Lecture Notes for Chapter 34: Images Disclaimer: These notes are not meant to replace the textbook. Please report any inaccuracies to the professor. 1. Spherical Reflecting Surfaces • Bad News: This subject is very heavy in notation! • Good News: There aren’t any new principles. Everything follows from “angle of incidence equals angle of reflection” and we will provide simple rules to avoid even having to use that! Types of Mirrors Figure 1 shows the three types of mirrors we will consider. All of them are segments of spheres centered on a horizontal axis. There is an object (O) being reflected and a human (on the same side) observing the reflection. The terms “concave” and convex” are from the perspective of the object: • A concave mirror caves in on the object; whereas • A convex mirror flexes away from the object. Each mirror has a radius of curvature r (which is infinite for the plane mirror) 1 and a focal length f = 2 r. By convention, distances are measured, along the central axis, as positive from the mirror in the direction of the object and negative away from the object. Hence the radius of curvature and the focal length are positive for concave mirrors and negative for convex mirrors. The Various Lengths Six lengths are relevant for mirrors: • The radius of curvature r, which is positive for concave mirrors and negative for convex ones; • The focal length f, which is positive for concave mirrors and negative for convex ones; • The object distance p, which is always positive; 1 Flat Mirror Concave Mirror Convex Mirror O O O C F F C Figure 1: The three types of mirrors. In each case the human (the filthy, diseased animal in green) stands to the left of the mirror, as does the object (O) being reflected. The image can form either on the same side as the human (in which case it is called a REAL image) or on the side opposite from the human (in which case it is called a VIRTUAL image). The center of each spherical mirror is C and its focal point is F. The radius of curvature r and focus f of the flat mirror are infinite; the concave mirror has r =2f > 0; and the convex mirror has r =2f < 0. • The object height h, which is always positive; • The image distance i, which is positive for REAL images (on the same side as the human) and negative for VIRTUAL images (on the opposite side as the human); and • The image height h′, which is always positive, even if the image is inverted. The Focal Point The key property of the focal point is that any light ray which approaches the mirror traveling parallel to the central axis is reflected back along a line passing through the focal point. Note that the time reversal invariance of electrodynamics therefore implies that any light ray passing through the focal point is reflected back along a line parallel to the central axis. Types of Images We distinguish images depending upon whether they form on the same 2 Concave Mirror Convex Mirror C F F C Figure 2: Incident light rays which are parallel to the central axis (red) reflect back along a line through the focal point. Light rays which are incident along a line through the focal point (blue) reflect back parallel to the central axis. For the case of the convex mirror note that neither of the rays actually reaches the focal point, but they nevertheless move along lines which pass through the focal point. side of the human or the opposite side: • REAL images form on the same side of the mirror as the human. They have i > 0 and they are INVERTED with respect to the object. The magnification for a REAL image is m = −h′/h. • VIRTUAL images seem (to the stupid human) as if they emerge from the opposite side of the mirror. They have i < 0 and they are NOT INVERTED with respect to the object. The magnification for a VIR- TUAL image is m =+h′/h. Figure 3 shows examples of the two types of images. How to find the image There is a geometrical method and and algebraic method. The geomet- rical method is to follow any two incident light rays which pass through the top of the object. Three rays which are easy to follow are: • One which is incident parallel to the central axis and hence reflects back through a line passing through the focal point; • One which is incident along a line which passes through the focal point, and hence is reflected back parallel to the central axis; and 3 Concave Mirror Convex Mirror O O I h h h′ C F F C h′ I Figure 3: The concave mirror on the left forms a REAL image which is INVERTED when the object lies outside the focal point. The magnifica- tion in this case is m = −h′/h. Note that the image distance i is positive. The convex mirror on the right forms a VIRTUAL image which is NOT IN- VERTED where ever the object is located. The magnification in this case is m =+h′/h. Note that the image distance i is negative. Mirror Object Image Image Image sgn(f) sgn(m) Type Location Location Type Orientation sgn(r) Plane Anywhere Opposite Virtual Not Inverted NA + Concave Inside F Opposite Virtual Not Inverted + + Concave Outside F Same Real Inverted + − Convex Anywhere Opposite Virtual Not Inverted − + Table 1: Table 34-1 with the entries filled in. • One which reflects off the mirror at the central axis, and hence reflects back symmetrically. Figure 4 illustrates the technique. Using these techniques we can fill out the entries in the text’s Table 34-1. which you should include in your formula sheet. Analyzing Mirrors the Easy Way There is unfortunately no way to avoid the complicated notation. How- ever, two simple formulae allow us to avoid the complicated graphical con- structions. The first of these relations allows us to determine the image 4 Concave Mirror Convex Mirror O O I C F F C I Figure 4: The geometrical technique for finding the image is to follow any two incident rays which pass through the top of the object. In each case the red ray is incident parallel to the central axis, so it reflects back along a line that passes through the focal point. The blue ray is incident along a line which passes through the focal point, so it reflects back parallel to the central axis. And the green ray reflects from the mirror at the central axis, so the reflected ray is symmetric about the central axis. distance i in terms of the object distance p and the focal length f: 1 1 1 + = . (1) p i f Note that this relation is valid no matter what are the signs of f and i. (The sign of p is always positive.) The second relation gives the magnification in terms of the object and image distances: i m = − . (2) p Note that this formula is valid no matter what the sign of i. For some examples, let’s work through the values used to construct Figure 3. The concave mirror has a focal length of f = +25 length units. The object O is at a distance of p = +40 length units. We can infer the location of the image from equation (1), 1 1 1 1 1 1 3 200 + = =⇒ = − = =⇒ i = ≃ 66.3 . (3) 40 i 25 i 25 40 200 3 Because i > 0 the image stands about 66.3 length units to the left of the mirror, so it is REAL and INVERTED. We can infer the magnification from 5 equation (2), 200 5 m = − 3 = − . (4) 40 3 So if the height of the object is h = +20 length units then the height of the ′ 100 ′ image is h =+ 3 ≃ 33.3 length units. Note that h is always positive, even if the image is inverted. The convex mirror in Figure 3 has a focal length of f = −25 length units, and the object is at p = +50 length units. We again employ equation (1) to find the location of the image, 1 1 1 1 1 1 3 50 + = − =⇒ = − − = − =⇒ i = − ≃ −16.7 . 50 i 25 i 25 50 50 3 (5) Because i < 0 the image forms on the other side of the mirror. Hence it is VIRTUAL and NOT INVERTED. The magnification is, − 50 1 m = − 3 =+ . (6) 50 3 ′ 20 So if the object is h = 20 length units high then the image is h = 3 ≃ 6.7 length units high. 2. Spherical Refracting Surfaces • Bad News: This subject is also very heavy in notation, and some of it disagrees with the notation for reflecting surfaces. • Good News: There aren’t any new principles. Everything follows from Snell’s Law and we will provide simple rules to avoid even having to use that. Further, the discordant notation was arranged to make the simple rules carry over from reflecting surfaces. Notation for Spherical Refracting Surfaces We will consider refraction between two media, one with index of re- fraction n1, which contains the object being viewed, and the other medium with index of refraction n2, which contains the human who observes the ob- ject. The two regions are joined along a spherical boundary whose radius of curvature is r. Most of the notation is the same as for mirrors, in particular: • Concave surfaces cave in on the object, whereas convex surfaces flex away from the object; 6 Concave Surface Convex Surface O O I I C C 3 3 n1 = 2 n2 =1 n1 = 2 n2 =1 Figure 5: The figure shows a typical concave surface (with r = −50) and a convex surface (with r = +50).