Chapter 35 MirrorsLenses Images

We will use geometrical optics: light propagates in straight lines until its direction is changed by reflection or refraction. When we see an object directly, light comes to us straight from the object. When we use mirrors and lenses, we see light that seems to come straight from an object but actually doesn’t. Thus we see an image (of the object), which may have a different position, size, or shape than the actual object. Image Formation

Images Formed by Plane Mirrors (the ray reaching your eye doesn’t really come from the image) object image virtual image But…. the brain thinks the ray came from the image. When we use mirrors and lenses, we see light that seems to come straight from the object but actually doesn’t. Thus we see an image, which may have a different position, size, or shape than the actual object.

Images Formed by Plane Mirrors You can locate each point on the image with two rays: 1. A ray normal to the mirror Image is reversed front to back object image

Images Formed by Plane Mirrors You can locate each point on the image with two rays: 1. A ray normal to the mirror 2. The ray that reaches the observer’s eye Image is reversed front to back object image

Images Formed by Plane Mirrors object image Image is reversed (front to back) You can locate each point on the image with two rays.

Images Formed by Plane Mirrors object image Image is reversed (front to back) You can locate each point on the image with two rays.

Images Formed by Plane Mirrors object image The distance from the image to the mirror equals the distance from the object to the mirror: p = i p i Also, the height of the image equals the height of the object

Parabolic Mirrors Shape the mirror into a parabola of rotation (In one plane it has cross section given by y = x 2 ). All light going into such a mirror, parallel to the para- bola’s axis of rotation, is reflected to pass through a common point - the focus. What about the reverse?

These present the concept of a focal point - the point to which the optic brings a set of parallel rays together. Parallel rays come from objects that are very far away (and, after reflection in the parabolic mirror, converge at the focal point or focus). Parabolas are hard to make. It’s much easier to make spherical optics, so that’s what we’ll examine next. Parabolic Mirrors

To analyze how a spherical mirror works we draw some special rays, apply the law of reflection where they strike the spherical surface, and find out where they intersect. Spherical Mirrors c f A ray parallel to the mirror axis reflects through the focal point f A ray passing through the focus reflects parallel to the axis A ray that strikes the center of the mirror reflects symmetrically A ray passing through the center of curvature c, returns on itself

When the object is beyond c, the image is: real (on the same side as the object), reduced, and inverted. Spherical Mirrors c f

c f Object between c and f. Spherical Mirrors - Concave Image is real, inverted, magnified

Object between f and the mirror. Spherical Mirrors c f Image is virtual, upright, magnified

c The Mirror Equation f i p Here f = R / 2

c f Magnification h h’ The magnification is given by the ratio M = h’ / h = - i/p

Curved Mirrors mirror equation focal length magnification Sign conventions: Distance in front of the mirror  positive Distance behind the mirror  negative Height above center line  positive Height below center line  negative

Positive and Negative Mirrors You can fill a positive mirror with water. You can’t fill a negative mirror. positive mirror is concave negative mirror is convex

Image With a Convex Mirror c f Here the image is virtual (apparently positioned behind the mirror), upright, and reduced. Can still use the mirror equations (with negative distances for f, c=R, and i).

A simple lens is an optical device which takes parallel light rays and focuses them to a point. This point is called the focus or focal point The Simple Lens Snell’s Law applied at each surface will show where the light comes to a focus.

Each point in the image can be located using two rays. Image Formation in a Lens Ray tracing: 1. A ray which leaves the object parallel to the axis, is refracted to pass through the focal point. 2. A ray which passes through the lens’s center is undeflected. 3. A ray passing through the focal point (on the object side) is refracted to end up parallel to the axis. f f

Some Simple Ray Traces 2f f f f f Object between 2f and f Image is inverted, real enlarged. Object between f and lens Image is upright, virtual, and enlarged.

Some Simple Ray Traces (diverging lens) 2f f f f f Object beyond 2f. Image is upright, virtual, reduced. Object between f and lens. Image is upright, virtual, reduced. A ray parallel to the axis diverges such that its extension passes through the focal point.

Sign Conventions 1. Converging or convex lens focal length is positive image distance is positive when on the other side of the lens (with respect to object) height upright is positive, inverted is negative 2. Diverging or concave lens focal length is negative image distance is always virtual and negative (on the same side of the lens as the object) height upright is positive, inverted is negative

The Thin Lens Equation 2f f f f p i h’ h M = h’/h = -i/p Magnification 1/p+ 1/i= 1/f The thin lens equation

About the Thin Lens Formula When p = f, i = infinity When p = 2f, i = 2f and the magnification is 1. When f > p > 0, i is negative – This means that the image is virtual, and so it is on the same side of the lens as the object. If f < 0, i is always negative – A negative lens can not produce a real image. It always produces a virtual image. 1/p + 1/i = 1/f

The lens formula gives the image distance as a function of the object distance and the focal distance [1/f -1/p = 1/i] The lensmaker’s formula gives the focal distance f as a function of R 1 and R 2, the radii of curvature of the two surfaces of the lens. The Lensmaker’s Formula Lensmaker’s Formula

The Eye: A Simple Imager A simple lens focuses the light onto the retina -- the photosensor The retina sends signals to the brain about which sensor is illuminated, what color the light is, and how much of it there is. The brain interprets. Your eye’s lens is a simple refracting lens which you can deform to change focus. The retina senses the light.

The Eye: Near & Farsightedness Far-sighted: Near-sighted: Eye too short: correct with a converging lens Eye too long: correct with a diverging lens

Image Forming Instruments Cameras are much like the eye except for having a shutter and better lenses. Binoculars and telescopes create magnified images of distant objects. Microscopes create magnified images of very small objects.

The UCF Robinson Observatory The telescope here is a 66 cm (26 in) dia. Schmidt-Cassegrain reflecting telescope. Telescopes are described by their diameter since that describes its ability to collect light from distant stars and glaxies; and that’s more important than magnification. Florida is not ideal for most observational astronomy. Why?

Image-Forming Instruments: the Telescope When we look at distant objects, we judge their sizes from their angular size. In the case of stars, what we observe is their angular separation.  A telescope magnifies the angular size (or separation) of objects.

The Telescope fofo  “Objective” fofo The “objective” lens first creates an image, in the focal plane, of a point at infinity.

The Telescope fofo  “Objective” fofo The “objective” lens first creates an image, in the focal plane, of a point at infinity. The “eypiece” is placed f o + f e from the objective, so that it produces parallel rays into the eye, at angle . fefe fefe “Eyepiece” EYE 

The Telescope fofo  “Objective” fofo fefe fefe “Eyepiece” EYE  The eye sees an image at infinity, but at an angle of  instead of . The magnification is therefore: M =  = f o / f e. h tan  / tan  = (h/f e )/(h/f o )