Relation between focal length and radius of curvature | Derivation
In ray optics, there are two types of cases. one is spherical mirror and other is lens. the relation between focal length and radius of curvature for both spherical mirror and lens are same. How? we discussed in later part of the article. Now first of all we have to know the meaning of focal length and radius of curvature in detail by diagram. So lets start with focal length.
Focal Length of spherical mirror
In above fig. Two different rays of light parallel to the principal axis incident on a polished concave portion. Due to the curve of mirror, rays reflect and passes through a point called focus(F). And the distance between pole on mirror to the focus of a point called focal length. In other words. Focal length is defined as the length between pole and focus on a principle axis. for example Let us find the focal length of a concave mirror. face a concave mirror to the sun. take a small piece of paper and fix it away from the mirror in such a way that the distance between the mirror and paper should be let 'f'. Now let the rays of the sun falls on a piece of paper passing through the mirror. After falling the rays on paper you will notice a small bright spot on the paper. Now mark that point where the bright spot form as 'O'. Now measure the distance between the pole of the mirror and to the point 'O'. This distance is approximately equal to the focal length. Focal length is denoted by 'f'. Similarly you can find focal length for convex mirror. But the difference is you have to take another object for demonstration instead of paper. focal length exist in spherical mirror as well as in lens. So lets find the focal length of a lens.
Focal length of a lens
If a beam of light is incident parallel to the principal axis on a convex lens, it converges to a point F2 on the principle axis on the other side of the lens (see fig. a). This point is known as second principal focus of convex lens. In another case if a parallel beam of light falls on the surface of concave lens. A beam of light diverges after passing through the lens (see fig. b). When we produce these diverges rays in backward direction. It meets at point F2 known as second principal focus.
As we see the incident rays after converges and diverges in both convex and concave lenses it passes through the point F2 known as focus of the lens. And if we measure the distance between optical center to the point F2 in both convex and concave it would be equal to focal length. So we can say that the distance between focus to the optical center called focal length.
Relation between focal length and radius of curvature
The general relation between focal length and radius of curvature for spherical mirror and for spherical lenses are same. That is, In both cases focal length is equal to the half of the radius of curvature. i.e, R = 2f. Apart from this relation there many similarities between them. the given following are some common similarities or relation between focal length and radius of curvature.
(1.) In concave mirror focal length is equal to the half of the radius of curvature.
(2.) In convex mirror focal length is equal to the half of the radius of curvature.
(3.) In concave lens and convex lens both focal length and radius of curvature exist on both the sides.
(4.) In convex lens and convex lens focal length is equal to the half of the radius of curvature.
(2.) In convex mirror focal length is equal to the half of the radius of curvature.
(3.) In concave lens and convex lens both focal length and radius of curvature exist on both the sides.
(4.) In convex lens and convex lens focal length is equal to the half of the radius of curvature.
Relation between focal length and radius of curvature derivation
In above fig. A ray of light AB is incident on a concave mirror and then reflect to passes through the focus F. Suppose AB is very-very close to the principal axis. Now from the law of reflection of light, BC will be the normal passing through the center of curvature.
As we know that, first law of reflection of light says that angle of incident is equal to the angle of reflection.
∠i = ∠r ------------------ (1)
As we can see in fig. AB is also parallel to the principal axis. So that,
∠i = ∠𝛂 ------------------ (2)
Now, from equation (1) and (2), we get,
∠r = ∠𝛂 ------------------ (3)
Now, we take triangle BFC,
As we know that angle r is equal to angle alpha. i.e, ∠i = ∠𝛂. therefore, ⧍BFC is an isosceles tringle. So that CF = BF. because we know that equal angles of opposite sides of an isosceles triangle are equal.
CF = BF -------------- (4)
Now, as we suppose that AB line is nearly close to the principal axis. So that the point B and P should be extremely close to each other. So we can say that BF is approx coincide to PF.
BF = PF ---------- (5)
From equation (4) and (5) we get,
CF = PF
CF = PF = focal length
CF + PF = Radius of curvature
PF + PF = R
2PF = R
2 focal length = Radius of curvature
Hence, we conclude from this derivation that twice of focal length is equal to the radius of curvature.