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Many modern technologies, such as computers and smartphones, are built on the principles of Physics Topics such as quantum mechanics and information theory.
Define the Term Optical Center. What is First Principal Focus and Second Principal Focus?
Definition: A lens is a portion of a transparent refracting medium bounded by two spherical surfaces or a spherical surface and a plane surface.
Lens is generally of two types:
- convex or con-verging lens and
- concave or diverging lens.
A lens which is thicker in the middle than towards its edges is called convex lens [Fig.(a)]. A lens which is thinner in the middle than towards the edges is called concave lens [Fig.(b)],
Different Types Of Lenses
Convex lens: Convex lenses may be of three types according to the shape of two surfaces forming it.
1. Bi-convex or double convex lens: It is one in which both the surfaces are convex [Fig.(a)]. The radii of curvature of both the surfaces may or may not be equal. If the radii of curvature are equal, the convex lens is called equi-convex lens.
2. Plano-convex lens: It is a lens with one surface plane and the other convex [Fig.].
3. Concavo-convex lens: Here one surface is concave and the other is convex [Fig.(c)], In this type of lens the radius of curvature of the convex surface is smaller than that of the concave surface.
Concave lens: Similarly concave lens may be of three types according to the shape of two surfaces forming it.
1. Bi-concave or double concave lens: This type of lens has both the surfaces concave [Fig.(a)].
The radii of curvature of both the surfaces may or may not be equal. If the radii of curvature are equal, the concave lens is called equicon-cave lens.
2. Plano-concave lens: This type of lens has one surface plane and the other concave [Fig.(b)],
3. Convexo-concave lens: Here one surface is convex and the other is concave [Fig.(c)]. The radius of curvature of the concave surface is smaller than that of the convex surface.
Action Of A Lens
Principal axis: The line passing through the centres of curvature of the two bounding surfaces of a lens is called the principal axis of the lens. If one surface of the lens is spherical and the other plane, then the perpendicular drawn from the centre of curvature of the spherical surface to the plane surface is the principal axis of the lens. [The definition of centre of curvature is given in the section 3.9]
Converging and diverging lenses: If the surrounding medium of a lens is rarer compared to the medium of the lens, then the parallel beam of rays after refraction through a convex or a concave lens appears to be converging or diverging respectively. Therefore, a convex lens is called a converging lens and a concave lens is called a diverging lens.
Convergence by convex lens: A convex lens may be imagined as being formed of two sets of truncated prisms arranged symmetrically on the opposite sides of a central parallel-faced rectangular slab [Fig.(a)], the prisms in each set being placed one above another with their bases turned towards the principal axis of the lens.
As we move further away from the principal axis the angle of refraction consequently keeps on increasing. Any parallel ray incident on a prism will bend by refraction through the prism towards its base. Since the refracting angles of the various prisms increase successively with their distance from the principal axis, the rays which fall on a prism at a distance from the axis are bent more than those which pass nearer to the axis. So a pencil of parallel rays is refracted by the combination of prisms i.e., by the convex lens so as to converge to a particular point on the principal axis. Hence, a convex lens is called a converging lens.
Divergence by concave lens: Let us refer to the Fig.(b). The concave lens may also be imagined as being formed of two sets of truncated prisms arranged symmetrically on the opposite sides of a central parallel-faced slab. The pile of the prisms on each side of the principal axis have their refracting angles turned towards the axis. So their bases are turned towards the edge of the lens. Therefore in this case a pencil of parallel rays after refraction through the prism will bend away from the axis being turned towards the bases of the prisms. So the emergent light will behave as a divergent beam. Hence, a concave lens is called a diverging lens.
Divergence by convex lens and convergence by concave lens: It is to be noted that usually a convex lens acts as a converging lens and a concave lens as a diverging one. These types of behaviour of the lenses are seen when the refractive indices of the material of the lenses are greater than that of the surrounding medium. But if the refractive index of the material of the lens is less than that of the surrounding medium i.e., the medium surrounding the lens is denser, the convex lens will diverge and a concave lens will converge the incoming parallel rays. [Fig(c) and (d)].
A Few Definitions
Centre of curvature: Generally the two surfaces of a lens are spherical. The two spherical surfaces are each a part of two spheres. The centres of the spheres are called the centres of curvature of the
If the two surfaces of a lens are spherical, the centres of curvature of the lens are at a finite distance. In [Fig.(a) and (b)], C1 and C2 are the centres of curvature of the lenses. If a surface of the lens is plane, the centre of curvature of that surface is at infinity.
Radius of curvature: The two spherical surfaces are each a part of two spheres. The radii of the spheres are called the radii of curvature of the lens.
If the two surface of a lens are spherical, each radius of curvature is finite. In Fig.(a) and (b) AC2 and BC1 are the radii of curvature of the lens. If one of the surfaces is plane, its radius of curvature is infinite.
Principal axis: For the lens having two spherical surfaces, the line passing through the centres of curvatures of the two bounding surfaces of a lens is called the principal axis of the lens. In Fig.(a) and (b), C1C2 is the principal axis of the lens. If one surface of the lens is spherical and the other is a plane, then the perpendicular drawn from the centre of curvature of the spherical surface to the plane surface is the principal axis of the lens [Fig.(c) and (d)].
Optical Centre: If a ray of light passes through a lens in such a way that the direction of emergence is parallel to the direction of incidence, the path of the ray inside the lens intersects the principal axis at a fixed point. This fixed point for a lens is called its optical centre.
In Fig. the incident ray AB and the emergent ray CD are parallel to each other. The refracted ray BC intersects the principal axis at O. So the point O is the optical centre of the lens.
It is to be noted that the incident ray AB and the emergent ray CD do not lie on the same straight line. The emergent ray CD is laterally displaced from the incident ray AB. The displacement will be small if the lens is a thin one. If the lens is very thin the displacement is so negligible that AB, BC and CD may be taken as same straight line.
So we can say that optical centre of a thin lens is such a point on its principal axis that a ray passing through it passes out straight without any displacement or deviation.
Optical centre is a fixed point: The optical centre of a lens is a fixed point on its principal axis. But the position of the point depends on its shape. It can be proved in the following way.
In Fig. LL’ is a Lens. C1 and C2 are the centres of curvature of the spherical surfaces LBL’ and LAL’ respectively. Q and R are two points on the spherical surfaces. If r1 and r2 are the
radii of curvature of the surfaces LBL’ and LAL’ then C1Q = C1B = r1 and C2R = C2A = r2
Let us join Q and R and let the line QR intersect the principal axis at O. Therefore, O is the optical centre of the lens. Thus the rays PQ and RS drawn in figure are parallel to each other.
Suppose, the thickness of the lens = AB = t.
Two tangent planes are drawn at Q and R of the two surfaces of the lens. We know that when a ray is refracted through a parallel glass slab the incident ray and the emergent ray are parallel. In this case the rays PQ and RS being parallel we can assume that the ray PQ is refracted through a parallel glass slab. So the tangent planes at Q and R will be parallel to each other.
The radius of curvature C1Q is perpendicular to the tangent plane at Q and the radius of curvature C2R is perpendicular to the tangent plane at R. Since the two tangent planes are parallel, therefore C1Q and C2R are parallel to each other.
So the triangles C1OQ and C2OR are similar.
∴ \(\frac{O C_1}{O C_2}\) = \(\frac{C_1 Q}{C_2 R}\) = \(\frac{r_1}{r_2}\)
So, the point O divides the thickness of the lens AB in a fixed ratio i.e., in ratio of the radii of curvature of the two surfaces.
Since t, r1 and r2 are constants, so the position of O is constant. i.e., the optical centre O of a lens is a fixed point.
In case of equi-convex and equi-concave lens: In this case since r1 = r2, therefore from equation (1), we get OB = OA. i.e., in this case the optical centre is situated on the principal axis within the lens and equidistant from both the surfaces.
In case of plano-convex and plano-concave lens: In this case one surface of the lens is plane. Therefore when r1 → ∞, then OA → 0 . Again when r2 → ∞, then OB → 0.
So, in this case the optical centre lies at the intersecting point of the spherical surface with the principal axis.
It is to be noted that the optical centre may be within the lens or outside, depending on the nature of the two surfaces. In case of concavo-convex and convexo-concave optical centre lies outside the lens [Fig.]. Wherever may be the position of the optical centre, its distance from any surface of the lens is proportional to the radius of curvature of the surface, because \(\frac{O B}{O A}\) = \(\frac{r_1}{r_2}\).
Principal focus: Suppose, a narrow beam of rays parallel to the principal axis is incident on a lens.
i) If the lens is convex, the beam of rays after refraction converges to a point on the principal axis [Fig.(a)]. This point is called the principal focus of the lens.
ii) If the lens is concave, the beam of rays after refraction appears to diverge from a point on the principal axis [Fig.(b)]. This point is called the principal focus of the lens.
In Fig.(a) and (b) the point F is the principal focus of the lens. A lens has two principal foci. Here in either case the point F is the second principal focus of the lens. In addition to this principal focus a lens has another principal focus which is called first principal focus.
First principal focus:
i) In case of convex lens, the first principal focus is a point on its principal axis such that the rays diverging from it emerge out parallel to the axis after refraction through the lens.
ii) In case of concave lens, the first principal focus is a point on its principal axis such that the rays directed towards it emerge out parallel to the axis after refraction through the lens.
In Fig.(a) and (b), the point F’ is the first principal focus.
The second principal focus is conventionally called the principal focus of a lens.
Focal length: The distance of the principal focus from the optical centre of a lens is the focal length F of that lens.
First principal focal length is the distance of first principal focus from the optical centre. Second principal focal length is the distance of second principal focus from the optical centre.
In Fig.’s the point O is the optical centre.
In Fig.(a) and (b), OF’ = first principal focal length of lens (f1)
In Fig.(a) and (b) OF = second principal focal length of lens (f2)
The second principal focal length is conventionally taken as the focal length of a lens.
The value of the focal length of lens depends on the colour of light, lens medium and also on the surrounding medium. If the media on both sides of the lens are same, then it can be proved that first principal focal length (f1) and second principal focal length (f2) are equal. But if the media are different on both sides, then these two lengths would be different.
Focal plane: A plane perpendicular to the principal axis of a lens drawn through the principal focus is known as the focal plane of a lens.
A lens has two focal planes corresponding to its two focal points. The focal plane through the first principal focus is called the first principal focal plane and the plane through the second principal focus is called the second principal focal plane.
Secondary focus: Suppose a beam of parallel rays inclined at a small angle with the principal axis of a lens is incident on it. The point on the focal plane to which the beam converges (in case of convex lens) and from which the beam appears to diverge (in case of concave lens) after refraction is called the secondary focus of the lens.
In Fig.(a) and (b) the point F’ is the secondary focus of the lens. It is to be noted that the principal focus of a convex or concave lens is a fixed point, but the secondary focus is not a fixed point. With the change of the angle of inclination of the incident rays with the principal axis of the lens, the position of the secondary focus changes. But the secondary focus always remains on the focal plane.
Aperture: The boundary line of the two planes of a lens is circular and the diameter of the circle is ordinarily called aperture of the lens. In Fig.(a) and (b) the diameter CD is the aperture of lens.
Thin lens: A thin lens is one of which the thickness at the principal axis is small compared with the radii of curvature of the two surfaces.