Mirror Formula (1/f = 1/v + 1/u = 2/R)

 Spherical Mirror

1. Spherical mirror. It is a polished surface which forms the part of a sphere. It is of two types :

     (a) Concave mirror. It is a spherical mirror which when looked from the reflecting side is depressed at the centre and bulging at the edges. It is shown in [Fig. 1(i)].

Fig. 1. Characteristics of a spherical mirror. 

     (b) Convex mirror. It is a spherical mirror which when looked from the reflecting side bulges at the centre and is depressed at the edges. It is shown in [Fig. 1(ii)].

2. Pole (P). The central point which is most depressed (in a concave mirror) and which is most bulging (in a convex mirror) is called pole of the mirror. 

3. Radius of curvature (R). Radius of curvature of a mirror is defined as the radius of that sphere of which the mirror forms a part. 

4. Centre of curvature (C). It is the centre of that sphere of which the surface forms a part. 

5. Principal axis. A straight line joining the pole of the mirror and its centre of curvature is called principal axis. 

6. Aperture (XY). The diameter of the circular outline of the mirror is called aperture of the mirror. 

7. Principal section. A section of the mirror by a plane passing through the centre of curvature and the pole is called principal section. 

8. Principal focus. Consider a beam of light coming parallel to principal axis and incident on a spherical mirror. In case if a concave mirror [Fig. 2(i)], the beam actually meets at F while in the case of convex mirror it only appears to diverge from F, when the rays are produced back [Fig. 2(ii)]. They meet at the point 'F' and is called principal focus. 

Fig. 2. Principal focus of a spherical mirror. 

     Principal focus is a point situated on the principal axis at which a beam coming parallel to principal axis meets or appears to meet after reflection from the mirror. 

9. Focal plane. It is a vertical plane passing through the principal focus and perpendicular to the principal axis. 

10. Focal length (f). Focal length of a spherical mirror is the distance of its principal focus from its pole. It is denoted by 'f' as shown in [Fig. 2(i) and (ii)]. Focal length is measured in terms of units of length i.e. cm or m. 

Sign Conventions

While handling derivations in optics we have to deal with measurement of certain distances like distance of object (u), distance of image (v), focal length (f) and radius of curvature (R) etc. We shall adopt the following sign conventions for their measurements :

     (i) All the ray diagrams will be drawn with light travelling from left to right. 

    (ii) The pole of the surface (in case of lenses) will be considered to be situated at the origin of a system of co-ordinate axes. All the measurements will be done from the origin. 

  (iii) The principal axis will always coincide with X-axis of the system of co-ordinate axis. 

   (iv) All the distances measured towards left of origin are taken as negative while, those measured towards right of origin are taken as positive. 

   (v) All transverse measurements done above the X-axis are taken as positive while those done below the X-axis are taken as negative. 

Assumption

While obtaining some relations, in ray optics, we shall make some assumptions given as follows. All those formulae will hold good only if these conditions are satisfied. 

     (i) The sources are considered to be point one. 

    (ii) The aperture of the surface/lens should be small. 

   (iii) Rays of light should make smaller angles with the principal axis. 

Relation between focal length and radius of curvature

Considered a ray of light coming from infinity (parallel to the principal axis), incident on a spherical mirror at A. If the mirror is concave [Fig. 3(i)], it meets at F after reflection. If the mirror is convex [Fig. 3(ii)], it appears to come from F after reflection. The angle of incidence and angle of reflection are shown in the diagram. 

     (i) Concave mirror

                ㄥSAC = ㄥACF = ㄥi   [alternate angles]

               ㄥCAF = ㄥr

     According to law of reflection

              ㄥi = ㄥr      ∴    ㄥCAF = ㄥFCA

    (ii) Convex mirror

              ㄥSAN = ㄥPCA = i  [corresponding angles]

             ㄥNAI = ㄥFAC = r  [vertically opposite angles]

     According to law of reflection

                      ㄥi = ㄥr

     ∴      ㄥCAF = ㄥFCA

     ∴        ㄥAF = FC 

     From [Fig. 3 (i) and (ii)], radius of curvature R is 

     R = PC = PF + FC = PF + AF = 2PF     ... (i) 

     [For very small aperture AF ≃ FP]

     According to sign convention for concave mirror [Fig. 3(i)]

Fig. 3. Reflection at a spherical mirror. 

     CP ≃ -R, PF = -f   or   -R = -2f   or  f = R/2

     According to sign convention for convex mirror [Fig. 3(ii)]

             CP = + R, PF = + f

     Substituting in (i), 

               R = 2f  or   f = R/2

     Thus, the focal length of a spherical mirror is half its radius of curvature. 

Mirror Formula (1/f = 1/v + 1/u = 2/R) 

It is a relationship connecting object distance, image distance and focal length of a spherical mirror. 

     Consider a spherical mirror having principal section XPY and focus F. AB is an object standing vertically on the principal axis at a point B. (In case of virtual image by concave mirror, the point B is within focus F). A ray AK parallel to principal axis after reflection from mirror either coverages at F (in case of concave) or appears to diverge from F (in case of convex). The ray passes along KR. Another ray AP is reflected from pole (P) and proceeds along PS. In [Fig. 4(i)], the two reflected rays KR and PS actually meet at a point A' giving the real image of a point A. In [Fig. 4(ii) and (iii)], the ray KR and PS are divergent forward apparently originating from point A' behind the mirrors, giving virtual image of the point A. Another ray BP from foot of object is incident normally and after reflection from mirror retraces the path (along PB). Draw a perpendicular from A' upon the principal axis. 

Fig. 4. Formation of image by a spherical mirror. 

     In [Fig. 4(i)], A'B' represents real and inverted image of AB. In [Fig. 4(ii)], A'B' represents virtual, erect and magnified image of AB. In [Fig. 4(iii)], A'B' represents virtual, erect and diminished image of AB. 

Deduction

     (i) Concave mirror producing real image [Fig. 4(i)]

        BP = distance of object from pole = u

       B'P = distance of image from pole = v

        PF = focal length = f

     

     ∆APB and ∆A'PB' are similar. 

     ∴         A'B'/AB = P'B/PB                       ... (ii) 

     Again for very small aperture KP is nearly straight. 

     ∆KPF, ∆A'B'F are similar. 

     ∴         A'B'/KP = B'F/PF

     ⇒       A'B'/AB = PB' - PF/PF    [∵  KP = AB]

     Using equation (ii), we get

             PB' - PF/ PF = PB' - PF/PF

     According to sign convention

            PB' = -v, PB = -u and PF = -f

            -v/-u = -v - (-f)/-f

     ⇒   vf = uv - uf      ⇒    uf + vf = uv

     Dividing by uvf, we get

            1/v +1/u = 1/f.

     (ii) Concave mirror producing virtual image [Fig. 4(ii)]

     ∆APB and ∆A'PB' are similar. 

     ∴         A'B'/AB = PB'/PB                      ... (iii) 

     Again ∆KPF and ∆A'B'F are similar. (KP is nearly straight for small aperture) 

     ∴         A'B'/KP = B'F/PF

     ⇒       A'B'/AB = PB' + PF/PF

     Using equation (iii), we get

             PB'/PB = PB' + PF/PF

     According to sign convention

             PB' = +v, PB = -u and PF = -f. 

     Making these substitutions

                   +v/-u = v - f/-f

or         -vf = -uv + uf       ⇒    uf + vf = uv

     Dividing by uvf, we get

           1/v + 1/u = 1/f.

     (iii) Convex mirror [Fig. 4(iii)]

     ∆APB and ∆A'PB' are similar. 

     ∴           A'B'/AB = PB'/PB

     Again ∆KPF, ∆A'B'F are similar. 

     ∴           A'B'/KP = B'F/PF

     (KP is straight for very small aperture) 

     ⇒         A'B' = PF - PB'/PF

     According to sign convention

             PB' = +v, PB = -u and PF = +f

     Making these substitutions

            v/-u = f - v/f    or    vf = -uf + uv

     ⇒   uf + vf = uv

     Dividing by uvf, we get

     ⇒       1/v + 1/u = 1/f.                            ... (iv) 

Magnification

Magnification gives us an idea about the relative change in size of image as compared to that of object. There are two types of magnification. 

1. Linear magnification or Transverse magnification

     Linear magnification produced by a spherical mirror is defined as the ratio between size of the image to the size of the object. 

     Let the size of the images and object be denoted by I and O respectively. The linear magnification produced by the spherical mirror is given by

                    m = I/O                                   ... (v) 

     Case (a). Magnification produced by a concave mirror

     ∆APB, ∆A'PB' [in Fig. 4(i)] are similar. 

     ∴        A'B'/AB = PB'/PB

     Using sign conventions, 

     A'B' = - I    (Measurement below the principal axis) 

     AB = +O    (Measurement above the principal axis) 

     PB' = -v     (Against incident ray) 

     PB = -u      (Against incident ray) 

     Making the substitution, 

           - I/O = -v/-u

or         I/O = -v/u                                      ... (vi) 

     From equations (v) and (vi), 

                 m = I/O = - (v/u)                      ... (vii)

     Case (b). Magnification produced by a convex mirror

     ∆APB and ∆A'PB' [in Fig. 4(iii)] are similar. 

     ∴         A'B'/AB = PB'/PB

     Using sign conventions, 

     A'B' = +I     (Measurement above the principal axis) 

     AB = +O     (Measurement above the principal axis) 

     PB' = +v     (Along incident ray) 

     PB = -u      (Against incident ray) 

     Making the substitution, 

           I/O = -v/u                                        ... (viii) 

     From equations (v) and (viii), we get,

           m = I/O = -v/u                                ... (ix) 

     It is clear from equations (vii) and (ix) that

     (i) Expression for linear magnification produced by a concave mirror and by a convex mirror is same. 

    (ii) Expression for magnification is independent of the nature of image (real/virtual). 

Relation between m and f

     (i) In term of v

     According to mirror formula, 

  

             1/v + 1/u = 1/f

     Multiplying throughout by v, 

             1 + v/u = v/f

     Since v/u = -m [from equation (ix)]

     ∴       1 - m = v/f      or      m = 1 - v/f

or                m = f - v/f                             ... (x) 

     (ii) In term of u

     Again, according to mirror formula

              1/v + 1/u = 1/f

     Multiplying throughout by u, 

             u/v + 1 = u/f

     Since u/v = - 1/m [from equation (ix)]

              -1/m + 1 = u/f

or                 1/m = 1 - u/f = f - u/f

     

     ∴                 m = f/f - u                             ... (xi) 

     Combining equations (ix), (x) and (xi), 

             m = I/O = -v/u = f - v/f = f/f - u

2. Longitudinal magnification (m') 

     When the object and the image possess finite size along the principal axis, it is desirable to compare them with each other. This is expressed in terms of longitudinal magnification. 

     Longitudinal magnification of a mirror is defined as the size of the image to the size of the object both measured along the direction of principal axis. 

     Let 'dv' and 'du' respectively be the distances occupied by the image and the object along principal axis. Longitudinal magnification m' is given by

                      m' = dv/du

     According to mirror formula, 

                    1/v + 1/u = 1/f

     Differentiate both side with respect to v. 

       - (1/v²) dv - (1/u²) du = 0    [∵  f = constant]

     ∴       m' = dv/du = - (v²/u²) = - [- (v/u)²]

     Since     - (v/u) = m   (transverse magnification) 

     ∴       m' = - m²

     or    Longitudinal magnification = - (transverse magnification) 

Relative positions, Size and Nature of images as object is brought from infinity to the pole of a concave mirror

As object is brought from infinity towards the pole of a concave mirror u changes and hence v also changes giving different position of images. 

      

(i) If the object is at infinity, u = ∞ (Fig. 5)

     

Fig. 5. Object at infinity. 

     For concave mirror, focal length = - f

   

     From mirror formula

                   1/v + 1/u = 1/f

     We get, 1/v + 1/∞ = 1/-f

     ⇒                     1/v = - (1/f). 

     Thus, image is obtained at the focus of focal plane. Linear magnification, 

               m = v/u = -f/∞ = 0

     Thus, image us very small in dimensions. Here rays from infinity are rendered into beam parallel to principal axis. After reflection rays coverage at focus giving real and point image at focus. 

    (ii) The object lies beyond centre of curvature (Fig. 6).

Fig. 6. Object beyond 2f.

i.e.,             ∞ > u > 2f

                   0 < 1/u < 1/2f

                   1/v + 1/u = 1/-f

     ⇒                    1/v = 1/-f + 1/u

     Since         u > 2f

     ⇒              1/v > - 1/f + 1/2f

                       1/v > - 1/2f    ⇒   v < - 2f

     Since          u < ∞

     Again        1/v < - 1/f - 0    ⇒     v > f. 

     Magnification,      m = (v/u) < 1

     Therefore, if        ∞ > u > 2f

                                    2f > u > f. 

     Thus, a real inverted and diminished image is formed in between focus and centre of curvature. 

   (iii) Object is at centre of curvature (Fig. 7) 

Fig. 7. Object at 2f.

i.e.,                 u = - 2f = - r

     ∴     1/v + 1/-2f = 1/-f

     ⇒                   v = - 2f.

     Magnification, 

                 m = v/u = - 2f/- 2f = 1.

     Thus, a real and inverted image of same size as that of object is formed at centre of curvature. 

    (iv) Object is in between a distance f and 2f, i.e., in between focus and centre of curvature (Fig. 8).

Fig. 8. Object between f and 2f.

     Here,          f < u < 2f; i.e., 1/f > 1/u > 1/2f

     Since, for     u = - f,  1/v = 1/- f + 1/u

                           1/v > 1/- f + 1/f   ⇒    1/v > 0

     ⇒                   v < 1/0 ; i.e.,  v < ∞

     Again, for      u = - 2f

                             1/v < 1/- f + 1/2f     

   

     ⇒                     1/v < - (1/2f) 

     ∴                      v > - 2f,    m = (v/u) > 1.

     Thus, a real inverted and magnified image is formed in between centre of curvature and infinity. 

     (v) Object is kept at focus (Fig. 9) 

Fig. 9. Object at f. 

i.e.,                         u = - f

                 1/v + 1/- f = 1/- f

     ∴                     1/v = 0

                               v = ∞

     The rays after reflection are rendered into a parallel beam meeting in infinity. 

    (vi) Object is kept within focus (Fig. 10) 

Fig. 10. Object between F and pole. 

i.e.,               u < - f

     ⇒           1/u > - (1/f) 

                    1/v = 1/- f + 1/u

     Since    -1/u > 1/f

     ∴   v is positive. 

     Thus, a virtual, erect and magnified image is formed on the other side of mirror. 

     All the cases discussed above are being put in a tabular form below :

S. No.	Position of object	Position of image	Size of image	Nature of image
1.	At infinity	At F	Very small	Real, inverted
2.	Beyond C	Between F and C	Diminished	Real, inverted
3.	At ac	At C	Equal in size	Real, inverted
4.	Between F and C	Beyond C	Magnified	Real, inverted
5.	At F	At infinity	Highly magnified	Real, inverted
6.	Between F and C	Behind the mirror	Magnified	Virtual, erect

Use of Spherical Mirror

 Spherical mirrors (concave and convex) have been put to use in daily life, in a number of ways. 

Use of Concave mirror

     1. When object is situated in between principal focus and pole of a concave mirror, through the image is virtual, it is highly magnified. As such a concave mirror can be used as a shaving mirror or make-up mirror. 

     2. Doctors use concave mirrors for focussing light on ear, nose, throat for their close examination. There is a hole at the centre of such a mirror. They wear it on their heads with the help of a belt in such a way that the hole comes directly in front of their eye. They can look through the hole. 

     3. Concave mirrors, when used as reflector, can be employed for constructing reflecting telescopes. 

Use of Convex mirror

     1. A convex mirror can be used as a rear view mirror in automobiles. No doubt the image obtained with a convex mirror is diminished, but the mirror covers a large field of view. Hence, the objects situated behind the automobile and on bothe the sides of road can be easily seen by the driver. 

     2. A convex mirror is also used as a street light reflector. 

Key words

1. Focal length. Distance between pole and the principal focus. 

2. Image (real). A point where reflected rays actually intersect. 

3. Image (virtual). A point where reflected rays appear to intersect. 

4. Magnification (linear). Ratio between size of image to the size of object measured perpendicular to the principal axis. 

5. Magnification (longitudinal). Ratio between size of image to the size of object measured along the direction of principal axis. 

6. Mirror (concave). Spherical reflecting surface buldging at the corners and depressed at the centre. 

7. Mirror (convex). Spherical reflecting surface buldging at the centre and depressed at the corners. 

8. Pole. Centre point of a spherical mirror. 

9. Principal axis. A line passing through the pole of the mirror and perpendicular to its surface. 

10. Principal focus. A point on the principal axis where a beam of light coming parallel to principal axis meets or appears to meet after reflection from the mirror. 

11. Principal section. A section of the mirror by a plane passing through the centre of curvature and the pole. 

12. Radius of curvature. Radius of the sphere of which the mirror forms a part. 

13. Ray. Path along which light travels. 

Key Formulae

1. Relation between focal length (f) and radius of curvature (R) of a mirror :

                        f = R/2

2. Mirror formula : 1/v + 1/u = 1/f

3. Magnification 'm' (linear) : m = I/O = ±v/u

4. Relation between m and f :

                m = f - v/f = f/f - u

5. Magnification (longitudinal) : m' = dv/du

6. Relation between m and m' : m' = - m².