PHYS-2020: General Physics IICourse Lecture NotesSection XIIDr. Donald G. LuttermoserEast Tennessee State UniversityEdition 3.3AbstractThese class notes are designed for use of the instructor and students of the course PHYS-2020:General Physics II taught by Dr. Donald Luttermoser at East Tennessee State University. Thesenotes make reference to the College Physics, 9th Edition (2012) textbook by Serway and Vuille.XII. Mirrors and LensesA. Plane Mirrors.1. Images formed by plane (i .e., flat) mirrors have the followingpropert ies:a) The image is as far behind the mirror as the object is infront.b) The image is unmagnified, virtual, and erect.2. Image orientation:a) Erect: Image is oriented the same as the object.b) Inverted: Image is flipped 180◦with respect to the ob-ject.3. Image classification:a) Real: Image is on the same sid e of mirror as the object=⇒ light rays actually pass through the image point.b) Virtual: Image is on the opposite side of mirror fromobject =⇒ light rays appear to diverge from image point.4. Image size is determined by the magnification of an object whichis given byM ≡image heightobject height=h0h. (XII-1)XII –1XII –2 PHYS-2020: General Physics I I|M| > 1 =⇒ Image is bigger than object (magnified ).|M| = 1 =⇒ Image is unmagnified (like a plane mirror).|M| < 1 =⇒ Image is smaller than object (demagnified) .M > 0 =⇒ Image is erect.M < 0 =⇒ Image is inverted.M = 0 =⇒ No image is formed .5. Ray Tracing Rules:a) Images form at the point where ray s of light actually in-tersect (for real images) or from which they appear tooriginate (for virtu al images).b) For plane mirrors, p ( the object distance from the mirror)= q (the image distance from the mirr or) and h = h0.c) The following diagram shows how images are constructedfor a plane mirror.i) One ray runs parallel to the optical axis (⊥ line tothe mirror surface at the center of the mirror) fromthe head of the object (e.g., Ray 1 in the figure).ii) One ray travels f rom the head through the mirrorat the point where the optical axis intersects themirror (e.g., Ray 2 in the figure) .MirrorOBJECThIMAGEh’Optical Axis(defined wrtobject base)p1q2θiθrDonald G. Luttermoser, ETSU XII –3B. Spherical Mirrors1. Spherical mirrors have the shape of a segment of a spher e.a) Concave mirror: Reflecting surface is on the “inside” ofthe curved surface.b) Convex mirror: Reflecting surface is on t he “outside”of the curved surface.ConcaveMirror(this side)ConvexMirror(this side)2. Construct ing the image. Consider t he following concave mirror:OhpOpticalAxis(defined wrtmirror center)CVertex(V)RIh’qααθiθrXII –4 PHYS-2020: General Physics I Ia) The line that is normal to the mirror surface at the exactcenter is called t he optical axis of the mirror.b) The point where the optical axis intersects the mirror sur-face is called the vertex (labeled ‘V’ in the preceedingdiagram).c) Point ‘C’ indicates the position of the center of curva-ture of the mirror =⇒ line CV is equal to the radius ofcurvature, R, of the mirror.d) Note that the lengths p, q, and R are all measured withrespect to the vertex position. Also note that t he objectposition is labeled with ‘O’ and the image position with‘I’ in the preceeding diagram.e) Construct the image using the law of reflection:θi= −θr, (XII-2)where θ is measured with respect to the norm al of themirror sur face. The ‘negative’ sign is introduced here tonote that the reflected angle sweeps away f rom the opticalaxis in the opposite ‘sense’ of the incident angle.i) All normal lines on spherical concave mirrors gothrough center of curvature point C (θi= θr= 0)!ii) Now reflect a ray off the vertex of the mirror V.f) Using trigonometry, we see thath0q= tan θr&hp= tan θi.i) Since θi= −θr, we get tan θi= − tan θr, and henceh0q= − tan θi= −hporDonald G. Luttermoser, ETSU XII –5M =h0h= −qp. (XII-3)ii) The magnifi cation also can be determined by theratio of the image to the object distance.g) Using the “α” triangles in the preceeding diagram, we canwritetan α =hp − R& tan α = −h0R − q,orhp − R= −h0R − q,orh0h= −R − qp − R.Finally, using Eq. (XII-3) givesqp=R − qp − R.Solving this above expression givesR − qq=p − RpRq− 1 = 1 −RpRq+Rp= 1 + 1 = 2 .Finally,1p+1q=2R, (XII-4)which is the mirror equation.i) If p  R, then 1/p  2/R, so we say that asp → ∞, 1/p → 0 and1q=2Ror q =R2XII –6 PHYS-2020: General Physics I I=⇒ the image is formed (i.e., comes to a focus)halfway out to th e center of curvature.ii) So when p  R, the focal length of the mirror isf =R2. (XII-5)iii) When p  R, the mirror appears “thin” to thedistant object, th erefore Eq. (XII-5) is called thethin mirror approximation and we rewrite Eq.(XII-4) as1p+1q=1f. (XII-6)3. Both convex and concave mirrors use Eq. (XII-6), except there isa “change” in sign for the radius and focal length of the mirror.Table XII-1 shows t he sign conventions used for the geometricoptics parameters for cur ved mirrors.4. Image location can either be determined algebraically from Eqs.(XII-3) & (XII-6) or by drawing ray diagrams. There are threeprinciple rays that d efine the image location (see figures on pageXII-8):a) Concave mirror: Ray 1 is d rawn parallel to the optical axisand is reflected back through the focal point, F.b) Concave mirror: Ray 2 is drawn through the f ocal point,F, and reflected parallel to the optical axis.c) Concave mirror: Ray 3 is drawn thr ough the center ofcurvature, C, and reflected back on itself.Donald G. Luttermoser, ETSU XII –7Table XII–1: Sign Conventions for Curved Mirrors+ SIGNS –p object left of mirror object right of mirror(real object) (virtual object)image same side of image opposite side ofq mirror as object mirror as obj ect(real image) (virtual image)h object is erect object is invertedh0image is erect image is invertedM image is in same i mage is invertedorientation as object with respect to objectR concave mirror convex mirrorf concave mirror convex mirrorsymbollight lightXII –8 PHYS-2020: General Physics I Id) Convex mirror: Ray 4 is d rawn parallel to the optical axisand is r eflected back away from the focal point, F, on theback side of t he mirror.e) Convex mir ror: Ray 5 is drawn toward the focal point, F,on the back side of t he mirror and reflect ed back, parallelto the optical axis.f) Convex mirror: Ray 6 is drawn toward the center of curva-ture on