3.7. Magnification by plane mirrors
• A plane mirror generates a virtual image that occurs as far
into the mirror as the object is placed in front of the
mirror and is the same size as the object .
• The magnification M of an optical system such as a mirror
or lens is defined as.
M=-
Im age dis tan ce
Object dis tan ce
M=-
Im age height
Object height
†
• Thus a plane mirror can only produce an image with unitary
magnification
†
3.8. Sign Convention
• The - sign in the expression for the magnification M is a consequence of
the sign convention we will adopt.
• We will use the real is positive sign convention.
• If an object or image is real then it is a positive distance from the
reflecting surface. This means that the image and object are formed on
the same side of the mirror.
• If an object or image is virtual then it is a negative distance from the
mirror.
• A real(virtua)l object and a virtual(rea)l image are always on opposite
sides of the mirror.
P
Q
i
H
i
P
i
Q
3.8. Sign Convention
• If we look at the image formed by a plane mirror it is evident that the
image is erect i.e., pointing in the same vertical direction as the object.
• The image is however virtual and so is given a negative distance.
• Hence we would expect the magnification to be negative, but a negative
magnification corresponds to an inversion of the image.
• Thus we must include the - sign to ensure that the virtual image
produced by a plane mirror is erect.
P
Q
i
H
i
P
i
Q
M=-
Im age dis tan ce
Object dis tan ce
3.9. Reflections from Curved surfaces
• The image size can be varied by use of a curved surface.
There are several types of curved surface for example
– Spherical
– Parabolic
– Elliptical
• We will look at spherical surfaces. These come in two forms
– Convex - like the back of a spoon
– Concave - like the bowl of a spoon.
• The curvature is set by the radius of curvature, R
– A concave mirror has a positive radius of curvature
– A convex mirror has a negative radius of curvature
• Any line that is drawn to the mirror surface from the
centre of curvature, a radius, strikes the surface at 90˚
and makes a normal to the surface.
Convex mirror
R
Direction of light
incident on mirror
Centre of
curvature
Concave mirror
Direction of light
incident on mirror
R
3.10. Reflections at Curved Surfaces
• To understand how a ray interacts with a curved surface we
will make us of the paraxial approximation. This assumes
that any ray propagating to a surface makes a small angle
with the optic axis of the surface.
• The optic axis is a line that pass through the centre of the
surface at right angles to the surface.
Ray
optic axis
3.11. Reflections at a Concave Surface
• Any normal drawn at a concave surface is directed towards
the optic axis.
• Any ray that is incident on the mirror as shown below is
reflected towards the optic axis.
Normal
3.11. Reflections at a Convex Surface
• Any normal drawn at a convex surface is directed away from
the optic axis.
• Any ray that is incident on the mirror as shown below is
reflected away from the optic axis.
Normal
3.12. The focal point of a curved surface
• Consider a concave surface. A set of rays travel parallel to
the optic axis as shown below.
• The rays are all reflected such that they cross the optic
axis at the same point. This point is known as the focal point
and occurs a distance f, the focal length from the curved
surface.
• For a concave mirror the
focal length is positive.
i
• For a convex mirror the
focal length is negative.
focal point
i
h
2i
Centre of
curvature
R
f
3.13. The relationship between the focal
length and radius of curvature
focal point
i
i
h
2i
Centre of
curvature
R
f
• A ray travels at a height h parallel to the optic axis.
• The ray is incident on a concave mirror at angle i and is reflected to the
focal point at an angle i.
• From the figure it is evident that tani = h/R and tan2i = h/f.
• By the paraxial approximation tani = i and so i = h/R and 2i = h/f.
• Thus R = 2f.
3.14. Image location by ray tracing
Consider a real object that is placed in front of a concave
mirror. The image location can be found by considering the
path of 4 rays drawn from the top of the object. The 4 rays
follow the following paths:
Ray 1: Parallel to the optic axis. This ray is reflected through
the focal point of the mirror.
Ray 2: Passes through the focal point of the mirror. This ray is
reflected parallel to the optic axis.
Ray 3: Passes through the centre of curvature of the mirror.
This ray strikes the mirror at normal incidence and is reflected
back on itself.
Ray 4: Strikes the centre of the mirror and is reflected
symmetrically about the optic axis.
The image is formed where the rays cross.
3.14 Ray Tracing
O
R
F
I
Ray 1, Ray 2, Ray 3, Ray 4
3.15 Mathematical method for finding image location
p
O H
1
R
H2
F
I
q
Let the object of height H1 be placed a distance p from the
mirror of focal length f. The resulting image of height H2 is
formed a distance q from the mirror.
3.15 Mathematical method for finding image location
p
a
i
O H
1
c
g
e
R
H2
d
2i
f
F
I
From triangle abc we get tans = H1/p
From triangle deb we get tans = H2/q
From triangle dfe we get tan2i = H2/(q-f)
From triangle gfb we get tan2i = H1/f
So H2/q = H1/p and H2/(q-f) = H1/f
b
s
q
3.15 Mathematical method for finding image location
So H2/q = H1/p and H2/(q-f) = H1/f
So H2/ H1 = q/p
and H2/ H1 = (q-f) /f
So q/p =(q-f) /f
X both sides by fp
qf = pq - pf
/ both sides by pfq
1/p = 1/f - 1/q
Re-arrange to give
This is the equation that links the focal length,
object and image distances.
NOTE
1
f
=
1
p
+
1
q
IS NOT EQUAL TO f = p + q
3.16 Examples of image location with a concave mirror
We know that the focal length f, object distance p and
image distance q are related by
1
f
=
1
p
+
1
q
Where is the image formed if:
•A real object is located at infinity?
•Here 1/p = 0. Thus q = f
†
•The image is real and inverted
•A real object is located at the radius of curvature, R.
•Here p = R and R = 2f.
•Thus p = 2f and so q = 2f.
•Object and image located at the same plane.
•The image is real and inverted.
•A real object is located at f.
•Here p = f and so 1/q = 0.
•Image formed at infinity
•The image is real and inverted
•A real object is located at f/2
•Here 1/p = 2/f. Thus q = -f
•So image formed is virtual.
•The image is erect
When a real object is placed between the focal point and the
concave mirror the image is virtual.
3.16 Why is an image real when p > f?
O
F
R
I
When p > f the reflections from the concave mirror
converge to a point on the same side of the mirror as
the object. Hence the energy in the rays passes
through the point where the image is formed.
3.16 Why is an image virtual when p < f?
O
I
F
When p < f the reflections from the concave mirror diverge.
To find the point where the rays appear to cross they must be
projected back into the mirror. Hence the image is virtual as
the energy in the rays appears to come from the image.
3.17. Image formation by a convex mirror
• For a convex mirror the focal length is negative. This means
that the focal point of the mirror is located behind the
curved surface.
• As a result a real object placed anywhere in front of the
mirror will produce a virtual image.
• Why does this happen?
3.17. Image formation by a convex mirror
O
I
F
• Here the rays always diverge and so image is found by
tracing the rays back to the point where they appear to
cross.
3.18. Defects caused by reflections
• We have assumed that the paraxial approximation applies in that all rays
make small angles with respect to the optical axis. We also assume that
the rays strike close to th e centre of the mirror.
• In real systems the rays hit the whole mirror. We find that rays that
hit near the outside of mirror are focused at a different point when
compared with those that are close to the centre.
• This causes a blurring of the image, and is known as spherical
aberration.
Note that to show the effect the
reflection of the outer rays has been
greatly exaggerated. The normals for
the outer rays come from the same
point as those for the inner rays.
This problem is eliminated by use of a
parabolic mirror.