`text(Reflection of light :)`

When a beam of light is incident on a boundary separating two media then some part of it may be transmitted and some is turned back in the medium from which it became incident (reflection of light).

`text(Specular and Diffuse Reflection :)`

The type of reflection which is usually invoked in a discussion of reflection at plane and spherical mirrors is known as specular or regular reflection. An incident parallel beam of light is reflected as a parallel beam in figure. The energy in the incident light is confirmed to one direction only on reflection.
Diffuse or irregular reflection is the most common type of reflection and no image formation takes place as the reflected light can not intersect at a common point.

`text(Plane Mirror :)`

A highly polished smooth surface is a mirror. To form a good mirror a thin layer of silver is chemically deposited on a glass surface for high reflectivity.

(i) The incident ray (AB), the reflected ray (BC) and normal (NN') to the surface (SC') of reflection at the point of incidence (B) lie in the same plane. This plane is called the plane of incidence (also plane of reflection).
(ii) The angle of incidence (the angle between normal and the incident ray) and the angle of reflection (the angle between the reflected ray and the normal) are equal

`anglei=angler`

`text(Laws of Reflection in Vector Form :)`

Let `hate_1 =` unit vector along incident ray
`hatn =` unit vector along normal
`hate_2 =` unit vector along reflected ray

Now `vece_(||) =` component of `hate` parallel to mirror `=hate_1 - (hate_1.hatn)hatn`

and `vece_(bot) =` component of `hate` perpendicular to mirror `=(hate_1.hatn)hatn`

Hence `hate_2=vece_(||) - vece_(bot)= hate_1-2hatn(hate_1.hatn)`

`text(Note :)`

Whenever reflection takes place, the component of incident ray parallel to reflecting Surface remains unchanged, while component perpendicular to reflecting surface (i.e., along normal) reverses in direction.
Consider incident ray along unit vector `hate_1` given `hate_1=-xhati-yhatj` unit vector along reflected ray will be given by `hate_2=-xhati+yhatj` similarly `hate_3=xhati+yhatj` diverge.

`text(Principle of reversibility of light :)`

According to this principle if the path of light is reversed then it will retrace its path. i.e., if BP would incident ray then PA would be corresponding reflected ray.

`text(Deviation produced by plane mirror :)`

`delta=180^o-2theta`

Angle of deviation produced by a single surface depends on the angle of incidence, Greater the angle of incidence lower will be the deviation and vice-versa.

(i) If incident ray is rotated (Mirror is kept fixed) in the plane of incidence by angle `theta` then reflected ray rotates by the same angle in the same in plane of incidence but in opposite sense.

(ii) If mirror is rotated (taking position of incident ray same) by angle `theta` such that normal at the point of incidence rotates in the plane of incidence then reflected ray rotates by `2theta` and in same sense.

`delta_1=` Deviation in first position of mirror `=pi-2phi`
`delta_2=` Deviation in second position of mirror `=pi-2(theta+phi)`

`therefore` `delta_1-delta_2=pi-2phi{pi-29theta+phi)}=2theta`

`text(Object :)`
The point of intersection of incident beam is called point object.

`text(Real object Point:)`
If the incident beam is diverging then its intersection point is called real object. It can be seen by human eye and can be photographed by camera.

`text(Virtual object point :)`
If the incident beam is converging then its intersection point is called virtual object. It cannot be seen by human eye and photographed by a camera.

`text(Image :)` The point of intersection of reflected or refracted beam is called image.
`text(Real image :)` If the reflected or refracted beam is converging then its intersection point is called real image. It can be seen by eye, photographed by a camera and can be taken on screen.

`text(Virtual image :)`
If the reflected or refracted beam is diverging then its intersection point is called virtual image. It can be seen by eye, photographed by a camera but can't be taken on screen.

`text(Image of a point object formed by plane mirror :)`

(i) Distance of object from mirror = Distance of image from the mirror.
(ii) The line joining a point object and its image is normal to the reflecting surface.
(iii) The size of the image is the same as that of the object.
(iv) For a real object the image is virtual and for a virtual object the image is real.

(i) Minimum height of a single mirror required for a man to see its complete image.

`DeltaEM_1M_2` and `DeltaEH^'F^'` are similar

`:.` `(M_1M_2)/(H^'F^')=z/(2z)`

or `M_1M_2=H^'F^'//2=HF//2`

`:.` the minimum size of a plane mirror, required to see the full image of an observer is half the size of that observer.

(il) Minimum height of a mirror required to see top as well as bottom of wall when man is mid of wall and mirror.

`DeltaEM_1 M_2` and `DeltaE`H'F' are similar

`:.` `(M_1M_2)/(A^'B^')=x/(3x)`

or `M_1M_2=A^'B^'//3=AB//3=H//3`

`:.` the minimum size of a plane mirror, required to see the full image of an observer is one third the size of wall if observer is standing exactly between wall and mirror.

`text(Aperture :)`
The edge of a spherical mirror is a circle. Part of the plane of circle, enclosed by the circle is called its aperture.

`text(Paraxial Ray :)`
A light ray incident on the mirror at very small angle then the ray is called paraxial ray.

`text(Marginal Ray :)`
A light ray incident on the mirror at finite angle then the ray is called marginal ray.

`text(Focus :)`
Suppose a light ray AQ parallel to x axis become incident on a concave mirror at angle of incidence `theta` (fig). After reflecrion we have reflected ray QF at angle of reflection e which intersects X axis at F. We want to calculate PF

In triangle CFQ
`angle(QCF)=angle(AQC)=theta` (alternate angle)

`=>` triangle CFQ is an isoscless triangle (CF = QF).
`=>` CN = QN = CQ/2 = R/2

In triangle NFQ

`costheta=(QN)/(QF)=(R//2)/(QF)`

`=>` `QF=R/(2costheta)`

`=>` `CF=QF=R/(2costheta)`

`:.` `PF = PC - CF = R - CF = R - R/(2costheta)`

For marginal rays `theta` is not small. Hence different light rays intersects x-axis at different points. But if we consider paraxial beam (`theta-> 0`)

`=>` `PF=R/2` (as `theta->0 => costheta=1`)

i.e., all the light rays intersects x-axis at single point. This single point is called focus of the spherical mirror.

As i increases `cosi` decreases. Hence CQ increases.
If i is a small angle `cos i approx1`
`:.` CQ=R/2

`text(Principal axis :)`
A line passing through focus and centre of curvature.

`text(Pole :)`
Point of intersection of principal axis and mirror.

`text(Focal length :)`
The distance between focus and pole is called focal length.

The following sign convention is used for measuring various distances in the ray diagrams of spherical
mirrors:

`=>` All distances are measured from the pole of the mirror.
`=>` Distances measured in the direction of the incident ray are positive and the distances measured in the direction opposite to that of the incident rays are negative.
`=>` Distances measured along y-axis above the principal axis are positive and that measured along y-axis below the principal axis are negative.

The fig. shows a convex mirror of focal length `f_0` in front of which an object O is placed at a distance x from the pole P.

According to Cartesian sign convention, the formula may be modified as

`u=-x` and `f=+f_0`

Thus `v=(xf_0)/(f_0 + x)`

The above expression shows that whatever may be the value of x, v is always positive and its value is always less than or equal to `f_0`.

The magnification formula may be modified as

`m=(f_0)/(f_0 + x)`

When the object is placed at infinity, a virtual, erect and very diminished image is formed at the focus.