MIRROR AND LENSES
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This explores how images are formed by reflection and refraction. First, we will
look at the image formation by reflection due to flat and spherical surfaces. This
section is concerned with three different types of mirrors: flat mirror; concave mirror;
and convex mirror. Then, we will consider the image formation by refraction due to
spherical surface, flat surface and then examine them under different types of thin lenses such as: converging and diverging lens.
Images Formed by Reflection:
Plane Mirrors Let us now apply what we have discussed about Reflection in Unit 1 to the important topic of Mirrors. First, we begin our discussion by examining the simplest form of mirror, the plane mirror (or the flat mirror). Consider an object O being placed at a distance p in front of a plane mirror.
The distance p is called the object distance. The observer will see the image of the
object O at point I at a distance q from the mirror. This distance q is called the image
distance. Image can be classified as real or virtual. A real image is the one in which
light actually passes through the image point whereas a virtual image (sometimes
called imaginary image) is the one in which the light does not pass through the image
point but appears to have originated from the image point.
We will examine some of the properties of images formed by plane mirror through
simple geometric construction. We begin by considering an object O being placed at
some distance p from the plane mirror as shown. To locate the image position formed,
we need to draw at least two light rays. A light ray is the imaginary line drawn to
visualize the path that the light will follow.
The first light ray, Ray 1, follows the horizontal path PQ and reflects back on itself (law of reflection). The second light ray, Ray 2, follows path PR and get reflected along path RS. An observer standing on the left of the mirror will trace the two reflected rays back to the point P´ from which they appear to have originated. If this process is continued for the point other than P on the object, the result will be the formation of a virtual image on the right side of the mirror. Geometric shows that the image formed by the plane mirror is at the same distance from the mirror as that of the object i.e. p = q. It also shows that the image is of the same height as that of the object i.e. h = h´. In optics, the ratio of the image height and the object height is termed as lateral magnification, M.
Mathematically, this is expressed as:
SPHERICAL MIRRORS
Consider the formation of an image due to spherical mirrors. Spherical
mirrors have a shape of a segment of a sphere. A spherical mirror has the shape of a
section of a sphere. The mirror focuses incoming parallel rays to a point.
If the segment has inner surface as the reflecting surface then such mirror is called a
concave mirror. On the other hand, if the reflecting surface is the outer surface then
such a mirror is called a convex mirror. [Note that for all the spherical mirrors we
will be drawing henceforth, the reflecting surface will be indicated by black curve]
A spherical mirror has a center of curvature and is at C. The distance between the
center of curvature and the mirror is called the radius of curvature R. Point V is at the
center of the mirror called the vertex of the mirror and the line CV is called the
principal axis or optic axis. The concave mirror is sometimes referred to as a
converging mirror and that the convex mirror is referred to as a diverging mirror. As
the names imply, light rays being reflected from a concave or convex surface tends to
converge or diverge from the mirror as shown respectively.
Ideally, all the incident rays that are moving parallel to the principle axis must
converge (in the case of concave mirror) or appear to be converging (in the case of a
convex mirror) to a single point called the focal point f or sometimes referred to as
principal focus. The focal length is the distance between the vertex of the mirror to
the focal point. This is equal to half the radius of curvature. A spherical mirror has a center of curvature and is at C. The distance between the
center of curvature and the mirror is called the radius of curvature R. Point V is at the
center of the mirror called the vertex of the mirror and the line CV is called the
principal axis or optic axis. The concave mirror is sometimes referred to as a
converging mirror and that the convex mirror is referred to as a diverging mirror. As
the names imply, light rays being reflected from a concave or convex surface tends to
converge or diverge from the mirror as shown respectively.
Ideally, all the incident rays that are moving parallel to the principle axis must
converge (in the case of concave mirror) or appear to be converging (in the case of a
convex mirror) to a single point called the focal point f or sometimes referred to as
principal focus. The focal length is the distance between the vertex of the mirror to
the focal point. This is equal to half the radius of curvature.
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Images Formed by Concave Mirrors (Converging Mirror)
There are three cases to consider when using the ray diagrams to locate image
position for concave mirrors:
1. The object is placed outside the focal point;
2. When the object is placed on the focal point; and
3. The object is placed between the focal point and the mirror.
Case 1: Object is placed outside the focal point.
Image Characteristics:
- The image will be real.
- The image will be inverted.
- The image will be smaller than the object (reduced) (diminished).
Case 2: Object is placed on the focal point.
Image Characteristics:
- When the object is located at the focal point of the mirror, no image will be formed (or it is located at infinity). The parallel reflected rays never cross.
Case 3 Object is placed between the focal point and
mirror.
- The image will be erect; i.e., same orientation as the object.
- The image will be virtual; that is, it seems to be located behind mirror.
- The image will be enlarged; bigger than the object.
Images Formed by Convex Mirrors
With the convex mirror, the image formed is always virtual and upright. As the object
to the mirror distance p increases, the image formed gets smaller and approaches the
focal point as shown.
Image Characteristics:
- All images are erect, virtual, and diminished. Images get larger as object approaches.
Mirror Equation
Geometrical method is not the only method of locating the images formed by
spherical mirrors. Mathematical equations, called mirror equations, can also be
conveniently used to locate the image position and determine its nature.
Consider the following figure for the derivation of the mirror equation.
Equating the two yields:
The mirror equation derived for concave mirror can also be used in the case of
a convex mirror if the following sign conventions are used. Let us call the region in
which the light rays move the front side and the other side where the virtual image is
formed the back side.
Images Formed by Refraction:
We have looked at mirrors to understand how images are formed by reflection. In this
section we will discuss how images are formed when the light ray is refracted at the
boundary between two transparent materials. First, we will look at how images are
formed by refraction at a spherical surface followed by flat refracting surfaces. Then
we will round off the discussion by looking at the different types of lenses and their
applications.
Consider a two transparent media with indices of refraction of n1 and n2 such that n1 <
n2 and that the boundary between the two is spherical of radius R as shown.
The rays originating from the object point O gets converged to the image point I after
being refracted at the spherical surface. Note that real image is formed on the side
opposite to the side from where the light rays come from. This is in contrast to the
mirrors.
We will use a simple geometric technique to study the relationship between the object
position p, image position q and the radius of curvature R. Consider the figure shown
below with a single ray being refracted from the spherical surface.
Snell’s law applied to this ray gives you:
Since θ1 and θ2 are assumed to be very small, we will use small-angle approximation
Sin θ ≈ θ (θ is in radians) to give:
Applying trigonometry rules to triangles OPC and PIC yields:
Combining these expressions and eliminating θ1 and θ 2, we have:
Note that the three triangles shown in the figure have a common vertical leg d. For
relatively small angles of α, β and γ, the horizontal legs of these triangles are
approximately p for triangle containing angle α, R for triangle containing β and q for
triangle containing angle γ. Using tangent line approximations we say that:
Substituting these in expression 2.7 and simplifying them yields:
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