Last Word on Chapter 22Geometric Optics

Images in a Plane Mirror

This chapter is a study of geometric optics.We will use “ray diagrams” to determine whereimages are formed from objects placed infront of mirrors and lenses. Some objectsand some images will be real; others will be“virtual.” By the end of this chapter, you’llunderstand the difference!

Mirrors & Lenses

Our goal is to understand and predict theway light reflects off of and refracts througha variety of materials.

To this end, it is helpful to follow the pathsof individual light rays...

Let’s start by studying what happens whenlight strikes a plane mirror (a 2D surface)..

pthe distanceof the object

from the mirror

qthe distanceof the image

from the mirror

h = h’ p = q

hobjectheight

object image

h’imageheight

What do we notice about the image of anobject in a plane mirror?

1) The image appears to be at a distancebehind the plane mirror equal to the distancethat 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 objectarrow is up, the image arrow is up as well).

4) The image is not real.

When we use this term, we’re referring toan image which forms at a location at whichno light from the object is received.

So in the case of our plane mirror, althoughthe object appears to be located behind themirror, if we look behind the mirror, we’llsee 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 placea screen backhere, no lightwould fall upon it!

T T

My eye

Real object Virtual image

The image appears behindthe mirror at a distanceequal to the distance ofthe object from the mirror.

It is useful to define a quantity which tellsus how the size of the image is related tothe size of the object. We call it

Mh

h

'

For a plane mirror, h’ = h, so M = 1.

While the most common mirrors we useare generally plane mirrors, not all mirrorsare flat.

Can you think of some mirrors you encounterin 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 caveon the side fromwhich lightapproaches.”

Convex:

“It must bethe other one!”

Whether we’re dealing with concave or convexmirrors, we define their center of curvature tobe the point which would be the center of thesphere if the sphere was complete.

Center ofcurvature

In 2D, it’s the place where you’d stick thecompass point to draw the circle.

Center ofcurvature

We next define the principle axis to be the linepassing through the center of curvature alonga radius through the center of the mirror.

Center ofcurvature

Principleaxis

The definition isthe same for the convexmirror (not shown).

Let’s call the distance from the center ofcurvature to the mirror R.

Now we can investigate what happens tolight when it strikes these spherical mirrors.

Let’s start with the concave case.

What happens when the purple beam hits thesurface of the mirror???

i r

The radius passes throughthe center of curvatureand is perpendicular

to the surface of the mirrorThe radius is the normal!

i

r

In fact, all parallel rays striking the surfaceof the concave mirror are reflected throughthe same point, known as the focus ofthe mirror.

Concave Mirror

Concave Mirror

Center ofcurvature

focus

Objects at infinity will have images at the focus.

We define the distance between the surfaceof the mirror and the focus along the principleaxis as the focal length (f).

How do we locate images for objects closerthan infinity?

Concave Mirror

Center ofcurvature

focus

For spherical mirrors, f and R are related bygeometry:

f = R / 2

Concave Mirror

1) A ray parallel to the principle axis of the mirrorwill be reflected through the focal point.

Concave Mirror

2) A ray passing through the center of curvaturewill be reflected back through the center ofcurvature.

3) A ray passing through the focal point willbe reflected back parallel to the principle axis.

Concave Mirror

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

The image formed by the concave mirroris inverted, but real (that is, if I put my eyeat the location of the image, I see the image).

Determining the magnification requires somegeometry...

h

h’

h

h’

Similar triangles

p qi

r

i r

Mh

h

q

p

'

h

h’

Similar triangles

p

q

A second pair of similar triangles producesthe mirror equation

R

(p - R)

(R - q)

h

p R

h

R q

'

With a littlealgebra, you get...

1 1 2 1

p q R f

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 arein front of the mirror (concave mirror).

f,R < 0 if the focus and center of curvature areare 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 directanalog for convex mirrors.

i r

i

r

Convex Mirrors

In fact, all parallel rays striking the surfaceof the convex mirror are reflected such thatthe the reflected rays all seem to originatefrom the same point behind the mirrorknown as the focus of the mirror.

Convex Mirrors

As was the case for the concave mirror,objects at infinity will have images at thefocus of convex mirrors.

Convex Mirrors

focus

center of curvature

How do we locate images for objects closerthan infinity?

For spherical mirrors, f and R are related bygeometry:

f = R / 2

Convex Mirrors

focus

center of curvature

1) A ray parallel to the principle axis of the mirrorwill be reflected such that the reflected rayappears to originate at the focal point.

Convex Mirrors

2) A ray headed directly toward the center ofcurvature of the mirror will be reflected backalong the path from whence it came.

Convex Mirrors

3) A ray headed directly toward the focal pointwill be reflected back parallel to the principle axis.

Convex Mirrors

Let’s try to figure out where the image of anobject in front of a convex mirror will be.

The image will appear here

Convex Mirrors

The image formed by the convex mirroris erect, but virtual (that is, if I put my eyeat the location of the image, I see the image).

We can determine the magnification using the same formula we derived for the concavecase.

h h’

Convex Mirrors

Mh

h

q

p

'

h h’

For the convex mirrorp > 0 but q < 0, so M > 0

The image will be alwaysbe virtual and erect in aconvex mirror.

Convex Mirrors

“negative

side”“positive

side”

“real

side”“virtu

al

side”

A 2.0 cm high object is placed10 cm in front of a mirror.What type of mirror and whatradius of curvature arerequired to create an uprightimage that is 4.0 cm high?

Mh

h

q

p

'M

cm

cm

q

cm

4

2 10

q = -20cm

f = R / 2

R = 40cm

Because f > 0, thismust be a concavemirror.

1 1 1

p q f

1

10

1

20

1

cm cm f

( )

1 1

20f

f = 20cm

Now that we’ve studies how images formthrough reflection, lets examine howrefraction processes can also lead to theformation of images.

What happensto light when itenters a pieceof glass fromair?

n n1 1 2 2sin sin

1

2

As was the case for spherical mirrors,a spherical piece of glass also has acenter of curvature.

n1

n2

object image

It turns out(using geometryand Snell’s Law)that:

n

p

n

q

n n

R1 2 2 1

p is the distance of the object from the surfaceq is the distance of the image from the surface

Unlike the mirror, however, for refractingsurfaces, 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 qare defined to be positive in the directionthe 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 therefracting surfaceis given by

Mh

h

n q

n p

' 1

2

What happens if the refracting surface is flat?That is, where does the image of a planerefracting surface form?

A plane has a radius of curvature equal toinfinity, so...

n

p

n

q

n n1 2 2 1 0

n

p

n

q1 2

So for a plane refracting surfaces, theimage formed is always on the sameside of the surface as the object.

Airn=1.00

Watern=1.33

object image