Chapter 34: Geometrical Optics Mirrors Plane Spherical (convex or concave) Lenses The lens equation Lensmaker s equation Combination of lenses E! Phys Phys 2435: 22: Chap. 34, 3, Pg
Mirrors New Topic Phys 2435: Chap. 33, Pg 2
Looking in the mirror E! The image by a plane mirror is virtual, upright, equal size, equal distance, and with left and right reversed. How to find the image? Ray diagram (or ray tracing) Click here for AP5.4 simulation Phys 2435: Chap. 34, Pg 3
How to find an image? Trace rays from object to mirror to eye. Brain assumes that light travels in a straight line! Extend rays back behind mirror to form image. This is a virtual image since it does not actually exist behind the mirror. The image by a plane mirror is virtual, upright, equal size, equal distance, and with left and right reversed. Phys 2435: Chap. 33, Pg 4
Spherical Mirrors In general spherical mirrors do not focus at a single point. (aberration) (Parabolic mirrors do have a perfect focus.) Paraxial approximation (small angle E approximation): mirror is small! compared to its radius of curvature so a single focal point exists. The focal length is f = r 2 Phys 2435: Chap. 34, Pg 5
Concave Spherical Mirrors: image formation E! If object is in between center and focal point, the image is real (light passes though), inverted, and enlarged. Phys 2435: Chap. 34, Pg 6
Example: Ray Tracing The object is placed beyond the center point. What is the image? C F E! The image is real, inverted, reduced. Phys 2435: Chap. 34, Pg 7
Mirror equation: Spherical Mirrors: calculations + d i = f Lateral magnification: m = h i h o =! d i Sign conventions: h i is positive if upright, negative if inverted. and d i are positive if on the reflecting side, negative if on the other side. Phys 2435: Chap. 34, Pg 8
Summary: Concave Mirrors (assume f = 2 cm, then r = 4 cm) + d i = f m = h i h o =! d i Object: inside focal point ( =.5 cm) Image: virtual, upright, enlarged.5 + d i = 2, d i =!6 cm m =!!6.5 = 4 Object: in between center and focal point ( =3 cm) Image: real, inverted, enlarged 3 + d i = 2, d i = 6 cm m =! 6 3 =!2 Object: outside center point ( =6 cm) Image: real, inverted, reduced 6 + d i = 2, d i = 3 cm m =! 3 6 =!0.5 Phys 2435: Chap. 34, Pg 9
Convex Spherical Mirrors Convex mirrors produce virtual, upright and smaller images. Mirror equation still holds, but f is negative. + d i = f Magnification equation also holds m = h i h o =! d i For example, a convex rearview car mirror has curvature r =6 m. The image of an object 0 m away from the mirror is 0 + = d i!8, so d i =!4.4m m =! d i =!!4.4 0 = 0.44 Warning: objects in mirror may be closer than they appear. Phys 2435: Chap. 34, Pg 0
Convex surface Refraction at a spherical surface n + n 2 = n! n 2 d i R Equation also holds for concave surface, but R is negative For example, A person looks vertically down into a -m deep pool. How deep does the pool appear to be?.33.0 + d i =!.33 " d i =!0.75 m Phys 2435: Chap. 33, Pg
ConcepTest 34.(Post) Plane mirror If you walk directly towards a plane mirror at a speed v, your image will move ) towards you with v 2) towards you with 2v 3) away from you with v 4) away from you with 2v Phys 2435: Chap. 33, Pg 2
Lenses New Topic Phys 2435: Chap. 33, Pg 3
Thin Lenses Thin in the sense that the thickness is much smaller than the diameter. They form the basis of optical instruments: eyeglasses, cameras, telescopes, Phys 2435: Chap. 33, Pg 4
Thin Lenses Focal point, focal length, focal plane. Optometrists like to use the reciprocal of the focus length, called power, P = /f. Unit is diopter (D). D = m -. For example, a 20-cm-focal-length converging lens has a power of P = /0.2 = 5.0 D. If it s diverging, P = -5.0 D Phys 2435: Chap. 33, Pg 5
Thin Lenses: Ray Diagrams Phys 2435: Chap. 33, Pg 6
The Lens Equation + d i = f m = h i h o =! d i Sign convention: ) f is positive for converging lens, negative for diverging lens. 2) is positive if it is on the same side of incident light, negative otherwise. 3) d i is positive if it is on the opposite side of incident light, negative otherwise 4) h i (or m) is positive if the image is upright and negative if inverted, relative to the object. (h o is always taken as positive) Click here for thin lens simulation AP5.9 Phys 2435: Chap. 33, Pg 7
Example: What is the position and size of the image of a 7.6-cm high flower placed m from a +50-mm-focal-length camera lens? 00 + d i = 5 d i = 5.26 cm m = h i h o =! d i =! 5.26 00 =!0.0526 h i = mh o =!0.0526 " 7.6 =!0.4 cm Real, inverted, reduced Phys 2435: Chap. 33, Pg 8
More Examples: Converging Lens (assume f = 20 cm) + d i = f m = h i h o =! d i Object: inside focal point ( =5 cm) Image: virtual, upright, enlarged 5 + d i = 20, d i =!60 cm m =!!60 5 = 4 Object: outside focal point ( =30 cm) Image: real, inverted, enlarged 30 + = d i 20, d = 60 cm i m =! 60 30 =!2 Object: outside focal point ( =60 cm) Image: real, inverted, reduced 60 + = d i 20, d i = 30 cm m =! 30 60 =!0.5 Phys 2435: Chap. 34, Pg 9
Combination of Lenses Basic idea: the image formed by the st lens is the object for the 2 nd lens, and so on. (Watch out for signs!) Example: f =20 cm, f 2 =25 cm, object is placed 60 cm in front. 60 + d i = 20, d i = 30 cm 80! 30 + = d i2 25, d i2 = 50 cm m =! d i =! 30 60 =!0.5 m 2 =! d i2 2 =! 50 50 =! m = m m 2 = (!0.5) " (!) = 0.5 Phys 2435: Chap. 34, Pg 20
Lensmaker s Equation f = (n!) " $! # R R 2 % ' & Symmetric in R and R 2 : so f is the same both ways. Example: double-convex, n=.52, absolute radius 0 cm for both surfaces, then the focal length is Example: double-concave, n=.52, absolute radius 0 cm for both surfaces, f = (.52!) "!0! % $ ' # 0& f =!9.6 cm f f & = (.52 ' ) $ % = 9.6 cm 0 ' ' 0 #! " Phys 2435: Chap. 34, Pg 2
ConcepTest 34.2(Post) Lens A light source is held near a lens and its light focuses at point B, as shown in the top figure. If the light source is moved closer to the lens, then at which of the indicated points will the light focus? Phys 2435: Chap. 34, Pg 22