# Reflection of Light , Laws of Reflection , Image formed by a plane mirror

Light : (i) Light is electromagnetic wave ( proposed by Maxwell ). It consists of varying electric field and magnetic field.
(ii)Light carries energy and momentum.

Ray optics:
Ray optics treats propagation of light in terms of rays and is valid only if the size of the obstacle is much greater than the wavelength of light. It concern with the image formation and deals with the study of the simply facts such as rectilinear propagation, laws of reflection and refraction by geometrical methods .

Ray:
A ray can be defined as an imaginary line drawn in the direction in which light is travelling. Light behaves as a stream of energy propagated along the direction of rays. The rays are directed outward from the source of light in straight lines .

### Reflection of Light at a Plane Surface:

We know that a ray of light is composed of packets of energy known as photons. The photons have the ability of colliding with any surface.  Thus the photons transfer momentum & energy in the same way as the transfer of momentum & energy take place between any two particles during collision.

For elastic collision a particle gets rebounded with same speed where as the direction of its motion changes due to the impact (impulsive) force offered by the other colliding particle.

Similarly when a photon is incident on a fine plane surface, it gets rebounded. This is known as Reflection.

Angle of incidence:
The angle which the incident ray makes with the normal at the point of incidence is called the angle of incidence. It is generally denoted by ‘i’.

Angle of reflection:
The angle which the reflected ray makes with the normal at the point of incidence is called the angle of reflection. It is generally denoted by ‘r’.

Glancing angle:
The angle which the incident ray makes with the plane reflecting surface is called glancing angle. It is generally denoted by ‘g’.

g = 90° – i

### Laws of Reflection :

There are two laws governing the physical phenomenon of reflection.

They are-

(i) The incident ray, normal to the interface and the reflected ray lie on the same plane.

(ii)  The angle of incidence is equal to the angle of reflection. ∠i = ∠r

These laws hold good for all reflecting surfaces either plane or curved.

##### Deviation produced by a Plane mirror:

Deviation is defined as the angle between directions of the incident ray and the reflected ray (or, the emergent ray). It is generally denoted by δ.
δ = 180° – 2 i

(iii) We can observe from the figure that two rays coming from a point object O are reflected at the points M & N. If the reflected rays are produced back they meet at a point P. This point P is known as image.

Since the image is formed at the back of the mirror, it is known as virtual image .
Geometrically we can see that object distance is equal to image distance from mirror surface   p = q

(As ΔOMP & ΔONP are isosceles triangles)

### Properties of image formed by plane mirror :

Properties of image formed by plane mirror

(a) the image is virtual erect and laterally reversed.

(b) the image size = object size

(c) object distance = image distance

(d) when the plane mirror is rotated through an angle θ, the reflected ray is rotated through an angle 2θ.

#### Numbers of images formed by the combination of two mirrors inclined at an angle θ

Number of image formed ,
(i) $\displaystyle N = (\frac{360}{\theta}- 1)$ ; If 360/θ is an even integer

(ii) $\displaystyle N = (\frac{360}{\theta})$ ; If 360/θ is an odd integer (If the object is not placed on the angle bisector)

(iii) $\displaystyle N = (\frac{360}{\theta} – 1)$ ; If 360/θ is an odd integer (If the object is placed on the angle bisector)

e.g. When two plane mirrors are placed parallel to each other , then θ = 0

⇒ N = ∞ ; Therefore infinite number of images are formed.

(iv) If 360/θ ≠ Integer ; then the number of images = nearest even integer

#### Relation between velocity of object and image :

Example : If an object moves towards a plane mirror with a speed V at an angle θ to the perpendicular to the plane of the mirror, find the relative velocity between the object & the image.

Solution :

Velocity of the object relative to the image is given as

$\displaystyle \vec{V_{OI}} = \vec{V_O} – \vec{V_I}$

$\displaystyle \vec{V_{OI}} = (Vcos\theta\hat{i}-Vsin\theta\hat{j}) – (- Vcos\theta\hat{i} – Vsin\theta\hat{j})$

$\displaystyle \vec{V_{OI}} = 2Vcos\theta\hat{i}$

$\displaystyle V_{OI} = 2Vcos\theta$

Exercise : Prove that if the plane mirror rotates through an angle θ, the reflected ray for any incident ray is deviated through an angle 2θ.

#### Minimum length of mirror to see the complete image of the body of a person:

The person must have to obtain a reflected ray one from his head & another reflected ray from the feet.

Geometrically, AC = AB + BC where

$\displaystyle AB = \frac{x}{2} , BC = \frac{H-x}{2}$

$\displaystyle AC = \frac{x}{2} + \frac{H-x}{2}$

$\displaystyle AC = \frac{H}{2}$

i.e. length of the mirror is just half the height of person.

#### Rotation of Mirror :

Q: Show that if a mirror is rotated by angle θ the reflected ray rotates through an angle 2θ When the mirror rotates by θ , the angle between the normals is also equal to θ.

Solution :

Incidence angle for M’ = i + θ

Angle of reflection for M’ = i + θ

i.e. ∠N2OR2 = i + θ

∠N1OR2 = ∠N1ON2 + ∠N2OR2

= θ + i + θ = i + 2 θ

or ∠R1OR2 =  ∠N2OR2 -∠N2OR1

= i + 2 θ – i = 2 θ

Hence the reflected ray rotates by an angle 2θ