Sir, how to prove R=2f and mirror formalae i.e 1/f=1/u+1/v in spherical  mirror

 

 To prove R=2f:     

 

  

 

 

Consider a ray of light AB, parallel to the principal axis, incident on a spherical mirror at point B. 

The normal to the surface at point B is CB and CP = CB = R, is the radius of curvature. 

The ray AB, after reflection from mirror will pass through F (concave mirror) or will appear to diverge from F (convex mirror) and obeys law of reflection, i.e., i = r.   

From the geometry of the figure, if the aperture of the mirror is small, B lies close to P. 

Therefore, BF = PF   

or FC = FP = PF   

or PC = PF + FC

         = PF + PF   

or R = 2PF 

or R = 2f

 

Mirror formalae in spherical mirror:

 

In the figure shown above, an object AB is placed at a distance u from the pole of the concave mirror of small aperture, just beyond the centre of curvature. Hence, its real, inverted and diminished image A’B’ is formed at a distance v in front of the mirror.

A/C to Cartesian sign convention,

Object distance (PB) = -u

Image distance (PB’) = -v

Focal length (PF) = -f

Radius of curvature (PC) = -R

It is clear from the geometry of figure, right angle ABP and A’B’P’ are similar.