A curve mirror is a smooth reflecting part (in any shape) of a symmetrical curved surface such as paraboloidal, ellipsoidal, cylindrical or spherical.

Concave Mirror

If the inner surface of the spherical mirror acts as a reflector then the mirror is called convex mirror.

Convex Mirror 

If the outer surface of the spherical mirror acts as a reflector then the mirror is called convex mirror.

TERMS RELATED TO SPHERICAL MIRROR:

Centre of curvature (C)

It is the centre of sphere of which the mirror is a part. (6)

Radius of Curvature (R) :

It is the radius of the sphere of which the mirror is a part.

Pole (P) :

It is the geometrical centre of the spherical reflecting surface.

Principal Axis :

It is the straight line joining the centre of curvature to the pole.

Focus (F) : When a narrow beam of rays of light, parallel to the principal axis and close to it ( known as paraxial rays ), is incident on the surface of a mirror, the reflected beam is found to converge (concave mirror) or appear to diverge (convex mirror) from a point on the principal axis. This point is called focus.

Focal Length (f)

It is the distance between the pole and the principal focus. For spherical mirrors, f=R/2

Aperture:

It is the effective diameter of light reflecting area of the mirror.

CARTESIAN SIGN CONVENTION

All distances are measured from the pole \((\mathrm{P})\) and let \(\mathrm{P}\) is the origin.

(2) Distances measured to the right of the pole are taken as positive.

(3) Distances above the principal axis are taken as positive.

(4) Angles measured from the normal in the anticlockwise sense are positive.

MIRROR FORMULAE:

(1) In terms of Cartesian sign convention mirror formula may be expressed as

1/v+1/u=1/f

Where (u, v) and f represents object distance, image distance and focal length respectively.

(2) Linear magnification:

Linear magnification is defined as the ratio of the size of image to the size of the object

m=1/o

(i) If one dimensional object is placed perpendicular to the principal axis, linear magnificaion is called transverse or lateral magnification and for mirrors becomes

m=negative, for real object-real image pair, for virtual object-virtual image pair

m=positive, for real-virtual pair

(iii) If a 2-D object is placed with its plane perpendicular to principal axis its magnification called superficial magnification, \(\left(\mathrm{m}_{\mathrm{S}}\right)\) will be

where m is transverse magnification

(iv) In case of more than one optical component, the image formed by first component will act as an object for the second and so on. So over all magnification

or

RULES FOR RAY DIAGRAMS :

1 Aray, initially parallel to the principal axis is reflected through the focus of the mirror.

2 Aray, initially passing through the focus is reflected parallel to the principal axis.

3 A ray passing through the centre of curvature is reflected back along itself.

4 Aray incident at the pole is reflected symmetrically.

Concave mirror:

The fig. shows a concave mirror of focal length \(\mathrm{f}_{0}\) in front of which an object \(\mathrm{O}\) is placed at a distance \(\mathrm{x}\) from the pole \(P\).

According formula may be modified as

\( u=-x; f=-f_0 \)

thus \( 1/v+1/-x=1/-f_0 \)

or

\( v=xf_0/ f_0-x \)

and, the magnification formula may be modified as

Convex mirror :

The fig. shows a convex mirror of focal length \(\mathrm{f}_{0}\) in front of which an object \(\mathrm{O}\) is placed at a distance \(\mathrm{x}\) from the pole \(P\).

An object \(\mathrm{O}\) placed in front of a convex mirror

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

\( u=-x and f=f+f_0 \)

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

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

Note :

(i) In case of spherical mirrors if we plot a graph between1/u and 1/v, the graph will be a straight line with intercept 1/f with each axis as 1/v+1/u=1/f becomes y+x=c with c=1/f . This is shown in figure

(ii) Position of real object and real image are interchangable.

if I_{1 and} I₂ is the height of the image 

NEWTON'S FORMULA:

If distance of object and image is measured from focus instead of pole then,

Where\(x_o, x_i\) and f are object distance, image distance and focal length respectively.