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Question

Write the relation between focal length and radius of curvature of a spherical mirror?

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Solution

For a concave mirror:
In figure (a),
BPC=PCF (alternate angles) and
BPC=PF (law of reflection,i=r)
Hence PCF=CPF
FPC is isosceles.
Hence, PF=FC
If the aperture of the mirror is small, the point P is very close to the point P,
then PF=PF
PF=FC=PC2
or f=R2

For a convex mirror:
In figure (b),
BPN=FCP (corresponding angles)
BPN=NPR (law of reflection, i=r) and
NPR=CPF (vertically opposite angles)
Hence FCP=CPF
FPC is isosceles.
Hence, PF=FC
If the aperture of the mirror is small, the point P is very close to the point P.
Then PF=PF
PF=FC=PC2
or f=R2
Thus, for a spherical mirror {both concave and convex), the focal length is half of its radius of curvature.

667035_629025_ans_9a9d507052de41e1a7cc90d38429fcd2.jpg

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