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Definitions in connection with spherical mirrors. In the given fig. , let APB be a principal section of a spherical mirror, i.e., the section cut by a plane passing through pole and centre of curvature of the mirror.

Fig. Characteristics of a concave mirror

(i) Pole: It is the middle point P of the spherical mirror.
(ii) Centre of curvature: It is the centre C of the sphere of which the mirror forms a part.
(iii) Radius of curvature: It is the radius R (= AC or BC) of the sphere of which the mirror forms a part.
(iv) Principal axis: The line passing through the pole and the centre of curvature of mirror is called its principal axis.
(v) Linear aperture: It is the diameter AB of the circular boundary of the spherical mirror.
(vi) Angular aperture: It is the angle ACB subtended by the boundary of the spherical mirror at its centre of curvature.
(vii) Principle focus: It is a point F on the principal axis where a beam of light parallel to the principal axis either actually converges to or appears to diverge from, after reflection from a mirror.

Fig. Principal focus of (a) a concave mirror (b) a convex mirror
As shown in Fig.(a), when a beam of light is incident on a concave mirror parallel to its principal axis, it actually converges to a point F on the principal axis. So a concave mirror has a real focus and it is called a converging mirror also. As shown in Fig.(b), when a beam of light is incident on a convex mirror parallel to its principal axis, after reflection it appears to diverge from a point F (lying behind the mirror) on the principal axis. So a convex mirror has a virtual focus and it is called a diverging mirror also.
(viii) Focal length. It is the distance f (= PF) between the focus and the pole of the mirror.
(ix) Focal plane. The vertical plane passing through the principal focus and perpendicular to the principal axis is called focal plane. When a parallel beam of light is incident on a concave mirror at a small angle to the principal axis, it is converged to a point in the focal plane of the mirror.
Note: A line joining any point of the spherical mirror to its centre of curvature is always normal to the mirror at that point.

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Relation between f and R for convex mirror: As shown in Fig, consider a ray AB parallel to the principal axis and incident at the point B of a convex mirror. After reflection from the mirror, the ray appears to come from focus F.


Fig. Relation between f and R for a convex mirror
If C is the centre of curvature, then PC = R, is the radius of the curvature and CB is the normal to the mirror at point B. According to the law of reflection,
∠i = ∠r
As AB is parallel to PC, so
∠i = ∠α (Corresponding angles)
∴ ∠ α = ∠ r
Thus ∆ BCF is isosceles. Hence, BF = FC.
If the aperture of the mirror is small, then B lies close to P, so that
                       
                  
or                        
or      

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Relation between f and R for a concave mirror: As shown in Fig, consider a ray AB parallel to the principal axis and incident at the point B of a concave mirror. After reflection from the mirror, this ray passes through its focus F, obeying the laws of reflectioin. If C is the cente of curvature, then CP = R, is the radius of curvature and CB is normal to the mirror at point B.

Fig. Relation between f and R for a concave mirror
According to the law of reflection,
∠ i = ∠ r
As AS is parallel to CP, so
∠α = ∠i (Alternate angles)
∴ ∠ α = ∠ r
Thus ∆ BCF is isosceles. Hence CF = FB.
If the aperture (or size) of the mirror is small, then B lies close to P, so that,
                                
                            
or                          
or        
Thus, the principal focus of a spherical mirror lies midway between the pole and the centre of curvature.

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Rules for drawing images formed by spherical mirrors: The position of the image formed by spherical mirrors can be found by considering any two of the following rays of light coming from a point on the object.
(i) A ray proceeding parallel to the principal axis will, after reflection, pass through the principal focus in the case of a concave mirrors [Fig.(a)], and appear to come from focus in the case of a convex mirror [Fig.(b)].

Fig.(a). A ray parallel to the principal axis through F after reflection from a concave mirror

Fig.(b) A ray parallel to the principal axis appears to come from F after reflection from a convex mirror.

(ii) A ray passing through the principal focus in the case of a concave mirror [Fig.(c)], and directed towards the principal focus in the case of a convex mirror will [Fig.(d)], after reflection, become parallel to the principal axis.

Fig. (c) A ray through F becomes parallel the principal axis after reflection from a concave mirror

Fig.(d) A ray directed towards F becomes parallel to the principal axis after reflection from a convex mirror

(iii) A ray passing through the centre of curvature in the case of a concave mirror [Fig.(e)] and directed towards the centre of curvature in the case of a convex mirror [Fig. F] falls normallly (∠i = ∠r = 0°) and is reflected back along the same path.

Fig.(e) A ray passing through C is reflected back along of same path after reflection from a concave mirror

Fig.(F) A ray directed towards C is reflected back along same path after reflection from a convex mirror

(iv) A ray incident obliquely to the principal axis, towards the pole P, on the concave mirror [Fig.(G)] or  a convex mirror [Fig.(H)] is reflected obliquely, following the laws of reflection at the point of incidence, i.e., the incident and reflected rays make equal angles with the principal axis.

Fig.(G) Incident and relfected rays follow the laws of reflection

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Formation of images by a concave mirror for different positions of the object.
(i) Object at infinity: When the object lies at infinity, the rays from the distant object fall on the concave mirror as a parallel beam, as shown in Fig.(a). The ray passing through F becomes parallel to the principal axis after reflection from the mirror. The ray through C is reflected back along its own path. The two reflected rays meet at point A' in the focal plane. Hence a real, inverted and highly diminished image A'B' is formed at the focus F of the concave mirror.

Fig. (a) Image formed by a concave mirror with object at infinity

(ii) Object beyond the centre of curvature: In Fig.(b) ,an object AB is placed on the principal axis of a concave mirror and beyond its centre of curvature C. A ray AM going parallel to the principal axis passes, after reflection, through the principal focus F. Another ray AN passing through F becomes, after reflection, parallel to the principal axis. The two reflected rays meet at point A'. Thus A' is the image of A. If we draw A'B' perpendicular to the principal axis, then A'B' is the complete image of object AB. Hence, the image is real, inverted, diminished in size and is formed between the focus and the centre of curvature.

Fig. (b) Image formed by a concave mirror with object beyond C

(iii) Object at the centre of curvature: In Fig.(c), an object AB is placed at the centre of curvature C of a concave mirror. A ray AM parallel to the principal axis passes after reflection through the focus F. Another ray AN passing through F becomes after reflection, parallel to the principal axis. The two reflected rays meet at point A'. Thus A' is the image of point A. The perpendicular A'B' drawn on the principal axis coincides with the position of AB. Hence a real, inverted and of same size image is formed at the centre of curvature.


Fig.(c). Image formed by concave mirror with object at the centre of curvature

(iv) Object between principal focus and centre of curvature: In Fig.(d), an object AB is placed between the focus F and centre of curvature C of a concave mirror. A ray AM parallel to the principal axis passes, after reflection, through the focus F. Another ray AN passing through F becomes, after reflection, parallel to the principal axis. The two reflected rays meet at point A'. The line A'B' drawn perpendicular to principal axis is the complete image of AB. Hence a real, inverted and enlarged image is formed beyond the centre of curvature.

Fig.(d).  Image formed by a concave mirror with object between its F and C

(v) Object at the principal focus: In Fig.(e), an AB object is placed at the principal focus F of a concave mirror. A ray AM parallel to the principal axis passes, after reflection, through the focus F. Another ray AM coming through C falls normally on the mirror and refraces its path after reflection. The two reflected rays are parallel to each other and meet at infinity. Hence a real, inverted and highly enlarged (or magnified) image is formed at infinity.



Fig.(e). Image formed by a concave mirror with object at F

(vi) Object between the principal focus and pole: In Fig.(f) , an object AB is placed between the focus F and the pole P of a concave mirror. A ray AM parallel to the principal axis passes, after reflection, through the focus F. Another ray AN coming from C follows its path back. Both the reflected rays appear to diverage from a common point A' behind the mirror. So A' is the virtual image of A. The normal A'B' upon the principal axis is the complete image of AB. Hence a virtual, erect and magnified image is formed behind the mirror.


Fig.(F).  Image formed by a concave mirror with the object between F and P