CURVED SPHERICAL MIRRORS

 

CURVED SPHERICAL MIRRORS

Curved spherical mirror are made by silvering a glass surface which is part of a sphere.

Types of curved mirror:

There are two types of curved mirror.

i.                       Concave mirror : If the outside surface is silvered and the inside surface is the reflecting part, the resulting mirror is known as a concave or converging mirror.

ii.                      Convex mirror: If the inside surface of this spherical part is silvered and the outside surface is the reflecting part, the resulting mirror is called a convex or diverging mirror.

 Terms associated with curved surface

Aperture: is the width of the mirror.

Principal Focus (F): of a curved mirror is that point on the principal axis to which incident rays parallel and close to the principal axis converge or from which they appear to diverge after reflection.

 Centre of curvature( C): is the centre of the sphere of which the mirror forms a part.

The  pole (P): is the centre of the reflecting surface of the curved mirror.

 Principal Axis: is the line from the pole to the centre of curvature.

Focal length (f): is the distance between from the principal focus to the pole. f = r /2.

The Radius of Curvature (r) : is the radius of the sphere of which the mirror forms a part.

  Experimental determination of focal length using a concave mirror.

An illuminated object is placed in front of a concave mirror mounted on a mirror holder. The position of the mirror is adjusted until a sharp image of the object is obtained on the surface at the same position as the object. The distance between the mirror and the illuminated object is measured. This distance gives the radius of curvature, r, of the mirror. The experiment is repeated two more times. Then focal length f is given by f = .

 Image formed on concave mirror.

By changing the position of the object from the concave mirror, different types of images can be formed. Different types of images are formed when the object is placed:

I. At the infinity        II. Beyond the centre of curvature              III. At the centre of curvature     IV. Between the centre of curvature and principal focus  V. At the principal focus             VI.  Between the principal focus and pole

Concave Mirror Ray Diagram   

When an object the is placed at infinity, a real image is formed at the focus. The size of the image is much smaller as compared to that of the object.

When an object is placed behind the center of curvature, a real image is formed between the center of curvature and focus. The size of the image is smaller as compared to that of the object.

When an object is placed at the center of curvature and focus, the real image is formed at the center of curvature. The size of the image is same as compared to that of the object.

When an object is placed in between the center of curvature and focus, the real image is formed behind the center of curvature. The size of the image is smaller as compared to that of the object.

When an object is placed at the focus, the real image is formed at infinity. The size of the image is much larger as compared to that of the object.

When an object is placed in between focus and pole, a virtual image is formed. The size of the image is larger as compared to that of the object.

Image Formation By Convex Mirror

The image formed in a convex mirror is always virtual and erect, whatever be the position of the object. In this section, let us look at the types of images formed by a convex mirror.

1. When an object is placed at infinity, a virtual image is formed at the focus. The size of the image is much smaller as compared to that of the object.

2.  When an object is placed at a finite distance from the mirror, a virtual image is formed between pole and focus of the convex mirror. The size of the image is smaller as compared to that of the object.

Parabolic mirrors:  is a special type of concave mirror which has the shape of a parabola. The parabolic mirror produces a wide parallel beam of light of constant intensity when a small light source is placed at its focus. Parabolic mirrors are used in car headlamps and as searchlights.

 

Mirror formulae

The object distance, u, the image distance, v, and the focal length, f, of spherical mirror are related by the equation

 1 /u + 1 /v  = 1 /f. The equation can be used to obtain the focal length of a concave mirror.

Sign convention

When the mirror formula is used in solving practical problems, it is necessary to add a positive(+) or a negative (-) sign to each of the distances according to a sign rule or convention.

Real is Positive	New Cartesian
1. Real objects and real images are considered to be at a positive distance from the mirror.	Distances measured to the right of the mirror are positive
2. Virtual images are at a negative distance from the mirror	Distances measured to the left of the mirror are negative
3. Focal length of a concave mirror is positive	Focal length of a concave mirror is negative
4. Focal length of a convex mirror is negative	Focal length of a convex mirror is positive

Linear magnification.

Linear magnification (M) produced by a  mirror given by M =v / u  = height of image / height of object

Example

A concave mirror of radius of curvature 40cm is to used to obtained a real image of an object. The image is to be one-third as large as the object. Where should the object be placed, and where is the image to be found?.

Solution

f = r/2  = f = 40/2 =20cm ; image height, u = 1/3 large as the object, v ( u =1/3v) ; 3u = v

f = vu / v+u =    =  ; 20cm =  ; v =  = 26.7cm;   u = 3v = 3 x 26.7cm = 80cm

The object should be placed, u= 80cm while the image is to be found, v = 26.7cm

Application of curved mirrors. 

I. Concave mirrors:- i. as shaving mirror                 ii. A Parabolic mirror

II. Convex mirror: as driving mirrors. They have a field of view

Assignment

An object 7 m high is placed 15 cm from a convex spherical mirror of radius 45 cm. describe its image.