Optics Concept Page - 3

Definition

Experimentally understand image formed by a plane mirror

A source of light O is placed in front of a plane mirror PQ. Two rays OA and OC are incident on it. Remembering the laws of reflection draw the direction of the reflected rays. Draw normal to the surface of the mirror PQ, at the points A  and C. Then draw the reflected rays AB and CD at the points A and C respectively. Extend them further. Do they meet? Extend them backwards. Do they meet now? Mark this point as I.
For a viewers eye at E the reflected rays appear to come from the point I. The reflected rays appear to meet at I. It is a virtual image and can not be caught on a screen. The image is also laterally inverted. That means left of the object will become right of the image and right of the object will become left of the image. The image on a plane mirror is erect and it is as far behind the mirror as the object is in front of it. In irregular reflection though the laws of reflection are followed, the reflected rays are scattered and image is not formed.

Definition

Field of view of an object for a given mirror

The field of view is the size of the largest object that can be seen through an optical system by a detector that is located at a fixed position. The size of the field can be expressed in units of length (such as meters), although it is more convenient to use an angular measures (e.g., degrees) in some situations.

For example, if you are looking at your own image in a plane mirror, then both the object (your shoes, for example) and the detector (your eye) are the same distance from the mirror. A ray of light can travel from your shoes to your eye by striking a mirror that is considerably smaller than your own height. In fact, in order to see your entire body at once, the mirror need be only one-half as tall as you are.

Example

Minimum height of a plane mirror to see the image of a given object

Example: A child of height H is standing in front of a straight plane mirror. His father is standing behind him, as shown. The height of the father is double the height of the child. What is the minimum length of the mirror required so that the child can completely  see his own image and his fathers image in the mirror?

Solution:
The length of mirror is the distance between points A and B.
To see the bottom most point of himself, the boy of height H$H$ needs to have mirror at least at a distance of H2${\displaystyle \frac{H}{2}}$ from the bottom.Thus the point B is established.
Also to see the top most point of his father, the child must have a point of mirror at a distance x$x$ from a point directly in front of him such that,2lH−x=lx${\displaystyle \frac{2l}{H-x}}={\displaystyle \frac{l}{x}}$⟹x=H3$⟹x={\displaystyle \frac{H}{3}}$
Hence the minimum length of mirror required=H2+H3=5H6${\displaystyle \frac{H}{2}}+{\displaystyle \frac{H}{3}}={\displaystyle \frac{5H}{6}}$

Example

Problems on Reflection of Light

Q. A light ray strikes a reflective plane surface at an angle of 56 with the surface. 
    a.) Find the angle of incidence. 
    b.) Find the angle of reflection.
    c.) Find the angle made by the reflected ray and the surface. 
    d.) Find the angle made by the incident and reflected rays.
Sol : a.) angle of incidence:
             ∠i=90−56=34o$\mathrm{\angle }i=90-56={34}^o$ 
        b.) angle of reflection
             ∠r=i=34o$\mathrm{\angle }r=i={34}^o$ (by the law of reflection) 
        c.) q=90−r=90−34=56o$q=90-r=90-34={56}^o$ 
        d.) ∠i+∠r=34+34=68o$\mathrm{\angle }i+\mathrm{\angle }r=34+34={68}^o$

Definition

Lateral inversion

Lateral inversion is the effect produced by a plane mirror in reversing images from left to right. This is the reason why ambulance is written laterally inverted so that its image is aligned properly.

Definition

Lateral Inversion in Plane Mirrors

Lateral Inversion:
In the mirror the right appears left and the left appears right. Note that only sides are interchanged; the image does not appear upside down.
For example in the given figure ambulance is written opposite so as to be seen correctly in the mirror of  the vehicle moving ahead.

Definition

Object moving perpendicular to a plane mirror

When the object moves with speed u$u$ towards (or away) from the plane mirror then image also moves toward (or away) with speed u$u$. But relative speed of image w.r.t. object is 2u$2u$.

Definition

Object moving inclined to a plane mirror

If an object moves towards a plane mirror with a speed V$V$ at an angle θ$\theta$ to the perpendicular to the plane of the mirror,velocity of the object relative to the image is given as
VOI→=VO→−VI→=(Vcosθi^−Vsinθj^)+(Vcosθi^+Vsinθj^)=2Vcosθi^$\overrightarrow{V_{OI}}=\overrightarrow{V_O}-\overrightarrow{V_I}=(V\cos \theta \hat{i}-V\sin \theta \hat{j})+(V\cos \theta \hat{i}+V\sin \theta \hat{j})=2V\cos \theta \hat{i}$
VOI=2Vcosθ

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