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Chapter 34
Geometric Optics
Lecture by Dr. Hebin Li
PHY 2049, Dr. Hebin Li
Goals of Chapter 34
To see how plane and curved mirrors form images
To learn how lenses form images
To understand how a simple image system works
PHY 2049, Dr. Hebin Li
Reflection at a plane surface
Light rays from a point radiate in all directions
Light rays from an object point reflect from a plane mirror as though they came from the image point
PHY 2049, Dr. Hebin Li
Refraction at a plane surface
Light rays from an object at P
refract as though they came
from the image point P .
PHY 2049, Dr. Hebin Li
Image formation by a plane mirror
Object distance and image distance
Sign rule for s: when the object is on the
same side of the reflecting or refracting
surface as the incoming light, s is positive;
otherwise, it is negative
Sign rule for 𝑠′: when the image is on the
same side of the reflecting or refracting
surface as the outgoing light, 𝑠′ is positive;
otherwise, it is negative
PHY 2049, Dr. Hebin Li
Characteristics of the image from a plane mirror
The image is just as far
behind the mirror as the
object is in front of the
mirror.
𝑠 = −𝑠′
The lateral magnification is
m = y/y.
The image is virtual (no real
light rays reach the image),
erect, reversed, and the same
size as the object 1
𝑠+1
𝑠′= 0
PHY 2049, Dr. Hebin Li
A flat mirror is simple, but …
A flat mirror is the simplest imaging optics.
Yeah! So simple… but wait a minute…
https://www.youtube.com/watch?v=vBpxhfBlVLU
I love physics!
PHY 2049, Dr. Hebin Li
The image is reversed
The image formed by a plane mirror is reversed
back to front.
PHY 2049, Dr. Hebin Li
Image formed by two mirrors
The image formed by
one surface can be the
object for another
surface.
This property can lead
to multiple images.
Question: What if the angle is not 90o?
PHY 2049, Dr. Hebin Li
Example:
An extended object is placed in front of a flat mirror and an image
of the object is formed by the mirror. Which of the following
statements is incorrect?
(A)The image is erect.
(B) The image is a virtual image.
(C) The image is reversed left and right.
(D)The image has the same size as the object.
PHY 2049, Dr. Hebin Li
Spherical mirror with a point object
Sign for R: when the center of
curvature C is on the same side as the
outgoing light, the radius of curvature
is positive; otherwise, it is negative.
PHY 2049, Dr. Hebin Li
Focal point and focal length
The focal length is half of the mirror’s radius of curvature: f = R/2.
(Lateral magnification)
PHY 2049, Dr. Hebin Li
Image of an extended object
Figure below shows how to determine the position, orientation and height of the image.
PHY 2049, Dr. Hebin Li
Example:
A spherical, concave mirror has a radius of curvature of 30 cm. An object
of is 90 cm to the left of the vertex of the mirror. The length of the object is
5 cm. Use the graphical method (i.e. use principal rays) to find the location
and length of the image. Compare your results by computing the location
and length of the image using the object and image relationship equation.
PHY 2049, Dr. Hebin Li
Focal point and focal length of a convex mirror
For a convex mirror, R is negative.
PHY 2049, Dr. Hebin Li
Image formation by a convex mirror
The same equations
describe the object-
image relationship.
𝑓 = 𝑅/2
PHY 2049, Dr. Hebin Li
Graphical methods for mirrors
Four principal rays for concave and convex mirrors.
Four principal rays cross at the object and image.
PHY 2049, Dr. Hebin Li
Concave mirror with various object distances
PHY 2049, Dr. Hebin Li
Example:
Rear-view Mirror: A mirror on the passenger side of your car is convex
and has a radius of curvature with magnitude 18.0 cm.
(a)Another car is behind your car, 9.00 m from the mirror, and this car
is viewed in the mirror by your passenger. If this car is 1.5 m tall,
what is the height of the image?
(b)What is the image distance? The mirror has a warning attached that
objects viewed in it are closer than they appear. Why is this so?
PHY 2049, Dr. Hebin Li
Image of a point object at a spherical surface
PHY 2049, Dr. Hebin Li
Image of a point object at a spherical surface
PHY 2049, Dr. Hebin Li
Example:
A Spherical Fish Bowl: A small tropical fish is at the center of a
water-filled, spherical fish bowl 28.0 cm in diameter.
(a)Find the apparent position and magnification of the fish to an
observer outside the bowl. The effect of the thin walls of the
bowl may be ignored.
(b)A friend advised the owner of the bowl to keep it out of direct
sunlight to avoid blinding the fish, which might swim into the
focal point of the parallel rays from the sun. Is the focal point
actually within the pool?
PHY 2049, Dr. Hebin Li
Thin converging lens
When a beam of rays parallel to the axis passes through a lens and converge to a point, such as a lens is called a converging lens (or positive lens).
The points 𝐹1 and 𝐹2 are called focal points. The distance from a focal point to the center of lens is called focal length.
The focal length is positive for a converging (positive) lens.
Any lens that is thicker at its center than at its edges is a converging lens with positive f.
PHY 2049, Dr. Hebin Li
Image formed by a thin converging lens
A thin converging lens can form an image of an extended object.
The object-image relationship is described as1
𝑠+1
𝑠′=1
𝑓
The lateral magnification can be calculated as
𝑚 = −𝑠′
𝑠
PHY 2049, Dr. Hebin Li
Types of lenses
Shown below are various types of lenses, both converging and diverging.
Any lens that is thicker at its center than at its edges is a converging lens with
positive f ; and any lens that is thicker at its edges than at its center is a
diverging lens with negative f.
PHY 2049, Dr. Hebin Li
Thin diverging lens
A beam of parallel rays incident on a diverging lens diverges.
The focal length of a diverging lens is a negative quantity, so a diverging lens is also called a negative lens.
Any lens that is thicker at its edges than at its center is a diverging lens with negative f.
The object-image relationship is described by the same equation:
1
𝑠+
1
𝑠′=
1
𝑓
The lateral magnification can be calculated as
𝑚 = −𝑠′
𝑠
PHY 2049, Dr. Hebin Li
Graphical methods for lenses
We can determine the position and size of an image formed by a thin lens by using three principal rays.
A ray parallel to the axis
A ray through the center of the lens
A ray through (or proceeding toward) the first focal point
PHY 2049, Dr. Hebin Li
Example: image formation by a converging lens
Find the image position and magnification for an object at each of
the following distances from a converging lens with a focal length
of 20 cm:
(a)50 cm;
(b)40 cm;
(c)15 cm.
PHY 2049, Dr. Hebin Li
Example: image formation by a diverging lens
A thin diverging lens has a focal length of 20 cm. You want to use
this lens to form an erect, virtual image that is 1/3 the height of the
object.
(a) Where should the object be placed?
(b) Where will the image be?
PHY 2049, Dr. Hebin Li
The effect of object distance
The object distance can have a large effect on the image.
1
𝑠+1
𝑠′=1
𝑓𝑠′ =
𝑓𝑠
𝑠 − 𝑓
PHY 2049, Dr. Hebin Li
Example:
A lens forms an image of an object. The object is 16.0 cm from the
lens. The image is 12.0 cm from the lens on the same side as the object.
(a)What is the focal length of the lens? Is it converging or diverging?
(b)If the object is 8.50 mm tall, how tall is the image? Is it erect or
inverted?
(c)Draw a principal-ray diagram.
PHY 2049, Dr. Hebin Li
An image of an image
The image formed by the first lens can act as an object for the second lens.
PHY 2049, Dr. Hebin Li
Example:
A 1.20-cm-tall object is 50.0 cm to the left of a converging lens of local
length 40.0 cm. A second converging lens, this one having a focal length
of 60.0 cm, is located 300.0 cm to the right of the first lens along the
same optic axis.
(a)Find the location and height of the image (call it I1) form by the lens
with a focal length of 40.0 cm.
(b)I1 is now the object for the second lens. Find the location and height
of the image formed by the second lens.
PHY 2049, Dr. Hebin Li
Cameras
When a camera is in proper focus, the position of the electronic sensor
coincides with the position of the real image formed by the lens.
© 2016 Pearson Education Inc.
PHY 2049, Dr. Hebin Li
Camera lens basics
The focal length f of a camera lens is the
distance from the lens to the image when the
object is infinitely far away.
The effective area of the lens is controlled by
means of an adjustable lens aperture, or
diaphragm, a nearly circular hole with
diameter D.
Photographers commonly express the light-
gathering capability of a lens in terms of the
ratio f/D, called the
f-number of the lens:
© 2016 Pearson Education Inc.
PHY 2049, Dr. Hebin Li
The eye
The optical behavior of the eye is similar to that of a camera.
© 2016 Pearson Education Inc.
PHY 2049, Dr. Hebin Li
Defects of vision
A normal eye forms an image on
the retina of an object at infinity
when the eye is relaxed.
In the myopic (nearsighted) eye,
the eyeball is too long from front
to back in comparison with the
radius of curvature of the cornea,
and rays from an object at
infinity are focused in front of
the retina.
© 2016 Pearson Education Inc.
PHY 2049, Dr. Hebin Li
Nearsighted correction
The far point of a certain myopic eye is 50 cm in front of
the eye.
When a diverging lens of focal length f = −48 cm is worn
2 cm in front of the eye, it creates a virtual image at 50 cm that
permits the wearer to see clearly.
© 2016 Pearson Education Inc.
PHY 2049, Dr. Hebin Li
Farsighted correction
A converging lens can be used to create an image far enough away
from the hyperopic eye at a point where the wearer can see it clearly.
© 2016 Pearson Education Inc.
PHY 2049, Dr. Hebin Li
Angular size
The maximum angular size of an object viewed at a comfortable
distance is the angle it subtends at a distance
of 25 cm.
© 2016 Pearson Education Inc.
PHY 2049, Dr. Hebin Li
The magnifier
The angular magnification of a simple magnifier is:
© 2016 Pearson Education Inc.
PHY 2049, Dr. Hebin Li
The compound microscope
© 2016 Pearson Education Inc.
PHY 2049, Dr. Hebin Li
The astronomical telescope
The figure below shows the optical system of an astronomical
refracting telescope.
© 2016 Pearson Education Inc.
PHY 2049, Dr. Hebin Li
The reflecting telescope
The Gemini North telescope
uses an objective mirror
8 meters in diameter.
© 2016 Pearson Education Inc.