Focal Length and Radius of Curvature
Focal Length and Radius of Curvature
Relation Between Focal Length and Radius of Curvature:
The focal length f of a spherical mirror of small aperture is half of the radius of curvature r
Proof: Let us suppose a ray OX parallel to principal axis is incident on a spherical mirror. After reflection it passes through the focus in case of a concave mirror or appears to come from in case of convex mirror.CXN is the normal at point X.
In case of Concave Mirror:
......I [According to laws of reflection angle of incidence is equal to angle of reflection]
But .......II [ OX is || CP and XC is the transversal and alternate angles are equal]
Therefore from I and II we can say that
CF = FX [Sides opposite to equal angles are equal]
In case of Convex Mirror:
......I [According to laws of reflection angle of incidence is equal to angle of reflection]
But .......II [ OX is || CP and XC is the transversal and corresponding angles are equal]
and .......III [ Vertically opposite angles are equal]
Therefore from I, II and III we can say that
CF = FX [Sides opposite to equal angles are equal]
Therefore in both the cases CF = FX
For a mirror of small aperture X will be quite nearer to P so FX = FP
therefore CF = FP or we can say that F is the midpoint of CP, so FP is half of CP
As FP is the focal length and CP is the radius of curvature so we can say that
Illustration : A concave mirror is made by cutting a portion of a hollow glass sphere of radius 24 cm. Find the focal length of the mirror.
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