25.4 Spherical Mirrors

Light rays near and parallel to the principal axis are reflected from the concave mirror and converge at the focal point.

1

The focal point of a spherical concave mirror is (approximately) halfway between the center of curvature of the mirror (C) and the middle of the mirror at B.

In Class Demo The focal length, f , is the distance between the focal point and the middle of the mirror.

Rf 21=

25.4 Spherical Mirrors

The effect is called spherical aberration.

2

Rays that lie close to the principal axis are called paraxial rays.

Rays that are far from the principal axis do not converge to a single point.

When paraxial light rays that are parallel to the principal axis strike a convex mirror, the rays appear to originate from the focal point behind the mirror.

Rf 21−=

Since the light rays do not really converge to this point, it is a virtual focus

The SIGN convention is to assign a negative value to virtual quantities

In Class Demo

25.3 The Formation of Images by a Plane Mirror

A ray of light from the top of the chess piece reflects from the mirror. To the eye, the ray seems to come from behind the mirror.

3

Because none of the light rays actually emanate from the image, it is called a virtual image.

The different rays from the top of the chess piece that enter the eye all appear to come from the same point behind the mirror: image.

(Virtual) Image formation by a plane mirror 1. The right hand of the person (object) becomes the

left hand of the image (inversion of handedness) 2. The image is upright. 3. The image is the same size as the object. 4. The image is as far behind the mirror as the

object is in front of it. oi dd −=oi hh +=

SƧ

Sign convention: negative image distance for virtual image

od idoh ih

25.3 The Formation of Images by a Plane Mirror

Conceptual Example Full-Length Versus Half-Length Mirrors What is the minimum mirror height necessary for her to see her full image?

4

Answer: the minimum height and location of the mirror required is the line segment BP

P

25.5 The Formation of Images by Spherical Mirrors Images from Concave Mirrors

[1] The ray that is initially parallel to the principal axis and passes through the focal point after reflection.

[2] The ray that initially passes through the focal point, then emerges parallel to the principal axis after reflection.

[3] The ray that travels along a line that passes through the center (i.e. along a radius) both before and after reflection

5

One can find images formed by a concave mirror by performing ray-tracing. The convention is to assume an upright object, and trace rays from the top of the object. In particular: Three special rays can be constructed with only a straight-edge:

Path of light rays are REVERSIBLE

Use a compass to draw the circle of the lens: here R=16 cm

Draw the Principle Axis between the Center and the middle of the mirror

Label the Center of Curvature (C) and the focus (F). Focal Length is f = R/2 = 8 cm

Draw object as an upward arrow

[1] The ray that is initially parallel to the principal axis and passes through the focal point after reflection.

[2] The ray that initially passes through the focal point, then emerges parallel to the principal axis after reflection.

[3] The ray that travels along a line that passes through the center (i.e. along a radius) both before and after reflection

The three rays converge at the tip of the inverted REAL image, which is located in front of the mirror

25.5 The Formation of Images by Spherical Mirrors

Image formation and the principle of reversibility

1

We start with an object between C and F

Tracing through three rays shown, they converge on an image in front of the mirror

This is a REAL image, because the rays arriving at the eye do in fact come from the point of the image

Real images are always inverted: hi < 0

If we place an object at the location, and of the size of the image in the previous case above, then: the new image appears where the old object was, and will be the same size

Note the two general cases we have already explored: 1. For an object located between C and F, you get an enlarged, real image located beyond C

2. For an object located beyond C, you get a shrunken, real image located between C and F

fdd io

111=+ Spherical Mirror

Equation Note the symmetry in image and object

Rays 1 and 2 are reversed

25.6 The Mirror Equation and Magnification Derivation: Mirror Equation

We start with the 4th special ray: [4] one that reflects symmetrically off the middle of the mirror (B)

The pink triangle (base d0 and height h0) is similar to the green triangle (base di, height hi) ...(1)

00 dd

hh ii =

−

B

length focal=f distanceobject =od distance image=idheightobject =oh height image=ih

B Here we use the ray that runs through the focus and reflects parallel to the principal axis.

The pink triangle (base d0-f , height h0) is similar to the green triangle (base f, height hi)

...(2)00 fd

fhhi

−=

−

fdf

dd

oo

i

−=

oi

o

oi

i

ddfd

ddfd=−→ 1fdfddd oioi =−→

io ddf111

=−→o

i

o

i

dd

hhm −==

fdd io

111=+ Mirror Equation Magnification

fdd io

111=+cm 0.82/ == Rfcm 0.16=R cm 0.20=od

oi dfd111

−=cm 0.20

1cm 0.8

1−= ( ) -1cm 050.0125.0 −= -1cm 075.0=

cm 3.131

=

cm 0.20=od cm 3.13=id cm 0.3=oho

i

o

i

dd

hhm −==

o

i

ddm −=

cm 0.20cm 3.13

−=

cm 0.2−=ih

67.0−=o

i

hhm =

cm 0.3cm 0.2−

= 67.0−=From Mirror Eqn. From graph

[4] ray that reflects symmetrically off the middle of the mirror (B)

B

Notice that ray [4] passes through the tip of the image

19

3. For an object located inside F (i..e between F and B) you get a enlarged, upright, virtual image located behind the mirror

Concave mirror: A Third Case

B

fdd io

111=+ Spherical Mirror Equation still applies

distance) (image idBut has a negative value for a virtual image

fdfd

oo

11 >→< 0111<

−=−=→

o

o

oi fdfd

dfd0 <→ id

o

i

ddm −=

−

−= oo

o dfd

fd)(

But here:

Magnification: oo df

ffd

f−

=−

−=

)(

fdo <<0 1 >−

=→odf

fmfdf <−<→ )(0 0

25.6 The Mirror Equation and Magnification

25.5 The Formation of Images by Spherical Mirrors

20

Images from Convex Mirrors

[1] The ray that is initially parallel to the principal axis and appears to originate from the focal point after reflection

[2] The ray that is initially headed toward the focal point, then emerges parallel to the principal axis after reflection.

[3] The ray that travels along a line that passes through the center of curvature (i.e. along a radius) both before and after reflection (back on itself)

Three special rays can be constructed with only a straight-edge: very similar to those for the concave mirror

For a convex mirror, with an object in front:

21

The virtual image is always diminished in size and upright.

Example A Virtual Image Formed by a Convex Mirror (similar to above)

A convex mirror is used to reflect light from an object placed 66 cm in front of the mirror. The focal length of the mirror is -46 cm. Find the location of the image and the magnification.

1cm 037.0cm 661

cm 461111 −−=−

−=−=

oi dfdcm 27 −=→ id

41.0 cm 66cm 27

=−

−=−=o

i

ddm

fdd io

111=+ still applies

00 0, >−=→<> fffdo 0 <+

−=→

fddf

do

oi

o

i

ddm −=

o

o

oi fdfd

dfd−

=−=111

o

o

dffd

−+

=

fdf

dfd

df

oo

o

o

+=

+−

−=

25.6 The Mirror Equation and Magnification

10 <<→ m

25.6 The Mirror Equation and Magnification

mirror. concave afor is +fmirror.convex afor is −f

mirror. theof front in isobject theif is +od

mirror. thebehind isobject theif is −od

image). (realmirror theof front in isobject theif is +id

image). (virtualmirror thebehind isobject theif is −id

imageupright anfor is +m

image. inverted anfor is −m

length focal=f

distanceobject =od

distance image=id

ionmagnificat=m

heightobject =oh

height image=ih

Definitions

(Textbook’s) Summary of Sign Conventions for Spherical Mirrors

fdd io

111=+ Spherical Mirror

Equation

22

25.6 The Mirror Equation and Magnification

mirror) theoffront (inimage andobject REALfocus,

afor are and , +io ddf

images and objects upwardfor are and +io hh

length focal=f

distanceobject =od

distance image=id

ionmagnificat=m

heightobject =oh

height image=ih

Definitions

Prof. Jui’s Conventions for Spherical Mirrors

fdd io

111=+ Spherical Mirror

Equation

mirror) theof (behind image andobject focus, VIRTUAL

afor are and , −io ddf

images and objects downwardfor are and −io hh

images realfor is −=o

i

hhm

images lfor virtua is +=o

i

hhm

2