Lens is a piece of glass bounded by two non-parallel curved surfaces, or by one surface curved and the other straight. The curved surfaces of the thin lenses we will consider are spherical.
Lenses were first used by the Chinese and Greeks and later, in medieval times, by the Arabs. Lenses of many different types play an important part in our lives. They are used in cameras, telescopes, microscopes and projectors, and they enable millions of people to read comfortably and see clearly.
1.1.1. TYPES OF LENSES
Lenses are of two types: ‘Converging (Convex) and Diverging (Concave)’. Converging lenses bring light rays together. Diverging lenses spread light rays apart. The two are easily distinguished by their shape. Converging lenses are thickest at the centre whereas diverging lenses are thinnest at the centre. Some common types of lenses are illustrated
Figure 9.56: Converging lenses
Figure 9.57: Diverging lenses
Imagine that a lens were made up of parts of prisms. Each of them would bend rays towards its thicker ends. Those prisms at the edges would bend the rays more than those in the middle. The rays would converge for a bi-convex lens and would diverge for a bi-concave lens.
The line passing through the centres of curvature of the two spherical surfaces of a lens is called the ‘principal axis’. The optical centre is the point on the principal axis where all the rays passing through it are not refracted but continue in a straight line.
For lens with both surfaces having equal curvature the optical centre is at the centre of the lens. The symbol for the optical centre is (O). The point to which rays parallel and close to the principal axis converge or from which they appear to diverge after refraction is the principal focus (F). The distance from the optical centre to the principal focus is called the focal length (f).
Figure 9.58: Refraction of light by lenses
Experiment 9.10: Measurement of the focal length of a lens.
Aim: To determine the focal length of a convex lens
Materials: convex lens, plane mirror, wire, pearl electric lamp, white screen
Procedure:
a. Place a lens in a stand close to a mirror as shown in the figure below. For the object use an illustrated crosswire placed in a hole in a white screen. Light passing through the lens is reflected back by the mirror. Adjust the position of the lens and mirror until the image of the crosswire comes into sharp focus next to the hole. Measure the distance between the lens and the screen. It is the focal length. Repeat the experiment five times and get the average
Figure 9.59: Focal length of a converging lens.
When the illuminated object is situated at the principal focus of the lens the rays from it emerge parallel to each other after refraction. They fall normally to the mirror and are refracted back and are therefore reflected back and converging again at the principal focus.
b. Set up the apparatus as follows:
Figure 9.60: Focal length of a converging lens
Adjust the position of the screen so that a sharp image of the object is formed on it. Measure the distance of the object to the lens u and that of the image to the lens v. Measure the size (height) of the object and the image. The focal length, object distance and image distance are related by the formula
Calculate the focal length using this formula.
The ratio of the size of the image (i) to the size of the object (O) is called the ‘magnification’. We could prove that it is also equal to the ratio of the image distance t the object distance. Magnification can be stated as
Figure 9.61: Magnification
Repeat the experiment several times, fill in a table, and find the average value of the focal length.
u (cm) v (cm) O (cm) l (cm) l/O u/v f (cm)
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1.1.3. LOCATING THE IMAGE BY MEANS OF DIAGRAMS
Images formed by lenses cans be located using ray diagrams. The following rules are applied when drawing ray diagrams:
1. Rays through the optical centre continue in a straight line;
2. Rays parallel to the principal axis are refracted through the principal focus;
3. Rays from the principal focus leave the lens parallel to the principal axis after refraction.
Note that a lens has two principal foci, one on each side)
Figure 9.62: Images from ray diagrams
a. The image of an object at an infinite distance is located as shown in the figure above (a). This shows that the image is in the focal plane of the lens, smaller than the object (diminished) real and inverted.
b. If the object is between infinite and 2F (twice the focal length) its image is located between F and 2F, diminished, real and inverted. The lens is used this way in a camera.
c. For an object at twice the focal length its image is located at 2F, same size as object, real and inverted.
d. If an object is between F and 2F its image is located beyond 2F, real and inverted and enlarged. A lens is used this way in a projector.
e. The object is at F, its image is located at infinite.
f. The figure shows the location of the image of an object at less than F. the image is on the same side as the object, enlarged, virtual and erect. In this case the lens is acting as a magnifying glass.
Figure 9.63: Image formed by a magnifying glass
1.1.4. THE THIN LENS EQUATION.
Just as we did for the curved mirrors, we can derive an equation for thin lenses, relating the image distance, the object distance, and the focal length.
Figure 9.64:
In the diagram, the triangles AOF and EDF are similar. Therefore
We now divide both sides by
This is the lens equation. It is exactly the same as the mirror equation. As was the case with the mirror equation, a sign convention must be used with the LE.
Image formed by concave lenses
The two standard rays have been used in the figure below to show how the image is formed by a concave lens:
Figure 9.65:
From the diagram, you can see that the image is virtual (not projected on the screen), erect (upright), and diminished (smaller than the object).
1.1.5. SIMPLE OPTICAL INSTRUMENTS
1. Camera
A conventional camera is a light-tight box in which a convex lens forms a real image on a film. The image is smaller than the object and nearer to the lens. The film contains chemicals that change on an exposure to light; it is developed to give a negative. From the negative a photograph is made by printing.In a digital camera the image is recorded electronically rather than on a film
Figure: 9.66: Structure of an optical camera
a. Focusing
In simple cameras the lens is fixed and all distant objects, i.e. beyond about 2 metres, are in reasonable focus. Roughly how far from the film will the lens be if its focal length is 5cm?
In other cameras exact focusing of an object at a certain distance is done by altering the lens position. For near objects it is moved away from the film, the correct setting being shown by a scale on the focussing ring.
b. Shutter
When a photograph is taken, the shutter is opened for a certain time and exposes the film to light entering the camera. Sometimes exposure times can be varied and are even in fractions of a second, e.g. 1/1000, 1/60, etc.. Fast-moving objects require short exposures.
c. Stop
The brightness of the image on the film depends on the amount of light passing through the lens when the shutter is opened and is controlled by the size of the hole (aperture) in the stop. In some cameras this is fixed but in others it can be made larger for a dull scene and smaller for a bright one.
The aperture may be marked in f-numbers. The diameter of an aperture with f-number 8 is
2. The film projector
The projector produces an enlarged image of a slide or film on a screen. The object (slide) is placed between the focal length and twice the focal length of the projector lens so that it produces a large image at quite some distance from the projector. This image is inverted and so the slide should be placed upside down in the projector.
The essential parts of the projector are shown in the figure below. The slide illuminated by a very powerful lamp which is situated at the centre of curvature of a concave mirror, which reflects any straying light rays.
Two Plano-convex lenses (condenser system) concentrate the light on the object to ensure a bright image. To focus the image, the distance of the projector lens from the object is adjusted by means of sliding tube in which the lens is mounted
3. Optical system of the Human eye
The action of the eye as an optical instrument can be compared to that of the simple lens camera. The front part of the eye is protected by the transparent cornea which owing to its curved shape, aids in focussing the rays of the light into the eye. The cornea is part of the Sclerotic, the white outermost layer of the three layers of the eye. Next comes the Choroid layer which completely encloses the eye except in front where there is a small opening called the pupil.The coloured part of the choroid is the iris,which surrounds the pupils. Muscles in the iris adjust the size of the pupil according to the intensity of light. The innermost layer is the retina on which images are formed. Electrical impulses are then relayed to the brain from the retina via the optic nerve.
Behind the pupil lies a convex crystalline lenssupported by the ciliary muscles fastened to the choroid layer. Between the lens and the cornea is a watery substance called the aqueous humour. The interior of the eyeball is filled with a jelly-like substance called the vitreous humour which keeps the eyeball firm. The Ciliary muscles vary the thickness and curvature of the lens, thus changing its focal length, so that both near and far objects may be focuses on the retina. This process is called “accommodation”.
Human eye
Figure: 9.68: Parts of the eye
1.1.6. DEFECTS IN VISION AND THEIR CORRECTION
1. Farsightedness/ Long sightedness
Farsightedness, or Hypermetropia, is a defect in the eye resulting in the inability to see nearby objects clearly. It usually occurs because the distance between the lens and the retina is too small, but it can occur if the cornea-lens combination is too weak to focus the image on the retina. This defect can be corrected by glasses or contact lenses that converge the rays of light so that the lens can focus the image clearly.
Figure: 9.69: Correction of the farsightedness
If a person grows older, the eye lenses lose some of their elasticity, resulting in a loss in accommodation. This kind of farsightedness is known as Presbyopia. It, too, can be corrected by glasses with converging lenses. Distant vision is usually unaffected, so bifocals are used. These have converging lenses in the lower portion of each frame, convenient for reading and other close work for which the eyes are lowered.
2. Nearsightedness/ Short sightedness
In nearsightedness, or Myopia, the distance between the lens and retina is too great or the cornea-lens combination is too strong. As a result, parallel light rays from distant objects are focuses in front of the retina. Correction is accomplished by means of glasses or contact lenses with diverging lenses. These diverge the light rays so that the eye lens can focus the image clearly on the retina.
Figure: 9.70: Correction of the farsightedness
3. Astigmatism
Astigmatism occurs when either the cornea or the lens of the eye is not perfectly spherical. As a result, the eye has different focal points in different planes. The image may be clearly focused on the retina in horizontal plane, for example, but in front of the retina in the vertical plane. Astigmatism is corrected by wearing glasses with lenses having different radii of curvature in different planes. They are commonly cylindrical lenses. Astigmatism is tested for by looking with one eye at the pattern below; since the astigmatismatic eye focuses rays in one plane at a shorter distance than another, sharply focused lines in the pattern will appear black whereas those out of focus will appear blurred.
Figure: 9.71: One test for astigmatism uses a wheel with numbered spokes. By noting which lines appear blurred to the patient, the oculist can determine what kind of astigmatism exists.
The power of a lens is simply the reciprocal or inverse of the focal length of a lens in metres. The quantity used by opticians to describe and prescribe lenses is the power of a lens (P) instead of focal length. The unit of measurement of the optical power of a lens is dioptre symbolised as dpt or (D)
For example, if your optometrist prescribes a corrective lens for farsightedness having a power of 2.5 D, he or she means a converging lens with a focal length of
To ‘see’ your blind spot, hold the book at arm’s length, cover your left eye, and focus your right eye on the apple. By changing the distance between your eye and the book you can make the orange disappear as its image falls on the point of the retina where the optic nerve begins. The blind spot is outside the area of normal vision and is usually not noticed.
