Optics 5 - Refraction of Light by a Single Thin Lens PDF

7.4 Optics 7.4.6 Refraction of Light by a Single Thin Lens Thin Lenses There are many types (and shapes) of lenses, but we shall only be considering: 1. Convex (or converging) lens 2. Concave (or diverging) lens Converging lens Diverging lens Biconve...

7.4 Optics 7.4.6 Refraction of Light by a Single Thin Lens Thin Lenses There are many types (and shapes) of lenses, but we shall only be considering: 1. Convex (or converging) lens 2. Concave (or diverging) lens Converging lens Diverging lens Biconvex Biconcave Figure 22 Consider a parallel beam of light incident on a convex lens: All rays parallel to the principal axis pass through the lens, i.e. the rays are refracted at both ends of the lens, after which the rays converge (meet at a point), which is F, on the principal axis. See figure 23. L. BONELLO 25  7.4 Optics Focal Plane Optical Centre Principal Axis Principal Focus (real) Convex Lens Focal Length f Figure 23 The principal axis is that horizontal line passing through the centre of the lens. The optical centre is the centre of the lens. F = Principal focus (or focal point) where all the rays converge after passing parallel to the principal axis. In the case of a convex lens, the focal point is real as the rays pass through this point. f = Focal length of the lens. This is the distance between the optical centre and the principal focus. The focal plane is a plane that is perpendicular to the axis of the lens and passes through the principal focus. L. BONELLO 26  7.4 Optics Consider a parallel beam of light incident on a concave lens: All rays parallel to the principal axis pass through the lens, i.e. the rays are refracted at both ends of the lens, after which the rays diverge (spread out). If these rays are produced backwards, they finally meet at F on the left-hand side, on the principal axis. See figure 24. Focal Plane Optical Centre Principal Axis Principal Focus F (virtual) Concave lens Focal Length f Figure 24 Note: The green dotted lines in figure 24 are not real, i.e. These are only imaginary because we take the real rays on the right and produce (elongate) these rays behind (on the left) until they finally meet at a point called the principal focus (which is considered virtual). A real image is one that is formed on a screen. A virtual image is one that cannot be formed on a screen. L. BONELLO 27 7.4 Optics Rules When Drawing Ray Diagrams 1. A ray, passing parallel to the principal axis, refracts through the lens and then passes through the principal focus (figure 25). F Figure 25 2. A ray, passing through the optical centre, passes straight through (figure 26). Figure 26 3. A ray, passing through the principal focus, refracts through the lens and then travels parallel to the principal axis. This is the reverse of Number (1). See figure 27. F Figure 27 L. BONELLO 28 7.4 Optics Ray Diagrams When drawing ray diagrams, it is best to draw them to scale. E.g. 1 cm represents 10 cm. A well sharpened pencil is required for accurate drawings and measurements. 1. Convex Lens (converging lens) 1.Object placed beyond 2F I O Figure 28 The image is: 1. Diminished (smaller than the object) 2. Inverted (upside-down) 3. Real 4. Between F and 2F on the opposite side of the lens E.g.: Camera The eye L. BONELLO 29 7.4 Optics 2. Object placed at 2F I O Figure 29 The image is: 1. Same size as the object 2. Inverted 3. Real 4. At 2F on the opposite side of the lens E.g.: Photocopier producing same size images L. BONELLO 30 7.4 Optics 3. Object placed between F and 2F II O Figure 30 The image is: 1. Magnified (larger than the object) 2. Inverted 3. Real 4. Beyond 2F on the opposite side of the lens E.g.: Projector Microscope objective lens L. BONELLO 31 7.4 Optics 4. Object placed at F O I at infinity Figure 31 The image is: 1. At infinity E.g.: Produces a parallel beam of light as in a spotlight with lamp at O. L. BONELLO 32 7.4 Optics 5. Object placed between F and optical centre of lens I O Figure 32 The image is: 1. Magnified (larger than the object) 2. Upright (or erect) 3. Virtual (not real) 4. On the same side as the object (behind the object) E.g.: Magnifying glass Instrument eyepieces Spectacles to correct long-sightedness L. BONELLO 33 7.4 Optics 6. Object placed at infinity O at infinity I Figure 33 The image is: 1. Diminished (smaller than the object) 2. Inverted 3. Real 4. At F on the opposite side of the lens E.g.: Objective lens of telescope L. BONELLO 34 7.4 Optics Note: O I Beyond 2F Between F and 2F (smaller than O) At 2F At 2F (same size as O) Between F and 2F Beyond 2F (larger than O) At F At infinity Between F and optical centre Behind O (larger than O) At infinity At F (smaller than O) Table 1 L. BONELLO 35 7.4 Optics 2. Concave lens (diverging lens) I O I Figure 34 The image is: 1. Diminished (smaller than the object) 2. Upright (or erect) 3. Virtual (not real) 4. On the same side as the object but nearer the lens E.g.: Eyepiece in some instruments Spectacles to correct short-sightedness L. BONELLO 36 7.4 Optics Lens Equation There are two conventions: 1. Real-is-Positive Convention 2. The New Cartesian Convention Both conventions can be used. However, it is important to learn only one. Real-is-Positive Convention u f I 2F O F F 2F v Figure 35 𝑢 = object distance (m) 𝑣 = image distance (m) 𝑓 = focal length (m) 1 1 1 = + 𝑓 𝑣 𝑢 where: 𝑢 = + for a real object 𝑢 = - for a virtual object = L.𝑣 BONELLO 37 7.4 Optics 𝑣 = + for a real image 𝑣 = - for a virtual image 𝑓 = + for a convex (converging) lens 𝑓 = - for a concave (diverging) lens Magnification Magnification is the ratio of height of image to height of object or it is the ratio of image distance to object distance. ℎ𝑖 𝑚 = ℎ𝑜 where: 𝑚 = magnification (No units since it is a ratio) ℎ𝑖 = height of image (m) ℎ𝑜 = height of object (m) 𝑣 𝑚 = 𝑢 L. BONELLO 38 7.4 Optics where: 𝑚 = magnification (No units since it is a ratio) 𝑢 = object distance (m) 𝑣 = image distance (m) To find an unknown value, we can use: ℎ𝑖 𝑣 𝑚 = = ℎ𝑜 𝑢 E.g. 1. An object is placed 20 𝑐𝑚 from: a) a converging lens b) a diverging lens of focal length 15 𝑐𝑚. Calculate the image position and magnification in each case. a) Converging (convex) lens: 𝑢 = + 20 𝑐𝑚 𝑣= ? 𝑓 = + 15 𝑐𝑚 Using the Lens Equation and applying the Real-is-Positive Convention: L. BONELLO 39 7.4 Optics 1 1 1 = + 𝑓 𝑣 𝑢 1 1 1 = + + 15 𝑣 + 20 1 1 1 = − 𝑣 15 20 1 4 − 3 1 = = 𝑣 60 60 𝑣 60 = 1 1 𝑣 = + 60 𝑐𝑚 (Real image) Or 1 1 1 = + 𝑓 𝑣 𝑢 1 1 1 = + + 15 𝑣 + 20 1 1 1 = − 𝑣 15 20 1 = 0.0667 − 0.05 𝑣 1 = 0.0167 𝑣 𝑣 1 = 1 0.0167 ∴ 𝑣 = 59.8 𝑣 = + 60 𝑐𝑚 (Real image) 𝑣 𝑚 = 𝑢 60 𝑚 = 20 𝑚 =3 L. BONELLO 40 7.4 Optics b) Diverging (concave) lens: 𝑢 = + 20 𝑐𝑚 𝑣= ? 𝑓 = − 15 𝑐𝑚 Using the Lens Equation and applying the Real-is-Positive Convention: 1 1 1 = + 𝑓 𝑣 𝑢 1 1 1 = + − 15 𝑣 + 20 1 1 1 = − − 𝑣 15 20 1 −4 − 3 −7 = = 𝑣 60 60 𝑣 60 = − 1 7 𝑣 = − 8.6 𝑐𝑚 (Virtual image) Or 1 1 1 = + 𝑓 𝑣 𝑢 1 1 1 = + − 15 𝑣 + 20 1 1 1 = − − 𝑣 15 20 1 = − 0.0667 − 0.05 𝑣 1 = − 0.1167 𝑣 𝑣 1 = − 1 0.1167 ∴ 𝑣 = − 8.56 𝑣 = − 8.6 𝑐𝑚 (Virtual image) L. BONELLO 41 7.4 Optics 𝑣 𝑚 = 𝑢 − 8.6 𝑚 = 20 𝑚 = − 0.43 Magnification of the virtual image is, therefore, 0.43. Note: When stating the magnification, it is normal to ignore the negative sign. E.g. 2 A beam of light converges on a wall forming a bright spot. A convex lens of focal length 15 𝑐𝑚 is now inserted in the path of the beam 20 𝑐𝑚 in front of the wall. The spot is seen to disappear. Where must a screen be placed, for the spot to become visible again? Wall I Virtual O 𝑣 20 𝑐𝑚 Figure 36 Originally, the beam was forming a spot on the wall. When the lens was inserted, the beam converged to a point in front of the wall. L. BONELLO 42 7.4 Optics The original spot of light disappeared as soon as the lens was placed between the light and wall. The spot can be regarded as a virtual object for the lens. Converging (convex) lens: 𝑢 = − 20 𝑐𝑚 (virtual object) 𝑣= ? 𝑓 = + 15 𝑐𝑚 Using the Lens Equation and applying the Real-is-Positive Convention: 1 1 1 = + 𝑓 𝑣 𝑢 1 1 1 = + + 15 𝑣 − 20 1 1 1 = + 𝑣 15 20 1 4 + 3 7 = = 𝑣 60 60 𝑣 60 = 1 7 𝑣 = + 8.6 𝑐𝑚 (Real image) A screen would have to be placed 𝟖. 𝟔 𝐜𝐦 away from the lens or (20 − 8.6 = 𝟏𝟏. 𝟒 𝒄𝒎) in front of the wall in order to see an image of the spot. L. BONELLO 43 7.4 Optics Method to Measure an Approximate Value for the Focal Length of a Convex Lens Figure 37 Aim: To determine an approximate value for the focal length of a convex lens. Apparatus: Screen Screen Rays from an I at F infinite distance f Focal length Figure 38 L. BONELLO 44 7.4 Optics Method: The image formed by a lens of a distant window is focused sharply on a screen. The distance between the lens and the screen is equal to the focal length 𝑓 of the convex lens. This is because rays of light coming from the window (i.e. an illuminated object at infinity) are parallel and form an image at the principal focus 𝐹. Method to Measure the Focal Length of a Thin Convex Lens using the Magnification Method: Aim: To measure the focal length of a convex lens using the magnification method. Apparatus: X wire screen Lens Lamp Screen 𝒖 𝒗 Figure 39 L. BONELLO 45 7.4 Optics Method: The height of the object ℎ𝑜 was measured at the beginning of the experiment. The convex lens was then moved to and fro until a clear image was obtained on the screen. For different values of object distance 𝑢, the corresponding values of image distance 𝑣 and height of image ℎ𝑖 were measured and recorded. The magnification for each image obtained was calculated. Table of Results: 𝑢/𝑐𝑚 𝑣/𝑐𝑚 𝑣 ℎ𝑖 𝑚= or 𝑚 = 𝑢 ℎ𝑜 Table 2 Graph: A graph of 𝑚 on the y-axis against 𝑣/𝑐𝑚 on the x-axis is plotted. See graph 1. L. BONELLO 46 7.4 Optics Magnification 𝑚 ∆𝒚 𝑮𝒓𝒂𝒅𝒊𝒆𝒏𝒕 = ∆𝒙 Image distance 𝑣 / 𝑐𝑚 𝑐 = −1 Graph 1 Calculations: Using the Lens Equation and applying the Real-is-Positive Convention: 1 1 1 = + 𝑓 𝑣 𝑢 Multiplying the equation by 𝑣: 𝑣 𝑣 𝑣 = + 𝑓 𝑣 𝑢 𝑣 =1+𝑚 𝑓 𝑣 𝑚= −1 𝑓 1 𝑚= 𝑣−1 𝑓 𝑦=𝑚𝑥 + 𝑐 1 ∴ 𝐺𝑟𝑎𝑑𝑖𝑒𝑛𝑡 𝑚 = and 𝐼𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 𝑐 = −1 𝑓 1 ∴ 𝐹𝑜𝑐𝑎𝑙 𝑙𝑒𝑛𝑔𝑡ℎ 𝑓 = 𝐺𝑟𝑎𝑑𝑖𝑒𝑛𝑡 L. BONELLO 47