2Objectives: After Completing This Module, You Should Be Able To: Determine the focal length of converging and diverging lenses.Apply the lensmaker’s equation to find parameters related to lens construction.Use ray-tracing techniques to construct images formed by converging and diverging lenses.Find the location, nature, and magnification of images formed by converging and diverging lenses.
3Refraction in PrismsTwo prisms base to baseIf we apply the laws of refraction to two prisms, the rays bend toward the base, converging light.Parallel rays, however, do not converge to a focus leaving images distorted and unclear.
4Refraction in Prisms (Cont.) Two prisms apex to apexSimilarly, inverted prisms cause parallel light rays to bend toward the base (away from the center).Again there is no clear virtual focus, and once again, images are distorted and unclear.
5Converging and Diverging Lens If a smooth surface replaces the prisms, a well-defined focus produces clear images.Converging LensDiverging LensReal focusVirtual focusDouble-convexDouble-concave
6The Focal Length of lenses Converging LensDiverging LensFocal length fFf-f+The focal length f is positive for a real focus (converging) and negative for a virtual focus.
7The Principal FocusSince light can pass through a lens in either direction, there are two focal points for each lens.Left to rightThe principal focal point F is shown here. Yellow F is the other one.FRight to leftNow suppose light moves from right to left instead . . .FF
8Types of Converging Lenses In order for a lens to converge light it must be thicker near the midpoint to allow more bending.Plano-convex lensDouble-convex lensConverging meniscus lens
9Types of Diverging Lenses In order for a lens to diverge light it must be thinner near the midpoint to allow more bending.Double-concave lensPlano-concave lensdiverging meniscus lens
10Lensmaker’s Equation The focal length f for a lens. R1 R2 R Surfaces of different radiusThe Lensmaker’s Equation:Negative (Concave)Positive (Convex)Sign conventionR
11Signs for Lensmaker’s Equation +-R1 and R2 are interchangeableR1, R2 = Radiin= index of glassf = focal lengthR1 and R2 are positive for convex outward surface and negative for concave surface.Focal length f is positive for converging and negative for diverging lenses.
12Example 1. A glass meniscus lens (n = 1 Example 1. A glass meniscus lens (n = 1.5) has a concave surface of radius –40 cm and a convex surface whose radius is +20 cm. What is the focal length of the lens.-40 cm+20 cmn = 1.5R1 = 20 cm, R2 = -40 cmf = 20.0 cmConverging (+) lens.
13Example 2: What must be the radius of the curved surface in a plano-convex lens in order that the focal length be 25 cm?R1=R2=?f = ?R1 = , R2 = 25 cmR2 = 0.5(25 cm)R2 = 12.5 cmConvex (+) surface.
14Terms for Image Construction The near focal point is the focus F on the same side of the lens as the incident light.The far focal point is the focus F on the opposite side to the incident light.Converging LensDiverging LensFFar focusFFar focusFNear focusFNear focus
15Image Construction:Ray 1: A ray parallel to lens axis passes through the far focus of a converging lens or appears to come from the near focus of a diverging lens.Converging LensDiverging LensRay 1Ray 1FF
16Image Construction:Ray 2: A ray passing through the near focal point of a converging lens or proceeding toward the far focal point of a diverging lens is refracted parallel to the lens axis.Converging LensDiverging LensFRay 1Ray 2Ray 2
17Image Construction:Ray 3: A ray passing through the center of any lens continues in a straight line. The refraction at the first surface is balanced by the refraction at the second surface.Converging LensDiverging LensFRay 1Ray 2Ray 3Ray 3
18Images Tracing PointsDraw an arrow to represent the location of an object, then draw any two of the rays from the tip of the arrow. The image is where lines cross.1. Is the image erect or inverted?2. Is the image real or virtual?Real images are always on the opposite side of the lens. Virtual images are on the same side.3. Is it enlarged, diminished, or same size?
19Object Outside 2F F 2F Real; inverted; diminished 1. The image is inverted, i.e., opposite to the object orientation.2. The image is real, i.e., formed by actual light on the opposite side of the lens.3. The image is diminished in size, i.e., smaller than the object.Image is located between F and 2F
20Object at 2F F 2F Real; inverted; same size 1. The image is inverted, i.e., opposite to the object orientation.2. The image is real, i.e., formed by actual light on the opposite side of lens.3. The image is the same size as the object.Image is located at 2F on other side
21Object Between 2F and F F 2F Real; inverted; enlarged 1. The image is inverted, i.e., opposite to the object orientation.2. The image is real; formed by actual light rays on opposite side3. The image is enlarged in size, i.e., larger than the object.Image is located beyond 2F
22Object at Focal Length F Parallel rays; no image formedWhen the object is located at the focal length, the rays of light are parallel. The lines never cross, and no image is formed.
23Object Inside F F 2F Virtual; erect; enlarged 1. The image is erect, i.e., same orientation as the object.2. The image is virtual, i.e., formed where light does NOT go.3. The image is enlarged in size, i.e., larger than the object.Image is located on near side of lens
24Review of Image Formations Parallel rays; no image formedF2FReal; inverted; enlargedF2FReal; inverted; same sizeObject Outside 2F RegionF2FReal; inverted; diminishedF2FVirtual; erect; enlarged
25Diverging Lens Imaging All images formed by diverging lenses are erect, virtual, and diminished. Images get larger as object approaches.Diverging LensFDiverging LensF
26Analytical Approach to Imaging F2Fpfqy-y’Lens Equation:Magnification:
27Same Sign Convention as For Mirrors 1. Object p and image q distances are positive for real and images negative for virtual images.2. Image height y’ and magnifi-cation M are positive for erect negative for inverted images3. The focal length f and the radius of curvature R is positive for converging lens or mirrors and negative for diverging lens or mirrors.
28Working With Reciprocals: The lens equation can easily be solved by using the reciprocal button (1/x) on most calculators:Possible sequence for finding f on linear calculators:P q1/x+=Finding f:Same with reverse notation calculators might be:Finding f:P q1/x+Enter
29Alternative Solutions It might be useful to solve the lens equation algebraically for each of the parameters:Be careful with substitution of signed numbers!
30Example 3. A magnifying glass consists of a converging lens of focal length 25 cm. A bug is 8 mm long and placed 15 cm from the lens. What are the nature, size, and location of image.Fp = 15 cm; f = 25 cmq = cmThe fact that q is negative means that the image is virtual (on same side as object).
31Example 3 Cont.) A magnifying glass consists of a converging lens of focal length 25 cm. A bug is 8 mm long and placed 15 cm from the lens. What are size of image.Fp = 15 cm; q = cmyy’Y’ = +20 mmThe fact that y’ is positive means that the image is erect. It is also larger than object.
32Example 4: What is the magnification of a diverging lens (f = -20 cm) the object is located 35 cm from the center of the lens?FFirst we find q then Mq = cmM =
33From last equation: q = -pM Example 5: Derive an expression for calculating the magnification of a lens when the object distance and focal length are given.From last equation: q = -pMSubstituting for q in second equation gives . . .Thus, . . .Use this expression to verify answer in Example 4.
34The principal focus is denoted by the red F. SummaryA Converging lens is one that refracts and converges parallel light to a real focus beyond the lens. It is thicker near the middle.FThe principal focus is denoted by the red F.FA diverging lens is one that refracts and diverges parallel light which appears to come from a virtual focus in front of the lens.
35Summary: Lensmaker’s Equation +-R1 and R2 are interchangeableR1, R2 = Radiin= index of glassf = focal lengthR1 and R2 are positive for convex outward surface and negative for concave surface.Focal length f is positive for converging and negative for diverging lenses.
36Summary of Math Approach qy-y’Lens Equation:Magnification:
37Summary of Sign Convention 1. Object p and image q distances are positive for real and images negative for virtual images.2. Image height y’ and magnifi-cation M are positive for erect negative for inverted images3. The focal length f and the radius of curvature R is positive for converging mirrors and negative for diverging mirrors.