Last Word on Chapter 22 Geometric Optics Images in a Plane Mirror

This chapter is a study of geometric optics. We will use “ray diagrams” to determine where images are formed from objects placed in front of mirrors and lenses. Some objects and some images will be real; others will be “virtual.” By the end of this chapter, you’ll understand the difference! Mirrors & Lenses

Our goal is to understand and predict the way light reflects off of and refracts through a variety of materials. To this end, it is helpful to follow the paths of individual light rays...

Let’s start by studying what happens when light strikes a plane mirror (a 2D surface).. p the distance of the object from the mirror q the distance of the image from the mirror h = h’p = q h object height object image h’ image height

What do we notice about the image of an object in a plane mirror? 1) The image appears to be at a distance behind the plane mirror equal to the distance that the object is in front of the mirror. 2) The image is the same size as the object. 3) The image is erect (that is, if our object arrow is up, the image arrow is up as well). 4) The image is not real.

When we use this term, we’re referring to an image which forms at a location at which no light from the object is received. So in the case of our plane mirror, although the object appears to be located behind the mirror, if we look behind the mirror, we’ll see nothing there! Hence, the image is called a virtual image.

Let’s look at our plane mirror problem again, this time from a top view perspective... If I were to place a screen back here, no light would fall upon it! TT My eye Real objectVirtual image The image appears behind the mirror at a distance equal to the distance of the object from the mirror.

It is useful to define a quantity which tells us how the size of the image is related to the size of the object. We call it For a plane mirror, h’ = h, so M = 1.

While the most common mirrors we use are generally plane mirrors, not all mirrors are flat. Can you think of some mirrors you encounter in your everyday life that aren’t plane mirrors? Another large class of mirrors are known as

There are two types of spherical mirrors:

Remembering which is which can be confusing! Concave: “It’s like a cave on the side from which light approaches.” Convex: “It must be the other one!”

Whether we’re dealing with concave or convex mirrors, we define their center of curvature to be the point which would be the center of the sphere if the sphere was complete. Center of curvature In 2D, it’s the place where you’d stick the compass point to draw the circle. Center of curvature

We next define the principle axis to be the line passing through the center of curvature along a radius through the center of the mirror. Center of curvature Principle axis The definition is the same for the convex mirror (not shown). Let’s call the distance from the center of curvature to the mirror R.

Now we can investigate what happens to light when it strikes these spherical mirrors. Let’s start with the concave case. What happens when the purple beam hits the surface of the mirror???

The radius passes through the center of curvature and is perpendicular to the surface of the mirror The radius is the normal! ii rr

In fact, all parallel rays striking the surface of the concave mirror are reflected through the same point, known as the focus of the mirror. Concave Mirror

Center of curvature focus Objects at infinity will have images at the focus. We define the distance between the surface of the mirror and the focus along the principle axis as the focal length (f).

How do we locate images for objects closer than infinity? Concave Mirror Center of curvature focus For spherical mirrors, f and R are related by geometry: f = R / 2

Concave Mirror 1) A ray parallel to the principle axis of the mirror will be reflected through the focal point.

Concave Mirror 2) A ray passing through the center of curvature will be reflected back through the center of curvature.

3) A ray passing through the focal point will be reflected back parallel to the principle axis. Concave Mirror

Now that we know how to find where the image will be located, let’s try to figure out how tall the image will be (i.e. let’s determine the magnification of the concave mirror).

The image formed by the concave mirror is inverted, but real (that is, if I put my eye at the location of the image, I see the image). Determining the magnification requires some geometry... h h’

h Similar triangles p q ii rr

h h’ Similar triangles p q A second pair of similar triangles produces the mirror equation R (p - R) (R - q)

With a little algebra, you get...

p > 0 if the object is in front of the mirror (real) p < 0 if the object is behind the mirror (virtual) q > 0 if the image is in front of the mirror (real) q < 0 if the image is behind the mirror (virtual) f,R > 0 if the focus and center of curvature are in front of the mirror (concave mirror). f,R < 0 if the focus and center of curvature are are behind the mirror (convex mirror). M > 0 means the image is erect. M < 0 means the image is inverted.

Convex mirrors are slightly less intuitive than concave mirrors, but everything we’ve done with concave mirrors has a direct analog for convex mirrors.

ii rr Convex Mirrors

In fact, all parallel rays striking the surface of the convex mirror are reflected such that the the reflected rays all seem to originate from the same point behind the mirror known as the focus of the mirror. Convex Mirrors

As was the case for the concave mirror, objects at infinity will have images at the focus of convex mirrors. Convex Mirrors focus center of curvature

How do we locate images for objects closer than infinity? For spherical mirrors, f and R are related by geometry: f = R / 2 Convex Mirrors focus center of curvature

1) A ray parallel to the principle axis of the mirror will be reflected such that the reflected ray appears to originate at the focal point. Convex Mirrors

2) A ray headed directly toward the center of curvature of the mirror will be reflected back along the path from whence it came. Convex Mirrors

3) A ray headed directly toward the focal point will be reflected back parallel to the principle axis. Convex Mirrors

Let’s try to figure out where the image of an object in front of a convex mirror will be. The image will appear here Convex Mirrors

The image formed by the convex mirror is erect, but virtual (that is, if I put my eye at the location of the image, I see the image). We can determine the magnification using the same formula we derived for the concave case. hh’ Convex Mirrors

hh’ For the convex mirror p > 0 but q 0 The image will be always be virtual and erect in a convex mirror. Convex Mirrors “negative side” “positive side” “real side” “virtual side”

A 2.0 cm high object is placed 10 cm in front of a mirror. What type of mirror and what radius of curvature are required to create an upright image that is 4.0 cm high? q = -20cm

f = R / 2 R = 40cm Because f > 0, this must be a concave mirror. f = 20cm

Now that we’ve studies how images form through reflection, lets examine how refraction processes can also lead to the formation of images. What happens to light when it enters a piece of glass from air?

11 22

As was the case for spherical mirrors, a spherical piece of glass also has a center of curvature. n1n1 n2n2 object image It turns out (using geometry and Snell’s Law) that: p is the distance of the object from the surface q is the distance of the image from the surface

Unlike the mirror, however, for refracting surfaces, we use the following sign conventions: p > 0 if the object is in front of the surface (real) p < 0 if the object is in back of the surface (virtual) q > 0 if the image is in back of the surface (real) q < 0 if the image is in front of the surface (virtual) R > 0 if center of curvature is in back of the surface (a convex surface). R < 0 if the center of curvature is in front of the surface (concave mirror).

Whether you’re dealing with mirrors, refracting surfaces, or lenses, p and q are defined to be positive in the direction the light actually travels. Real objects will have p > 0. Real images will have q > 0. Virtual objects will have p < 0. Virtual images will have q < 0.

The magnification provided by the refracting surface is given by What happens if the refracting surface is flat? That is, where does the image of a plane refracting surface form? A plane has a radius of curvature equal to infinity, so...

So for a plane refracting surfaces, the image formed is always on the same side of the surface as the object. Air n=1.00 Water n=1.33 object image