Question

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.
A) 24 cm
B) 12 cm
C) 6 cm
D) 18 cm

Answer 1

Hint: Focal length of a mirror is the distance between pole and focal point of mirror.
Focal point is the point where all parallel rays converge or appear to converge.
As we know that for a small size spherical mirror the focal length is equal to half the radius of curvature.
So, we can write $f = R/2$
Where
$f$ = focal length
$R$ = Radius of sphere or sometimes known as radius of curvature

Complete step by step answer:

Consider a concave mirror as shown in figure
According to the second law of reflection i.e. angle of reflection is always equal to angle of incidence.
So, we can write
$\angle r = \angle i$
Where,
$\angle i = $ angle of incidence
$\angle r = $angle of reflection

We know that AB is parallel to the principal axis, $\angle i$ and $\angle \theta $ are alternate angles. So, they must be equal
$\because \angle i = \angle \theta - - - (1)$
We know in a triangle sides opposite to equal angles are equal.
So, In $\vartriangle CFB$
$FC = FB - - - (2)$
Since we know that the mirror is small and focused, it is very near to the principal axis.
So, we can write
$FB = FP - - - (3)$
Form (2) & (3) we get
$FC = FB = FP - - - (4)$
From figure we can see
$CP = FC + FP - - - (5)$
Further substituting the $FC = FP$ from (4)
$
  CP = FP + FP \\
  CP = 2FP \\
 $
We know $CP = R$ and $FP = f$
$
  R = 2f \\
  f = R/2 \\
 $
So,
$f = R/2 = 24/2 = 12$ cm

Note: $f = R/2$ is true even when the mirror is immersed in liquid as focal length doesn’t depend on the medium in which it is kept but, when $R$ is large enough then this formula is not valid.
Using this relation and triangle properties, we can also derive formula
 $\dfrac{1}{f} = \dfrac{1}{u} + \dfrac{1}{v}$
Where,
$f = $ focal length of mirror
 $u = $ position of object from pole
$v = $position of image from pole
Above formula is known as mirror formula