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Chapter 34: Q. 78 (page 994)

A spherical mirror of radius R has its center at C, as shown in FIGURE P34.78. A ray parallel to the axis reflects through F, the focal point. Prove that f = R/2 if∅<<1rad.

Short Answer

Expert verified

The statement is proved.

Step by step solution

01

Given information

We have given,

Radius of the spherical mirror=R

∅<<1rad.

A ray incident parallel to the principle axis and reflects through F.

we have to provef=R2.

02

Simplify

According to the law of reflection

incident angle = Reflected angle

∠i=∠r

As incident angle is parallel to the principle axis.

then using the figure, we can say that the alternative angles are equal.

∠i=∠∅=∠r

since two angle of the triangle △CFAare equal then we can say that it is a isosceles triangle.

CF=FA

and CM=MA=R2

Cos∅=R2×CFCF=Rsec∅2

since pole to C length is R.

R=f+R2sec∅f=R-R2sec∅

since,

∅<<1rad∅≃0sec∅=1

then we write,

f=R-R2f=R2

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