Curved Lenses (Ray Diagrams)
The two main types of curved lenses are concave and convex lenses. Light rays reflect and
converge when they strike a convex lens, while they reflect and diverge when they strike a
concave lens.
The following is a diagram of a convex lens, with some of its characteristics labelled:
Object
optical centre
focal point
(f)
principal axis (PA)
Draw two rays on the above diagram:
1. From the object to the mirror, parallel to the principal axis. This ray refracts down through
the focal point.
2. From the object through the optical centre of the lens. This ray keeps going in a straight line.
Now draw an arrow from the principal axis to where the lines intersect. This arrow represents
the image.
There are two types of images that can be formed: real and virtual.
A real image is an image that can be projected onto a screen. If an image formed is a real
image, you can tell from your ray diagram by either of two observations: if the image is
formed on the opposite side of the lens as the object, the image is real, and if the image is
formed from two solid lines intersecting, the image is real.
A virtual image is an image that cannot be projected onto a screen. A person viewing a
virtual image must physically look into the lens in order to see it. If an image formed is
virtual, you can tell from your ray diagram by either of two observations: if the image is
formed on the same side of the lens as the object, the image is virtual, and if the image is
formed from one or more dotted lines (backwards extensions of the solid lines)
intersecting, the image is virtual.
Each image formed has three characteristics:
1) If the image is on the same side of the principal axis as the object, we say it is upright but
if it is on the other side, we say it is inverted.
2) If the image is larger than the object, we say it is larger, but if it is smaller than the
object, we say it is smaller.
3) If the image is drawn at the point where solid lines intersect, we say the image is real. If
it is drawn where extensions of solid lines (dotted lines) intersect, we say the image is
virtual.
The image formed in the ray diagram above is smaller, inverted, and real.
The following is a diagram of a concave lens, with some of its characteristics labelled:
optical centre
object
focal point
(f)
principal
axis
(PA)
Draw two rays on the above diagram:
1. From the object to the lens, parallel to the principal axis. This ray refracts away from the
focal point. Draw a dotted extension of the reflected ray through the lens all the way to the
focal point.
2. From the object through the optical centre of the lens. This ray keeps going in a straight line.
Now draw an arrow from the principal axis to where the lines intersect. This arrow represents
your image.
The image formed in the ray diagram above is smaller, upright, and virtual.
Example
For the diagrams on the following two pages, do the following:
1. Draw the ray diagram.
2. Identify the type of lens and the characteristics of the image.
Type of lens: _______________
Type of lens: _______________
Type of lens: _______________
Type of lens: _______________
Type of lens: _______________
Type of lens: _______________
Type of lens: _______________
Type of lens: _______________
Type of lens: _______________
Type of lens: _______________
Curved Lenses (Mathematics)
The distance from the lens to the focal point is called the focal length. Its symbol is f. Its typical
units are cm.
The distance from the lens to the object is called the object distance. Its symbol is do. Its typical
units are cm.
The distance from the lens to the image is called the image distance. Its symbol is di. Its typical
units are cm.
The height of the object is called the object height. Its symbol is ho. Its typical units are cm.
The height of the image is called the image height. Its symbol is hi. Its typical units are cm.
The relative size of an image to an object is called magnification. Its symbol is M. It has no
units.
The following table shows the signs that are given to the above variables when using them to
solve problems:
f
dO
di
hO
hI
M
REAL
positive
positive
positive
VIRTUAL
negative
negative
negative
UPRIGHT
INVERTED
positive
positive
positive
negative
negative
negative
The equation that relates together focal length, object distance, and image distance of a curved
lens system is the following:
1
1
1


f
do di
The equation that relates together magnification, image height, object height, image distance, and
object distance of a curved lens system is the following:
M
hi
d
 i
ho
do
Example
A 6.0 cm tall candle is placed 6.0 cm in front of a lens. An upright 2.0 cm tall virtual image is
produced. What is the focal length of the lens and what type of lens is used?
When analyzing a problem with two lenses in the system, find the position of the image for the
first lens and use it as the object for the second lens. The object of the second lens will be real if
the image from the first lens is in front of the second lens. The object distance should therefore
be positive. The object of the second lens will be virtual if the image from the first lens is behind
the second lens. The object distance should therefore be negative.
The total magnification of a two-lens system can be found by multiplying the magnification of
both lenses together.
Example
Two convex lenses, having focal lengths of 2.0 cm and 5.0 cm are 14.0 cm apart. A 4.0 cm tall
object is placed 3.0 cm in front of the first lens.
a. Draw a scale ray diagram showing the final image formed by this combination of lenses.
b. Calculate the position and magnification of the final image formed by this combination of
lenses.