,2. Figure b shows cross section of two mirrors at right angles to each other. Mirror 1 is placed horizontally at the top and mirror 2, vertically, to the right. The object is a human face, upright and facing right, towards mirror 2. Image 1 is above mirror 1, upside down and facing right. Image 2 is to the right of mirror 2, upright and facing left. Image 1,2 is at the top right corner, upside down and facing left.], alt: "Figure a shows mirror 1 and mirror 2 placed at right angles to each other and a lego man in front of them. Mirror 1 shows image 1, mirror 2 shows image 2 and the image of image 1, labeled image 1,2. Figure b shows cross section of two mirrors at right angles to each other. Mirror 1 is placed horizontally at the top and mirror 2, vertically, to the right. The object is a human face, upright and facing right, towards mirror 2. Image 1 is above mirror 1, upside down and facing right. Image 2 is to the right of mirror 2, upright and facing left. Image 1,2 is at the top right corner, upside down and facing left.", caption: [Two mirrors can produce multiple images. (a) Three images of a plastic head are visible in the two mirrors at a right angle. (b) A single object reflecting from two mirrors at a right angle can produce three images, as shown by the green, purple, and red images.]) === Summary - A plane mirror always forms a virtual image (behind the mirror). - The image and object are the same distance from a flat mirror, the image size is the same as the object size, and the image is upright. === Conceptual Questions What are the differences between real and virtual images? How can you tell (by looking) whether an image formed by a single lens or mirror is real or virtual? #solutionbox[ Virtual image cannot be projected on a screen. You cannot distinguish a real image from a virtual image simply by judging from the image perceived with your eye. ] Can you see a virtual image? Explain your response. Can you photograph a virtual image? #solutionbox[ Yes, you can photograph a virtual image. For example, if you photograph your reflection from a plane mirror, you get a photograph of a virtual image. The camera focuses the light that enters its lens to form an image; whether the source of the light is a real object or a reflection from mirror (i.e., a virtual image) does not matter. ] Can you project a virtual image onto a screen? Is it necessary to project a real image onto a screen to see it? #solutionbox[ No, you can see the real image the same way you can see the virtual image. The retina of your eye effectively serves as a screen. ] Devise an arrangement of mirrors allowing you to see the back of your head. What is the minimum number of mirrors needed for this task? If you wish to see your entire body in a flat mirror (from head to toe), how tall should the mirror be? Does its size depend upon your distance away from the mirror? Provide a sketch. #solutionbox[ The mirror should be half your size and its top edge should be at the level of your eyes. The size does not depend on your distance from the mirror. ] === Problems Consider a pair of flat mirrors that are positioned so that they form an angle of 120#math.equation(block: false, alt: "°")[$"°"$]. An object is placed on the bisector between the mirrors. Construct a ray diagram as in to show how many images are formed. Consider a pair of flat mirrors that are positioned so that they form an angle of 60#math.equation(block: false, alt: "°")[$"°"$]. An object is placed on the bisector between the mirrors. Construct a ray diagram as in to show how many images are formed. #solutionbox[ #figure(figph[Figure shows cross sections of two mirrors placed at an angle of 60 degrees to each other. Six small circles labeled object, I1, I2, I3, I4 and I5 are shown. The object is on the bisector between the mirrors. Line 1 intersects mirror 1 perpendicularly connecting the object to I1 on the other side of the mirror. Line 2 intersects the mirror 2 perpendicularly connecting the object to I2 on the other side of the mirror. Lines parallel to these respectively connect I2 to I3 and I1 to I4. Lines parallel to these respectively connect I4 to I5 and I3 to I5.], alt: "Figure shows cross sections of two mirrors placed at an angle of 60 degrees to each other. Six small circles labeled object, I1, I2, I3, I4 and I5 are shown. The object is on the bisector between the mirrors. Line 1 intersects mirror 1 perpendicularly connecting the object to I1 on the other side of the mirror. Line 2 intersects the mirror 2 perpendicularly connecting the object to I2 on the other side of the mirror. Lines parallel to these respectively connect I2 to I3 and I1 to I4. Lines parallel to these respectively connect I4 to I5 and I3 to I5.", caption: none) ] By using more than one flat mirror, construct a ray diagram showing how to create an inverted image.