Question

How will you determine the focal length of a concave mirror?

Answer 1

Let's assume the aperture of the mirror to be very small. Let a ray AB of light parallel to the principal axis be incident on the concave mirror at B. the ray makes the angle of incidence, i with the normal BC at B, C being the center of curvature of the mirror. The ray is reflected along with BF with the angle of reflection, r so that
< i = < r

In accordance with the laws of reflection. As the incident ray AB is parallel to the principal axis PC, so the reflected ray Bf passes through the principal focus, F. in figure (i)

< ABC = < CBF
But < ABC = alternate< BCF
Therefore< CBF = < BCF
And the ΔFBC is isosceles
BF = FC ... (i)

Since the aperture is assumed to be very small, so the point of incidence B is close to P And BF = PF (approx.) ... (ii )

From (i) and (ii )
PF = FC

Adding PF to both sides
PF + PF= PF + FC 2PF = PC
Now since PF = f, the focal length of the mirror
And PC = R, the radius of curvature of the mirror

Therefore 2f = R
From here we can determine the focal length of the concave mirror i.e. half of radius of curvature