We shall now study about the
reflections that take place in spherical surfaces.
A spherical surface is a part cut
from a hollow sphere. Spherical mirrors are generally constructed from glass.
One surface of the glass is silvered. The reflection takes place at the other
polished surface. If the reflection
takes place at the convex surface, it is called a convex mirror and if the reflection takes place at the concave
surface, it is called a concave mirror. These
are shown in Figure 6.6.
We shall now become familiar with
some of the terminologies pertaining to spherical mirrors.
Centre of curvature: The centre of the sphere
of which the mirror is a part is called the center
of curvature (C) of the mirror.
Radius of curvature: The radius of the sphere
of which the spherical mirror is a part is called the radius of curvature (R) of the mirror.
Pole: The middle point on the spherical surface of the mirror (or) the
geometrical center of the mirror is called pole
(P) of the mirror.
Principal axis: The line joining the pole and the
centre of curvature is called the principal
axis of the mirror. The light ray travelling along the principal axis
towards the mirror after reflection travels back along the same principal axis.
It is also called optical axis
Focus (or) Focal point: Light rays travelling parallel and
close to the principal axis when incident on a spherical mirror, converge at a
point for concave mirror or appear to diverge from a point for convex mirror on
the principal axis. This point is called the focus or focal point (F)
of the mirror.
Focal length: The distance
between the pole and the focus is called the focal length (f) of the mirror.
Focal plane: The plane through the focus and perpendicular to the principal
axis is called the focal plane of the
All the above mentioned terms are
shown in Figure 6.7 for both concave and convex mirrors.
The rays travelling very close to the principal axis and make
small angles with it are called paraxial
rays. The paraxial rays fall on the mirror very
close to the pole of the mirror. On
the other hand, the rays travelling far
away from the principal axis and fall on the mirror far away from the pole are
called as marginal rays. These two
rays behave differently (get focused at different points) as shown in Figure
6.8. In this chapter, we shall restrict our studies only to paraxial rays. As
the angles made by the paraxial rays are very small, this helps us to make some
approximations with the angles in ray optics.
Let C be the centre of curvature of the mirror. Consider a light ray
parallel to the principal axis is incident on the mirror at M and passes through the principal focus
F after reflection. The geometry of
reflection of the incident ray is shown in Figure 6.9(a). The line CM is the normal to the mirror at M. Let i be the
angle of incidence and the same will be the angle of reflection.
If MP is the perpendicular from M
on the principal axis, then from the geometry,
The angles ∠MCP = i and ∠MFP = 2i
From right angle triangles ∆MCP and ∆MFP,
PF is focal length f and PC is the radius of
Equation (6.4) is the relation
between f and R. The construction is shown for convex mirror in figure 6.9(b)
The image can be located by graphical
construction. To locate the point of an image, a minimum of two rays must meet
at that point. We can use at least any two of the following rays to locate the
image point as shown in Figure 6.10.
(i) A ray parallel to the principal
axis after reflection will pass through or appear to pass through the principal
focus. (Figure 6.10(a))
(ii) A ray passing through or appear
to pass through the principal focus, after reflection will travel parallel to
the principal axis. (Figure 6.10(b))
(iii) A ray passing through the centre
of curvature retraces its path after reflection as it is a case of normal
incidence. (Figure 6.10(c))
(iv) A ray falling on the pole will
get reflected as per law of reflection keeping principal axis as the normal.
While tracing the image, we would
normally come across the object distance u,
the image distance v, the object
height h, the image height ( h’ ), the focal length f and the radius of curvature R. A system of signs for these
quantities must be followed so that the relations connecting them are
consistent in all types of physical situations. We shall follow the Cartesian
sign convention which is now widely used as given below and also shown in
(i) The Incident light is taken from
left to right (i.e. object on the left of mirror).
(ii) All the distances are measured
from the pole of the mirror (pole is taken as origin).
(iii) The distances measured to the
right of pole along the principal axis are taken as positive.
(iv) The distances measured to the
left of pole along the principal axis are taken as negative.
(v) Heights measured in the upward
perpendicular direction to the principal axis are taken as positive.
(vi ) Heights
measured in the downward perpendicular direction to the principal axis, are
taken as negative.
The mirror equation establishes a
relation among object distance u,
image distance v and focal length f for a spherical mirror.
An object AB is considered on the principal axis of a concave mirror beyond
the center of curvature C. The image
formation is shown in the Figure 6.12. Let us consider three paraxial rays from
point B on the object. The first
paraxial ray BD travelling parallel
to principal axis is incident on the concave mirror at D, close to the pole P.
After reflection the ray passes through the focus F. The second paraxial ray BP
incident at the pole P is
reflected along PB´. The third
paraxial ray BC passing through
centre of curvature C, falls normally
on the mirror at E is reflected back
along the same path. The three reflected rays intersect at the point B’. A perpendicular drawn as A’B’
to the principal axis is the real, inverted image of the object AB.
As per law of reflection, the angle
of incidence ∠BPA is equal to the angle of reflection ∠ B’PA’.
The triangles ∆BPA and ∆ B’PA’ are similar. Thus, from the rule of
The other set of similar triangles
are, ∆DPF and ∆ B’A’F . (PD is almost a
straight vertical line)
As, the distances PD = AB
the above equation becomes,
From equations (6.5) and (6.6) we
PA′ / PA = A′F / PF
As, A’F= PA’ - PF , the above equation becomes,
We can apply the sign conventions
for the various distances in the above equation.
PA = −u,
PA′ = −v,
PF =− f
All the three distances are negative
as per sign convention, because they are measured to the left of the pole. Now,
the equation (6.7) becomes,
The above equation (6.8) is called mirror
equation. Although this equation is derived for a special situation
shown in Figure (6.12), it is also valid for all other situations with any
spherical mirror. This is because proper sign convention is followed for u, v
and f in equation (6.7).
The lateral or transverse
magnification is defined as the ratio of the height of the image to the height
of the object.The height of the object and image are measured perpendicular to
the principal axis.
magnification (m) = heightof theimage (h′)
/ height of theobject (h)
Identify the type of mirror used in each
of the application shown above.
Applying proper sign conventions for
A′B′ / AB = PA′ / PA
A′B′ = −h, AB = h, PA′ = −v, PA = −u
−h′ / h = −v / −u
On simplifying we get,
Using mirror equation, we can
further write the magnification as,
The students are advised to refresh
themselves with the image tracing for the concave and convex mirrors for
various predetermined positions of the object and the position of image, nature
of image etc. studied in 9th standard (Science, Unit 6. Light).
An object is placed at a distance of
20.0 cm from a concave mirror of focal length 15.0 cm.
(a) What distance from the mirror a
screen should be placed to get a sharp image?
(b) What is the nature of the image?
Given, f = –15 cm, u = –20 cm
As the image is formed at 60.0 cm to
the left of the concave mirror, the screen is to be placed at distance 60.0 cm
to the left of the concave mirror.
As the sign of magnification is
negative, the image is inverted.
As the magnitude of magnification is
3, the image is enlarged three times.
As the image is formed to the left
of the concave mirror, the image is real.
A thin rod of length f /3 is placed along the optical axis of
a concave mirror of focal length f such
that its image which is real and elongated just touches the rod. Calculate the
Image of one end coincides with the
Thus, the coinciding end must be at
center of curvature.
Hence, u’ = R = 2f