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Derive R=2f for a spherical mirror, where the symbols have their usual meaning.

Answer
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Hint :First law of reflection states that the incident ray, reflected ray and normal, all lie in the same plane. Second law of reflection states that the angle of reflection is always equal to the angle of incidence. The angles are measured relative to the perpendicular to the surface at the point where the ray strikes the surface.

Complete Step By Step Answer:
Let OA be the incident ray and CA be the line normal to the surface at the point A where the ray (OA) strikes the surface of a spherical mirror. Let f and R be the focal length and radius of curvature. Also let i and r be the angle of incidence and the angle of reflection. Then we can draw the figure as follows
seo images

Hence FP=f and CP=R .
Also C is the centre of the circle and R is also the radius of the sphere.
According to second law of reflection
 i=r ----(1)
Since OA is parallel to CP we get |ACP=|OAC=i ---(2)
From the equation (1), the equation (2) becomes
 |ACP=i=r=|CAF
Then ΔACF is an isosceles triangle. Therefore CF=FA ---(3)
For mirror of small aperture we get FAFP ---(4)
From the equation (3) and (4), we get
 CF=FP ---(5)
then CP=CF+FP=FP+FP=2FP
 R=2f .
Hence the proof.


Note :
Myopia(near-sightedness) is corrected by using a Concave Lens of suitable power. Hypermetropia(farsightedness) is corrected by using a convex lens of suitable power. A spherical mirror is a mirror which has the shape of a piece cut out of a spherical surface. There are two types of spherical mirrors: concave, and convex.