Focal Length and Radius of Curvature


 
 
Concept Explanation
 

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

f=frac{r}{2}

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:

angle OXC = angle CXF                ......I    [According to laws of reflection angle of incidence is equal to angle of reflection]

But angle OXC = angle XCF         .......II   [ OX is || CP and XC is the transversal and alternate angles are equal]

Therefore from I and II we can say that

angle CXF = angle XCF

CF = FX      [Sides opposite to equal angles are equal]

In case of Convex Mirror:

angle OXN = angle NXY                ......I    [According to laws of reflection angle of incidence is equal to angle of reflection]

But angle OXN = angle XCF         .......II   [ OX is || CP and XC is the transversal and corresponding angles are equal]

and angle NXY = angle FXC        .......III   [ Vertically opposite angles are equal]

Therefore from I, II and III we can say that

angle CXF = angle XCF

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

focal;length;(f)=frac{Radius;of;curvature(r)}{2}

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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