Question

For a spherical mirror show that r=2f.

Answer 1

Hi,
Here is the answer to your query:-

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, BF = PF

or FC = FP = PF

or PC = PF + FC = PF + PF

or R = 2 PF = 2f


or f=R/2

or 2f=R
or R=2f


Thanks
@SHIVAM