 Introduction
 Types of Thin lenses
 Images Formed by Thin Lenses
 Linear Magnification
 The Lens Formula
 Uses of Lenses on Optical Devices
 Power of Lens
Introduction
 A lens is conventionally defined as a piece of glass which is used to focus or change the direction of a beam of light passing through it.
 They are mainly made of glass or plastic.
 Lenses are used in making spectacles, cameras, cinema projectors, microscopes and telescopes.
Types of Thin Lenses
 A lens which is thicker at its centre than at its edges converges light and is called convex or converging lens
 A lens which is thicker at its edges than at its centre diverges light and is known as concave or diverging lens.
Properties of Lenses
 Optical centre
 This is the geometric centre of a lens which is usually shown using a black dot in ray diagrams.
 A ray travelling through the optical centre passes through in a straight line.
 Centre of curvature
 This is the geometric centre of the circle of which the lens surface is part of.
 Since lenses have two surfaces there are two centres of curvature.
 C is used to denote one centre while the other is denoted by C^{1} .
 Principal axis
 This is an imaginary line which passes through the optical centre at right angle to the lens.
 Principal focus
 this is a point through which all rays travelling parallel to the principal axis pass after refraction through the lens.
 A lens has a principal focus on both its sides.
 F is used to denote the principal focus
 Focal length
 this is the distance between the optical centre and the principal focus. It is denoted by ‘f’ .
 this is the distance between the optical centre and the principal focus. It is denoted by ‘f’ .
Note:
 The principal focus for a converging lens is real and virtual for a diverging lens.
 The principal focus is not always halfway between the optical centre and the centre of curvature as it is in mirrors.
Images Formed by Thin Lenses
 The nature, size and position of the image formed by a particular lens depends on the position of the object in relation to the lens.
Construction of Ray Diagrams
Three rays are of particular importance in the construction of ray diagrams.
 A ray of light travelling parallel to the principal axis passes through the principal focus on refraction through the lens. In case of a concave lens the ray is diverged in a way that it appears to come from the principal focus.
 A ray of light travelling through the optical centre goes undeviated along the same path.
 A ray of light travelling through the principal focus is refracted parallel to the principal axis on passing through the lens. The construction of the rays is illustrated below.
Images Formed by a Converging Lens
 Object between the lens and the principal focus.
 Image formed behind the object
 Virtual
 Erect
 Magnified  Object at infinity.
 Image formed at the principal focus of the lens
 Real
 Inverted
 Diminished  Object at the principal focus (at F).
 Image is at infinity.  Object between the principal focus (F) and 2F.
 Image situated beyond 2 F
 Real
 Inverted
 Magnified  Object at 2F.
 Image is formed at 2F
 Real
 Inverted
 Same size as the object  Object beyond F.
 Image moves nearer to F as object shifts further beyond 2F
 Real
 Inverted
 Diminished
Images Formed by a Diverging Lens
 Images formed by diverging lens are always erect, virtual and diminished for all positions of the object.
Linear Magnification
 The linear magnification produced by a lens defined as the ratio of the height of the image to the height of the object, denoted by letter ‘m’ ,therefore;
m = ^{height of the image}/_{height of the object}.  Magnification is also given by = distance of the image from the lens/ dist. of object from lens.
m = v/u
Example
 An object 0.05 m high is placed 0.15 m in front of a convex lens of focal length 0.1 m. Find by construction, the position, nature and size of the image. What is the magnification
Solution
Let 1 cm represent 5 cm. hence 0.05 m = 5 cm = 1 cm – object height
0.15 m = 15 cm = 3 cm
0.1 m = 10 cm = 2 cm – focal length. Image formed is – image is beyond 2F
 Inverted
 Real
 Magnified  Magnification = ^{v}/_{u} = ^{30 cm}/_{15 cm} = 2.
 Image formed is – image is beyond 2F
The Lens Formula
 Let the object distance be represented by ‘u’ ,the image distance by ‘v’ andthe focal length by ‘ f ’, then the general formula relating the three quantities is given by;
^{1}/_{f} = ^{1}/_{u} + ^{1}/_{v} – this is the lens formula.
Examples
 An object is placed 12 cm from a converging lens of focal length 18 cm. Find the position of the image.
Solution
Since it is a converging lens f = +18 cm (realispositive and virtualisnegative rule)
The object is real therefore u = +12 cm, substituting in the lens formula, then ^{1}/_{f} = ^{1}/_{u} + ^{1}/_{v} or ^{1}/_{v} = ^{1}/_{f} – ^{1}/_{u} = ^{1}/_{18} – ^{1}/_{12} =  ^{1}/_{36}
Hence v =  36 then the image is virtual, erect and same size as the object.  The focal length of a converging lens is found to be 10 cm. How far should the lens be placed from an illuminated object to obtain an image which is magnified five times on the screen?
Solution
f = + 10 cm; m = ^{v}/_{u} = 5 hence v = 5u
Using the lens formula ^{1}/_{f} = ^{1}/_{u} + ^{1}/v → ^{1}/_{10} = ^{1}/u + ^{1}/_{5u} ( replacing v with 5u ) ^{1}/_{10} = ^{6}/_{5u}, hence 5u = 60 giving u = 12 cm ( the lens should be placed 12 cm from the illuminated object)  The lens of a slide projector focuses on an image of height 1.5m on a screen placed 9.0 m from the projector. If the height of the picture on the slide was 6.5 cm, determine,
 Distance from the slide (picture) to the lens
 Focal length of the lens
Solution
a) Magnification = ^{height of the image}/_{height of the object} = ^{v}/_{u} = ^{150}/_{6.5} = 900/uu = 39 cm (distance from slide to the lens). m = 23.09
b) ^{1}/_{f} = ^{1}/_{u} + ^{1}/_{v} = ^{1}/_{39} + ^{1}/_{90} = 0.02564 + 0.00111
^{1}/_{f} = 0.02675 ( reciprocal tables ) f = 37.4 cm.
Determining Focal Lengths

Determining Focal Length of a Converging Lens
Experiment : To Determine the Focal Length of a Converging Lens Using the Lens Formula.
Procedure Set up the apparatus as shown below
 Place the object at reasonable length from the screen until a real image is formed on the screen. Move the lens along the metre rule until a sharply focused image is obtained.
 By changing the position of the object obtain several pairs of value of u and v and record your results as shown.
u v uv ^{uv}/_{u+v}
 The value ^{uv}/_{u + v} is the focal length of the lens and the different sets of values give the average value of ‘f’.
 Alternatively the value ‘f’ maybe obtained by plotting a graph of ^{1}/_{v} against ^{1}/_{u}.
 When plotted the following graph is obtained.
 Since ^{1}/_{f} = ^{1}/_{u} + ^{1}/_{v} , at the yintercept ^{1}/_{u} = 0 , so that ^{1}/_{f} = ^{1}/_{v} or f = v.
 The focal length may therefore be obtained by reading off the yintercept and finding the reciprocal. Similarly at the xintercept, ^{1}/_{v} = 0 , therefore ^{1}/_{f} = ^{1}/_{u} or f = u hence the focal length can also be obtained by reading off the xintercept and finding the reciprocal.
 Set up the apparatus as shown below
Uses of Lenses on Optical Devices
 Simple microscope – it is also referred to as magnifying glass where the image appears clearest at about 25 cm from the eye. This distance is known as the least distance of distinct vision (D) or near vision.
Magnification in a Simple Microscope Magnification produced depends on the focal length of the lens. Lens of short focal give greater magnification than those of long focal length.
 The angle β subtended by the image at the eye is much greater than α which is the angle that the object would subtend at the eye when viewed without the lens.
 The ratio of the β toα is known as angular magnification or magnifying power of an instrument.
 The angular magnification is equal to linear magnification.
 To study the features of small animals in biology
 To look closely at small print on a map
 To observe crystals in physics and chemistry
 For forensic investigation by the police
 Compound microscope  It consists of two lenses with one nearer the object called the objective lens and the other nearer the eye called the eyepiece lens.
Uses of Compound Microscope Used to observe Brownian motion in science
 To study microorganisms and cells in biology
 Analyze laboratory tests in hospital.
 The astronomical telescope – It is used to view distant stars. It consists of two lenses; objective and eyepiece lenses. The objective lens has a large focal length while the eyepiece lens has a much shorter focal length.
 The camera – consists of a converging lens system, clicking button, shutter, diaphragm and a mounting base for the film all enclosed in a light proof box. The distance is adjusted to obtain a clear focus. The diaphragm has a hole called the aperture with an adjusting control knob to control the amount of light entering the camera. The shutter opens to allow light and close at a given time interval.
Uses of a Camera The sine camera is used to make motion pictures
 High speed cameras are used to record movement of particles
 Close circuit television cameras (CCTV) are used to protect high security installations like banks, supermarkets etc.
 Digital cameras are used to capture data that can be fed to computers.
 Human eye – It consists of a transparent cornea, aqueous humour and a crystallike lens which form a converging lens system. The ciliary muscles contract or relax to change the curvature of the lens. Though the image fo the image as upright. For distant objects ciliary muscles relax while near objects it contracts to control the focal length and this is known as accommodation. When at 25 cmaway an object appears clearest and this is known as least distance of vision or near point.
Common Eye Defects Short sightedness or myopia – result of a bulging cornea or an elongated eyeball. Images of distant objects form at locations in front of the retina. The defect is corrected by placing a concave (diverging) lens infront of the eye
 Long sightedness or hypermetropia – images are formed beyond the retina. The defect is corrected by placing a converging (convex) lens in front of the eye
 Presbyopia – this is the inability of the eye to accommodate and this occurs as the eye ages due to the weakening of the ciliary muscles. It can be corrected by the use a pair of spectacles.
 Astigmatism – this is a defect where the eye has two different focal lengths as a result of the cornea not being spherical. Corrected by the use of cylindrical lens.
 Colour blindness – caused by deficiency of colour detecting cells in the retina.
 Short sightedness or myopia – result of a bulging cornea or an elongated eyeball. Images of distant objects form at locations in front of the retina. The defect is corrected by placing a concave (diverging) lens infront of the eye
Power of Lens
 The power of a simple lens is given by the formula: Power = ^{1}/_{f} . The unit for power of a lens is diopter (D).
Example
Find the power of a concave lens of a focal length 25 cm.
Solution
Power = ^{1}/_{f} = ^{1}/_{0.25} = 4 D.
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