Law of Reflection (Smooth Surface): Recap: Reflection incident wave fronts reflected wave fronts incident ray reflected ray Plane waves reflecting in a mirror at an angle: The light rays (wavefronts) strike mirror at an incident angle θi and are reflected off the mirror at same speed at angle θr . Law of Reflection (Smooth Surface): The angle the reflected ray makes with the normal to the surface of reflection equals the angle of incidence: θi = θr (Note: This is because the light waves travel at same speed before and after reflection.) The reflected ray always lies in same plane as incident ray and the surface normal.

Images in Plane Mirrors By extending the reflected rays backwards from the mirror, they all intersect at a point behind the mirror. Your eye sees the reflected rays and you perceive an image that appears to lie at this point of intersection. (I.e. the light appears to come from this point.) image (virtual) object mirror reflected rays i o This situation holds for every point on the object…and your face seems to lie behind the mirror (a virtual image). By geometry, the distance of image behind mirror ‘i’ equals the distance of object in front of the mirror ‘o’. i = o

virtual image shows right hand Question: How big does a mirror need to be in order to see your whole body? mirror Answer: Mirror needs to be half your height with its upper edge lowered by half distance between your eye and top of your hat! Images formed in a plane mirror are virtual images (as the light does not pass through the image). Images are upright (right side up) and same size as object (no magnification) but are laterally (left ↔ right) inverted. left hand mirror virtual image shows right hand Lateral Inversion: Left hand becomes a life-size image of a right hand!

Refraction What happens to light waves when they enter a transparent material such as glass, H20, plastic etc? Individual “photons” collide with atoms and are absorbed and immediately re-emitted (i.e scattered). Typically there are billions upon billions of photons absorbed, re-emitted, and absorbed again and again as light beam makes its way through the medium. The net effect of this process is that the light waves effectively propagate at a speed lower than ‘c’ (even though individual photons do not exist at any speed other than ‘c’!). The difference in speed of light in different materials is called the index of refraction ‘n’: Typical values of n = 1.5 or 1.6 (glass) which means light speed is ~ two thirds of speed in air /vacuum. v c n = c = speed of light v = speed in medium

Normal Incidence: The reduced speed results in a decrease in wavelength of the light in the higher ‘n’ medium. Effect of reduced speed and wavelength on light ray propagation in medium: ray wavefronts incident wavefronts refracted wavefronts incident ray refracted ray Result: Wave fronts are bent (refracted) at the surface due to difference in propagation speed. Refracted ray no longer parallel to incident ray (except at normal incidence)

θ sin n = Amount of bending depends on: - angle of incidence - refractive index of medium A large difference in refractive index produces a large bend in the light ray. Snell’s Law: When light passes from one transparent medium to another, the rays will be bent towards the normal if the refractive index of medium is larger. n1. sin θ1 = n2 . sin θ2 or Note: For small angles: sin θ → θ (in radians). When light travels from glass to air, the bending is in the opposite direction (i.e. rays bend away from normal when going from high to low ‘n’ medium). Remember: Light rays are reversible! 1 2 θ sin n =

Summary: Refraction Amount of bending depends on: - angle of incidence - refractive index of medium Snell’s Law: When light passes from one transparent medium to another, the rays will be bent towards the normal if the refractive index of medium is larger. n1. sin θ1 = n2 . sin θ2 θ1 λ1 n2 n1 (n2 > n1) θ2 λ2 θ2 θ1 λ1 When light travels from glass to air, the bending is in the opposite direction (i.e. rays bend away from normal when going from high to low ‘n’ medium).

Example of refraction: Viewing objects under water… image i o θ1 θ2 n1 n2 eye Due to refraction the image of the fish will appear closer to the surface than it actually is. Relationship for apparent depth: E.g. If n2 (water) = 1.33 what is the apparent depth of a fish at 2 m depth? (provided n2 > n1) ÷ ø ö ç è æ = 2 1 n o i The fish is 0.5 m below its image (virtual image) and is safe!

Total Internal Reflection When light travels from a high to a lower refractive index medium (as with the fish looking at us) the ray is bent away from normal. Depending on ‘n’, a critical angle of incidence (θc) can be reached where the angle of refraction = 90º. critically reflected ray transmitted ray 90º refracted ray increasing angle When the angle of refraction equals 90º, the ray is no longer transmitted but is instead totally internally reflected at the interface. At angles equal or greater than critical value (~42º for glass, n=1.5) 100% of light is reflected creating a perfect mirror! Note: On transmission some light is always lost to reflection within the medium.

Dispersion and Prisms: A right angle prism cut with 45º angles makes a perfect mirror using total internal reflection. (As angle of incidence > 42º). Dispersion and Prisms: “White light” comprises a range of E-M waves from 400 to 700 nm wavelength (in air). prism reflector Right angle reflector Light is bent as it enters prism and again as it leaves prism (by Snell’s law). prism dispersion of light Refractive index ‘n’ depends on color (i.e. freq. of light). It is larger for blue, which is bent most - creates dispersion of light.

Example: Rainbow Formation From experience we all know that a rainbow is usually seen in the late afternoon when the Sun is at low elevation and there is rain nearby. lower elevation higher elevation red blue Secondary rainbow (inverted) Sunlight at low elevation enters raindrop and is refracted (blue refracted most). Some of the light that hits the back of the raindrop is reflected back towards front. This light is again refracted as it exits the raindrop. Net effect is light is dispersed into its spectrum and rainbow appears with red at top (larger angle from arc center) and blue /violet at bottom. Sun must be at your back to see a rainbow.

convex spherical surfaces Lenses Question: How do lenses form images? Lenses are made of a transparent material: glass, quartz, etc. Refraction (bending) of light rays as they pass through lens is responsible for the resultant size and nature of the image. Two types of lenses: positive and negative. Positive lenses (convex): refracted rays A positive lens causes the light rays to converge. Lens acts as a set of prisms. Prism angle larger at top of lens. convex spherical surfaces Light at top of lens is bent more than light passing through it near the middle of the lens. Parallel rays are brought to a single point ‘F’ called the “focal point”.

Negative Lens (concave): Distance from center of lens to focal point is called focal length ‘f’. Focal length is a property of an individual lens and depends on its curvature and index of refraction. There are two focal points, one on either side of the lens. Light is reversible: (a) Parallel light brought to a focus. (b) Point light at focal point creates a parallel beam of light (flash light). f F (a) f F (b) Negative Lens (concave): Acts like a set of upside down prisms bending light away from the optic axis. Diverging rays appear to come from a common focal point to the left of lens.

Image Formation Using Ray Tracing Simple ray tracing techniques can be used to tell us the position and size of the image formed by different lenses. Example: (Convex lens) Method: Draw a ray from top of object parallel to axis and then bend it so it passes through focal point. Draw a ray passing through the focal point on the object (near) side and then make it emerge from lens parallel to axis. Draw a ray from top of object passing straight through the center of the lens (undeviated). object real image formed upside down ho hi Forms: An inverted real image on the opposite side of lens.

Determination of Image Position and Size (i.e. Magnification) o = object distance from lens i = image distance from lens f = focal length of lens Then: And: Example: Object 5 cm in height located 40 cm to left of positive (convex) lens of focal length 25 cm. Image distance ‘i’: Magnification ‘m’: ho hi Note: ‘i’ is -ve if a virtual image ‘f’ is -ve if diverging lens Note: if ‘m’ +ve, image upright if ‘m’ –ve, image inverted 67 cm

Eye Sight Nearsighted: Farsighted: The eye contains two positive lenses (cornea) and accommodating lens. A real, inverted, minified image is formed. Nearsighted: Parallel light focuses in front of retina Negative lens introduces divergence to correct focus. Farsighted: Parallel light focuses behind retina Positive lens introduces convergence to correct focus.