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Consider a Concave mirror as shown in the figure above.
A ray of light ABAB traveling parallel to the principal axis PCPC is incident on a convex mirror at BB. After reflection, it goes through the focus FF. PP is the pole of the mirror. CC is the center of curvature.
The distance PF=PF= focal length ff.
The distance PC=PC= radius of curvature RR of the mirror.
BCBC is the normal to the mirror at the point of incidence BB.
∠ABC = ∠CBF∠ABC=∠CBF (Law of reflection, ∠i=∠r∠i=∠r)
∠ABC = ∠BCF∠ABC=∠BCF (alternate angles)
=> ∠BCF = ∠CBF⇒∠BCF=∠CBF
∴ Delta FBC is an isosceles triangle.
Hence, sides BF = FC
For a small aperture of the mirror, the point B is very close to the point P,
=> BF = PF
∴ PF = FC= 1/2 PC
=> f = 1/2 R
Now consider a Convex mirror as shown in the figure below.
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A ray of light AB traveling parallel to the principal axis PC is incident on a convex mirror at B. After reflection, it goes to D and appear to be coming from the focus F.
The distance PF= focal length f.
The distance PC= radius of curvature R of the mirror.
Straight line NBC is the normal to the mirror at the point of incidence B.
∠ABN = ∠NBD (Law of reflection, ∠i=∠r)
∠CBF = ∠DBN (vertically opposite angles)
angleNBA=angleBCF (corresponding angles)
=> ∠BCF = ∠CBF
∴ Delta FBC is an isosceles triangle.
Hence, sides BF = FC
For a small aperture of the mirror, the point B is very close to the point P,
=> BF = PF
∴ PF = FC= 1/2 PC
=> f = 1/2 R
Thus, for a spherical mirror (both for a concave and for convex), the focal length is half of radius of curvature.
