Question

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$\varnothing <<1\mathrm{rad}.$

Short answer

The statement is proved.

Step-by-step solution

Given information

We have given,

Radius of the spherical mirror=R

$\varnothing <<1rad.$

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

we have to prove$\mathrm{f}=\frac{\mathrm{R}}{2}.$

Simplify

According to the law of reflection

incident angle = Reflected angle

$\angle \mathrm{i}=\angle \mathrm{r}$

As incident angle is parallel to the principle axis.

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

$\angle \mathrm{i}=\angle \varnothing =\angle \mathrm{r}$

since two angle of the triangle $△\mathrm{CFA}$are equal then we can say that it is a isosceles triangle.

CF=FA

and CM=MA=$\frac{\mathrm{R}}{2}$

$$\begin{gathered} \mathrm{Cos}\varnothing =\frac{\mathrm{R}}{2\times \mathrm{CF}} \\ \mathrm{CF}=\frac{\mathrm{R}\sec \varnothing }{2} \end{gathered}$$

since pole to C length is R.

$$\begin{gathered} \mathrm{R}=\mathrm{f}+\frac{\mathrm{R}}{2}\sec \varnothing \\ \mathrm{f}=\mathrm{R}-\frac{\mathrm{R}}{2}\sec \varnothing \end{gathered}$$

since,

$$\begin{gathered} \varnothing <<1\mathrm{rad} \\ \varnothing \simeq 0 \\ \sec \varnothing =1 \end{gathered}$$

then we write,

$$\begin{gathered} \mathrm{f}=\mathrm{R}-\frac{\mathrm{R}}{2} \\ \mathrm{f}=\frac{\mathrm{R}}{2} \end{gathered}$$