BPHCL-138 Waves and Optics: Laboratory - PDF

MPDD/IGNOU/P.O./12K/JUNE, 2022 BPHCL-138 Indira Gandhi National Open University...

MPDD/IGNOU/P.O./12K/JUNE, 2022 BPHCL-138 Indira Gandhi National Open University WAVES AND OPTICS: School of Sciences LABORATORY BPHCL-138 WAVES AND OPTICS: LABORATORY ISBN:  BPHCL-138 WAVES AND OPTICS: Indira Gandhi National LABORATORY Open University School of Sciences EXPERIMENT 1 Refractive Index of the Material of a Prism using a Spectrometer 5 EXPERIMENT 2 Investigations with Polarised Light using a Polarimeter 21 EXPERIMENT 3 Cauchy’s Constants of the Material of a Prism 35 EXPERIMENT 4 Wavelength of Sodium Light using Fresnel’s Biprism 43 EXPERIMENT 5 Wavelength of Sodium Light using Newton’s Rings 55 EXPERIMENT 6 Wavelength of Sodium / Mercury Light using a Plane Diffraction Grating 65 EXPERIMENT 7 Dispersive Power of a Prism 73 EXPERIMENT 8 Resolving Power of a Prism 83 EXPERIMENT 9 Diffraction from a Wire 93 EXPERIMENT 10 Study of Single Slit Diffraction of a Laser using Photo Sensor 103 Experiment 1 Refractive Index of the Material of a Prism using a Spectrometer EXPERIMENT 1 REFRACTIVE INDEX OF THE MATERIAL OF A PRISM USING A SPECTROMETER Structure 1.1 Introduction 1.4 Experimental Procedure Expected Skills Vernier Constant of Spectrometer Measurement of the Angle of Prism 1.2 Refractive Index Refraction of Light Measurement of the Angle of Minimum Deviation Refraction of Light through a Prism Calculations and Result 1.3 Spectrometer Parts of a Spectrometer Setting up the Spectrometer 1.1 INTRODUCTION In your school physics, you have studied reflection and refraction of light. When a ray of light is incident on a boundary separating two optically different media, a part of it is reflected at the boundary and the remaining part bends from its original path as it enters the second medium. The light is then said to have refracted. The extent of refraction is given by Snell’s law and it is characterised by a parameter called refractive index of the medium. Higher the refractive index, greater is the bending of light. In your +2 class physics, you must have studied several phenomena associated with refraction of light in everyday life. The rainbow in the sky is the most vivid example of refraction in nature. Similarly, appearances of an oasis in a desert and water on a coal tar road on a hot summer day are other familiar examples. You should list a few more examples of refraction and discuss with your counsellor. A prism is a very useful and versatile optical device that is used in a variety of optical instruments such as binoculars, cameras, telescopes and submarine periscopes. A prism has a three-dimensional (3D) shape with two identical faces, which are called bases. In the physics laboratory, you will get a prism with equilateral triangular bases, though in the market 5 BPHCL-138 Waves and Optics: Laboratory polygon base prisms are also available. The other faces of a prism are rectangular. No dispersion or refraction takes place through the base as it is grounded. The refractive index of the material of the prism plays an important role in the design and manufacturing of optical instruments. Newton showed that a prism disperses or breaks up white light into its seven constituent colours. Can you name these colours? [Remember, VIBGYOR (Violet, Indigo, Blue, Green, Yellow, Orange and Red).] The dispersion of light due to prism depends on the extent of refraction, which, in turn, depends on the wavelength of different colours constituting white light. It means that if we use a white light, the light emerging from a prism will show seven colours (wavelengths). However, in this experiment, we will use a sodium lamp, which is considered mono- chromatic (but, not strictly due to being a doublet of wavelengths 589.0 nm and 589.6 nm), and learn to determine the refractive index of the material of a given prism. For determining the refractive index, we use a spectrometer to measure angles of dispersion, angle of minimum deviation of refracted rays and angle of the prism. A spectrometer is an optical instrument which enables us to observe spectrum of light given out by a source of light. However, in the present experiment, we shall make use of the spectrometer for measuring angles with high degree of precision. Expected Skills After performing this experiment, you should be able to: ™ identify the refractive faces of a prism: ™ identify the main components of a spectrometer; ™ set up the spectrometer for experiment; ™ determine the angle of the prism and angle of minimum deviation; and ™ calculate the refractive index of the prism. You will require the following apparatus for this experiment. Apparatus Required Spectrometer, prism, spirit level, sodium lamp and a reading lens (magnifying glass). 1.2 REFRACTIVE INDEX The refractive index is a property of the material which determines the extent of refraction/bending of a ray of light passing through it. This parameter plays an important role in image formation by optical devices. From your school physics, you are familiar with the phenomenon of refraction of light and the concept of refractive index. But, for the sake of completeness, we briefly recall the basic concepts related to the refraction of light in the following paragraphs before you learn how to perform the experiment for determining the refractive 6 index. Experiment 1 Refractive Index of the Material of a Prism using a Spectrometer 1.2.1 Refraction of Light When light travels from one medium to the other, refraction of light refers to bending of a ray of light at the interface separating two optically different media. You may recall that when light travels from an optically rarer medium to an optically denser medium, it bends towards the normal. On the other hand, when light travels from an optically denser medium to an optically rarer medium, it bends away from the normal. The two laws governing refraction of light are i) The incident ray, the refracted ray and the normal at the point of incidence lie in the same plane. ii) The ratio of the sine of the angle of incidence to the sine of the angle of refraction is a constant for any two media. This is also known as Snell’s law. Refer to Fig. 1.1. It shows a ray of light passing from medium a to medium b and if we denote the angle of incidence and angle of refraction by i and r respectively, then accordingly to Snell’s law, we can write sin i constant (1.1) sin r Fig. 1.1: Refraction of light at the interface of two optically different media. The constant in Eq. (1.1) is referred to as the refractive index of medium b with respect to medium a and is denoted as aP sin i b (1.2) sin r If the first medium (the medium of incidence) is air, the refractive index of the second medium is simply denoted as P. In this case, Eq. (1.2) is written as sin i P (1.3) sin r The value of P for air is taken as unity. Snell’s law is an empirical law based on observations. The concept of refractive index was put on a sound theoretical foundation by Maxwell when 7 BPHCL-138 Waves and Optics: Laboratory he gave the theory of electromagnetic waves. According to this theory, the refractive index in a medium is given as the ratio of the velocity of light in vacuum and the velocity of light in that medium. Mathematically, we can write Velocity of light in vacuum P (1.4) Velocity of light in the specified medium Snell’s law is a natural consequence of Maxwell’s electromagnetic theory. However, at present, we make use of the definition of the refractive index given in Eq. (1.1) because it is easier to measure the angles of incidence and refraction required to determine refractive index. 1.2.2 Refraction of Light through a Prism A prism is a transparent wedge shaped structure (usually) made of glass with three rectangular and two triangular surfaces. The triangular faces are equilateral triangles and one of the rectangular surfaces is grounded. A prism can be made from any material that is transparent for the light for which it has been designed. The materials, other than glass, used for making prisms are plastic and fluorite. Fig. 1.2: A Prism To understand the unique geometrical shape of a prism, refer to Fig. 1.2. The triangles ABC and DEF are equilateral triangles and are parallel to each other. Each of these triangular faces is called base of the prism. The side faces Though the most ABED, ACFD and BCFE of the prism are parallelograms and are called sides commonly used prisms of the prism. in physics laboratory is triangular in shape, it Now, refer to Fig. 1.3 which shows the top view of the triangular prism ABC. can have a variety of The angle ‘A is called angle of prism. A ray of light PQ incident on the face shapes such as right AB gets refracted along QR inside the prism. At R (located on the face AC), it angle prism (used in again undergoes refraction and emerges out along RS. medical equipment, endoscope), penta Let i and e denote the angle of incidence and angle of emergence, prism (used in display systems), wedge prism respectively. The respective angles of refraction at Q and R are r1 and r2. (used in lasers), etc. depending upon the If PQ and SR are extended within the prism, they would meet at G (Fig. 1.3). requirements. The angle HGS (‘D) is known as the angle of deviation and it is denoted by G. Note that the angle of deviation is the angle through which the incident ray PQ has been deviated (refracted or bent) by the prism from its original direction PQGH. 8 Experiment 1 Refractive Index of the Material of a Prism using a Spectrometer Fig. 1.3: Refraction of light due to a prism. In order to determine the refractive index of prism material by any method using Snell’s law [Eq. (1.3)], we need to measure the values of i and r. As such, measurement of the angle of refraction, r is practically difficult, so we look for some alternative method. An alternative and relatively easy method is by measuring a quantity called the angle of minimum deviation. Let us now learn what we mean by the angle of minimum deviation and how it helps in determining the refractive index of the prism material. Minimum Deviation For thick prisms, that is, a prism whose prism angle is relatively large (a 60 degrees), the angle of deviation (G) is large for small angle of incidence (i). As the value of i is gradually increased, the angle of deviation decreases progressively, till it reaches one particular value of angle of incidence, for which the value of angle of deviation becomes minimum. If the angle of incidence is increased beyond this value, the angle of deviation begins to increase. This angle for which deviation of the incident ray is minimum, is known as the angle of minimum deviation and it is denoted by G m. When the angle of incidence is such that the angle of deviation has its minimum value, the incident ray passes through the prism symmetrically. That is, the refracted ray QR inside the prism becomes parallel to the base BC of the prism and i = e. Before proceeding further, you should answer the following SAQ. SAQ 1 - Angle of incidence and angle of emergence of a prism Show that the angle of incidence and the angle of emergence of a prism are equal to each other when the angle of deviation is minimum. Relation between Refractive Index, Angle of Prism and Angle of Minimum Deviation To derive the relation between refractive index, angle of the prism and angle of minimum deviation, we note from Fig. 1.3: i e A  Gm (1.5) and A r1  r2 (1.6) 9 BPHCL-138 Waves and Optics: Laboratory SAQ 2 - Angle of minimum deviation of a prism Using Fig. 1.3, establish the results contained in Eqs. (1.5) and (1.6). Now, when the prism is in the position of minimum deviation, the refracted ray passes symmetrically through the prism and we can write from Fig. 1.3 that i e i (1.7) and r1 r2 r (1.8) Using Eq. (1.7) in Eq. (1.5), we can write 2i A  G m A  Gm ? i (1.9) 2 Further, using Eq. (1.8) in Eq. (1.6), we can write A r1  r2 2r A or r (1.10) 2 From Eq. (1.3), we have sini P sinr Substituting for i and r from Eqs. (1.9) and (1.10), we get the expression for refractive index in terms of easily measurable quantities (namely, angle of prism and angle of minimum deviation): A  Gm sin P 2 (1.11) sin(A / 2) From Eq. (1.11), we note that we can easily determine the value of P of the material of the given prism once we determine the values of the angle of the prism A and, the angle of minimum deviation G m. To determine the value of A and G m , we use a prism spectrometer. So, we now discuss the construction and use of a basic laboratory spectrometer to make measurements. 1.3 SPECTROMETER Spectrometer is an optical instrument used to study the spectra of different sources of light and to determine the refractive indices of materials. A typical laboratory spectrometer is shown in Fig. 1.4. Fig. 1.4: A typical laboratory spectrometer. 10 Experiment 1 Refractive Index of the Material of a Prism using a Spectrometer 1.3.1 Parts of a Spectrometer A spectrometer consists of three main parts: (i) A collimator, (ii) a telescope and (iii) a prism table. We now discuss their construction and working. i) Collimator The collimator is a device used to produce a parallel beam of light. It consists of a long cylindrical tube having a vertical slit S of adjustable length and width at the outer end and a convex lens at the inner end of the tube. The distance between the slit and the lens can be so adjusted that the slit is at the focus of the lens. Then the length of the tube becomes equal to the focal length of the lens of the collimator. The slit is so kept that it faces the source of light. The collimator is rigidly fixed to the base of the spectrometer. The eye piece of an ii) Telescope optical instrument is a The telescope in a spectrometer is a simple astronomical type telescope simple magnifier. However, a single lens with an eye piece of Ramsden type provided with cross wires at one end of is inadequate as it the tube and an objective lens at the other end placed coaxially. The gives rise to aberration. distance between the objective lens and the eyepiece can be adjusted In Ramsden eye piece, using screw to obtain a clear image at the cross wires when a parallel two plano-convex beam of light coming out of the collimator is incident on the objective of the lenses made of same telescope. The telescope can rotate about the same vertical axis as the material and having prism table. The telescope is also provided with radial screws for fixing it in same focal length are used to minimise the desired position. aberration effects. The base of the telescope is fitted with two vernier scales, which move over a circular graduated main scale. This arrangement enables us to measure the angle of rotation of the telescope very accurately. iii) Prism Table The prism table is a circular table of adjustable height and can rotate about the same vertical axis as the telescope. The prism table carries three screws at its bottom. These are used to level it. The prism table is provided with a vertical stand so that it can be moved up or down. The prism is placed on the table such that its refracting surfaces are perpendicular to the plane of the table. To summarise, light enters the collimator through an adjustable slit and the collimating lens produces a parallel beam of light, which is then made to pass through a prism (or diffraction grating) placed on the prism table. On passing through the prism (or grating), the light bends through some angle and is then viewed through the telescope that can be moved about a vertical axis. The angle through which light bends can be very accurately measured using a vernier scale, which moves on a circular graduated main scale attached to the telescope. Before using a spectrometer for measurements, certain adjustments have to be made. A proper setting up of the spectrometer is very important for accurate measurement in an experiment with spectrometer. You must master it for obtaining accurate results. We now discuss how to set up a spectrometer, step by step. 11 BPHCL-138 Waves and Optics: Laboratory 1.3.2 Setting up the Spectrometer The basic objective of setting up the spectrometer for experiment is to align its different components with each other. The light beam emerging from the prism should be incident on the telescope objective in such a manner that a clear image is formed on the cross wires of the telescope. The adjustments required before working with a spectrometer are: i) the axis of the spectrometer is to be made vertical so that it coincides with the vertical axis of rotation of the prism table; ii) the axes of the collimator and the telescope should be horizontal so that they are perpendicular to the axis of the prism table; iii) the refracting faces of the prism should be vertical so that these are parallel to the axis of rotation of the telescope; and iv) the collimator and the telescope should be adjusted for parallel rays. While working with spectrometer, you should keep in mind that all adjustable parts of the spectrometer should move with very little effort; do not force any part of the spectrometer for movement. If you move a part by force, you may deform it or even break it. Some parts can probably be tight as it may be clamped. In such a situation, check it and locate the appropriate knob to loosen it. The procedure to adjust different components of the spectrometer is as follows: i) Levelling: To level the telescope, take a spirit level and keep it on the telescope tube along its length. Use the screws provided at the base of the spectrometer to bring the bubble of the spirit level at the centre. Rotate the telescope tube by 180 degrees and again use the base screws to bring the bubble at the centre. Repeat this process until the spirit level bubble remains at the centre for different positions of the telescope. This levelling ensures that the telescope is perpendicular to the vertical axis of the prism table. Similarly, you can level the collimator tube using the spirit level and the screws provided with the collimator tube. Further, to level the prism table, you can use one of the following two methods: a) Prism table can be levelled using a spirit level. Place the spirit level at the centre of the prism table and bring the bubble of the spirit level at the centre by adjusting the screws provided at the bottom of the prism table. Change the position of the spirit level on the prism table and again, use the prism table screws to bring the bubble at the centre. Repeat this process for different positions of the spirit level on the prism table. By this adjustment, you have made the prism table horizontal. Thus, when you place the prism on this table, its refracting surfaces will be perfectly vertical, that is, they will be perpendicular to the collimator and telescope axes. b) Sometimes, the levelling of the prism table by spirit level is not sufficient. In such a situation, the prism table should be levelled 12 optically. This consists of the following steps: Experiment 1 Refractive Index of the Material of a Prism using a Spectrometer x Illuminate the collimator slit by sodium light. (Do you know the mechanism of emission of light by a sodium lamp? Discuss with fellow students as well as with your academic counsellor, if you are not able to get correct reference.) Place the prism at the centre of the prism table such that one of its faces AB is perpendicular to the line joining the two screws P1 and P2 (Fig. 1.5). Fig. 1.5: Prism on a prism table. Fig. 1.6: Optical levelling of prism table. x Rotate the prism table so that the refracting edge points towards the collimator and light falls on both the refracting surfaces of the prism, as shown in Fig. 1.6. x Turn the telescope till you see the image of the slit due to light reflected from the AB side of the prism. Is the image symmetrical with respect to the horizontal cross wire of the telescope? If not, adjust the screws P1 and P2 by moving them in opposite directions so that the image is exactly at the centre of the field of view of the telescope. x Next, rotate the telescope to see the image of the slit due to light reflected from side AC (Fig. 1.6). Again ensure that the image is symmetrical with respect to the horizontal cross-wire of the telescope. If not, adjust the screw P3. Turn back the telescope towards face AB and repeat the earlier process, if the symmetry has been disturbed. x Repeat the process till the slit image is symmetrical with respect to horizontal cross-wire in both the positions of the telescope. With these adjustments, you have made the collimator, telescope and the prism table horizontal and perpendicular to the vertical axis of the prism table. ii) Focussing the cross-wire: Keep the telescope objective towards any illuminated background and move the eye piece inward or outward until you see the cross wires more clearly. iii) Adjustment of the slit: Remove the prism from the prism table and place the telescope in line with the collimator and see through the eye-piece of 13 BPHCL-138 Waves and Optics: Laboratory the telescope. Obtain a sharp image of the (collimator) slit by turning the focussing screw of the telescope and of the collimator. The slit can be made vertical by turning it in its plane and its width should be adjusted to about 1 mm using the attached screw. (The slit should be narrow.) iv) Adjusting the collimator and the telescope for parallel rays: The telescope and collimator can be focussed for parallel rays in two ways: a) Take the spectrometer out of the dark room and focus the telescope on a distant object like a tree or a street light and obtain the best distinct image of the object by adjusting the focussing screw. b) By Schuster’s method: This is a better and more scientific method for focussing the telescope and collimator for parallel rays. It involves the following steps: x Illuminate the collimator slit with sodium light. Bring the telescope in line with the collimator and adjust the slit and levelling screws of the apparatus so as to obtain the image of the slit at the centre of the field of view of the telescope. x Adjust the telescope by rotating it so that vertical cross-wire coincides with the slit. Adjust the eye-piece so that the cross wires are distinctly visible. x Place the prism on the table and adjust its height to receive collimated light beam on one of its refracting surfaces. If you look through the other refracting surface of the prism and by moving towards its base, you will see the image of the slit through the prism by unaided eye. x Now, rotate the prism table in such a direction that the image of the slit approaches the direct path of the rays from the collimator. x Bring the telescope to this position of the image. This is the approximate position of minimum deviation which is indicated by the fact that around this position, the slit image moves to only one side (away from the direct path from collimator) in the field of view of telescope irrespective of the direction of rotation of the prism table, clockwise or anti-clockwise. Fix the telescope in this position (Fig. 1.7). x Now rotate the prism table slightly so that the angle of incidence on its refracting surface is greater than that corresponding to the minimum deviation position (Fig. 1.7). Focus the telescope using the adjustment of its eye piece till the slit image is sharp. x Rotate the prism table in the opposite direction so that the angle of incidence is slightly less than that corresponding to minimum deviation position (Fig. 1.7). Focus the collimator by turning the screw attached with it and get a sharp image of the slit. 14 Experiment 1 Refractive Index of the Material of a Prism using a Spectrometer Fig. 1.7: Schuster’s method for focussing telescope and collimator for parallel rays. x Now again turn the prism in opposite direction to come back to initial position (angle of incidence greater than that for minimum deviation). The image of the slit gets blurred. Focus the image by adjusting the screw of the telescope to get the sharp image. x By repeating these two steps a few times, a sharp image of the slit in both positions of the prism will be obtained. This ensures focussing of collimator and telescope for parallel rays. x In order to avoid confusion, remember that, when the refracting edge of the prism is nearer to you (observer), the image is focussed by the telescope (which is nearer to you) and when the refracting edge of the prism is farther from you, the image is focussed by the collimator (which is farther from you). x If these steps are not followed in order, the image will worsen instead of improving. Source of Light The source of light used in an experiment is decided by the objective of the experiment. For example, if we have to determine wavelengths of various colours of light obtained due to dispersion, we use a mercury lamp. However, to determine the refractive index of the material of a prism using a spectrometer, we need a monochromatic source of light. The sodium vapour lamp is commonly used as a monochromatic source. However, the fact is that this source emits a doublet of wavelengths 589.0 nm and 589.6 nm. Since the difference in the wavelengths is extremely small, it is taken as a monochromatic source for all practical purposes and the wavelength of emitted light is taken as 589.3 nm, the average of the two. 1.4 EXPERIMENTAL PROCEDURE First of all, you should set up the spectrometer as described in the previous section. This is a necessary requirement for making any measurement with it. 15 BPHCL-138 Waves and Optics: Laboratory 1.4.1 Vernier Constant of the Spectrometer As mentioned earlier, a spectrometer has two circular vernier scales attached to its base which enables us to determine the angle by which the telescope has been rotated. You are familiar with the concept of vernier scale and vernier constant or least count from your school physics. You also got an opportunity to calculate the least count of a vernier in the first semester laboratory course entitled ‘Mechanics: Laboratory’ (BPHCL 132). You know that the difference between the value of one main scale division (MSD) and one vernier scale division (VSD) is called the vernier constant or the least count (LC) of the instrument and it is the smallest measurement that can be done accurately using a vernier scale. In case of spectrometer, the circular verniers are used to measure angles up to accuracy in minute. In ordinary laboratory spectrometers, each main scale division is equal to half a degree and the vernier scale is such that 30 VSD coincide with 29 MSD. (You must verify that it is true for the spectrometer with which you are doing the experiment.) Thus, we write 29 1 VSD MSD 30 So, Least Count = 1MSD  1 96' 1 MSD  29 §1  29 · MSD 1 MSD ¨ ¸ MSD 30 © 30 ¹ 30 Now 1 1 MSD q 30c 2 1 § 1 ·$ § 1· $ ? /& u¨ ¸ ¨ ¸ 1c 30 © 2 ¹ © 60 ¹ In some spectrometers, 40 VSD may coincide with 39 MSD and accordingly, the LC for such spectrometers will have different value. So, you must check the spectrometer vernier scale before calculating its LC. Working Formula For determining the refractive index of the material of the prism, you will use the formula given by Eq. (1.11): A  Gm sin P 2 sin A / 2 where, A is angle of prism and G m is angle of minimum deviation. So, we need to measure the values of A and G m. The procedure for these 16 measurements is given below. Experiment 1 Refractive Index of the Material of a Prism using a Spectrometer 1.4.2 Measurement of the Angle of Prism i) Switch on the sodium vapour lamp. ii) Set up the spectrometer following the procedure explained in Sec. 1.3.2. iii) Place the prism on the prism table with its refracting edge AB and AC faces the collimator as shown in Fig. 1.8. In this position, the parallel beam of light coming from the collimator will fall on the refracting surfaces of the prism. You must note that the slit is visible from both the faces with unaided eye. Fig. 1.8: Set up for measuring the angle of prism. iv) Move the telescope to a position, say P, so as to receive light after reflection from face AB and you can see the image of the slit. v) Adjust the vertical cross-wire of the eye piece so that it coincides with the slit image. vi) Note the reading of main scale and vernier scale on both the vernier windows V1 and V2 in the Observation Table 1.1. vii) Now, move the telescope and bring it to a position, say Q (Fig. 1.8) so as to receive light after reflection from face AC of the prism and you can see the image of the slit. viii) Note the reading of main scale and vernier scale on both the vernier windows V1 and V2 in the Observation Table 1.1. ix) Take three independent set of readings for telescope positions at P and Q each. x) The angle between these two positions gives 2A, twice the angle of the prism. Calculate the mean value of A. For measuring angles of prism, the prism table should be set in such a position so that the reading in V1 has small initial value, say between 0q and 30q. By doing so, you ensure that, after turning the telescope towards other face, an addition of 120q (usually A ~ 60q) will not exceed 150q and correspondingly the reading on V2 will not exceed 360q. This avoids the confusion while taking the difference of two readings. 17 BPHCL-138 Waves and Optics: Laboratory 1.4.3 Measurement of the Angle of Minimum Deviation The steps to set-up the experimental arrangement to measure the angle of minimum deviation are given below: i) Place the prism on the prism table with one of its refracting surface AB facing the collimator and the centre of the prism coinciding with the centre of the table as shown in Fig. 1.9. ii) Look through the other refracting surface AC of the prism to see the image of the slit, formed due to refraction of light, with unaided eye. iii) Rotate the prism table slowly in such a direction that the image seen by the unaided eye moves as close as possible to the direct ray from the collimator (shown by the dotted line CD in Fig. 1.9). If you continue rotating the prism table slowly in the same direction further, you will observe that, at some point, the image will begin to move away from the direction of the direct ray from the collimator. The position of the prism where the image just begins to move away from the direct ray is the approximate position of the prism for minimum deviation. P Fig. 1.9: Set up for measuring the angle of minimum deviation. iv) Bring the telescope to position P as shown in Fig. 1.9. Adjust the vertical cross-wire of the eye piece so as to coincide with the image of the slit. v) For fine tune the position of minimum deviation, rotate the prism table slightly with the help of tangent screw so that the image moves in the direction of decreasing deviation (that is, closer to the direct ray). vi) Rotate the telescope using the tangent screw to align its cross-wire with the new position of the slit image. This is the precise position of the prism for minimum deviation. vii) Continue with these slow adjustments of the prism table and the telescope till the slit image just begins to move in the opposite direction (that is, 18 moves in the direction of increasing deviation). Experiment 1 Refractive Index of the Material of a Prism using a Spectrometer viii) Note down both the vernier readings in the Observation Table 1.2. ix) Now remove the prism from the prism table. Align the telescope with the direct ray from the collimator so as to see the image of the slit. Adjust the vertical cross-wire of the eye piece with the slit image. x) Note the vernier readings for this position of the telescope in the Observation Table 1.2. This is the direct ray reading. xi) The difference between the mean readings for the minimum deviated ray and the direct ray gives the angle of minimum deviation G m of the prism. xii) Take two sets of readings for G m and calculate the mean value of G m. Keep either telescope or the prism table clamped while adjusting the other for proper readings. Observations: Least Count (LC) of the vernier of spectrometer =.......... Wavelength of the light used (Na-light) = 589.3 nm Observation Table 1.1: Angle of Prism, A st nd No. of Vernier 1 Position of 2 Position of Telescope Difference Angle of Observation Telescope (Y) (Y  X) Prism (=2A) A (X) MSR VSR Total MSR VSR Total V1 1 V2 V1 2 V2 Note that you need to multiply the vernier scale reading (VSR) by vernier constant or least count (LC) of the vernier before adding it to the main scale reading (MSR) to get Total (X or Y). Mean A =............ 19 BPHCL-138 Waves and Optics: Laboratory Observation Table 1.2: Angle of Minimum Deviation, G m No. of Vernier Minimum Deviation Direct Ray Difference, Observ- Ray (Y) (Y  X) ations (X) = Gm MSR VSR Total MSR VSR Total V1 1 V2 V1 2 V2 Mean G m =.............. 1.4.4 Calculations and Result You can calculate the refractive index of the material of the prism by substituting the values of A and G m in Eq. (1.11): A  Gm · sin §¨ ¸ P © 2 ¹ A sin 2 Refractive index of the medium of the prism for sodium light =............... Experiment al value  Standard value Percentage error u 100 Standard value = % Discuss your result with your academic counsellor. 20 Experiment 3 Cauchy’s Constants of the Material of a Prism EXPERIMENT 3 CAUCHY’S CONSTANTS OF THE MATERIAL OF A PRISM Structure 3.1 Introduction 3.4 Calculations Expected Skills Refractive Index for Different Colours 3.2 Theoretical Background of Light 3.3 Measurement of the Angle of Prism and Calculation of Cauchy’s Constants the Angle of Minimum Deviation Angle of Prism Angle of Minimum Deviation for Different Colours of Light 3.1 INTRODUCTION Reflection and refraction are the basic properties of light that you have studied in detail in your school physics classes. A brief introduction to the phenomenon of refraction has been provided in Experiment 1 of this course where you learnt how to obtain the refractive index of the material of a prism using sodium light. It is interesting to explore what happens to refractive index when composite light such as that from a mercury lamp source is made to fall on a prism. In this experiment, you will use the skills developed in Experiment 1 to set the spectrometer, determine the angle of the prism and the angle of minimum deviation for each wavelength of mercury light. Using these observations, you can determine the refractive index for each wavelength. This data will then be utilized to determine the Cauchy’s constants for the material of the prism. Expected Skills After performing this experiment, you should be able to: ™ set-up the spectrometer and calculate its least count; 35 BPHCL-138 Waves and Optics: Laboratory ™ determine the angle of the prism and angle of minimum deviation for light of given wavelength (colour); ™ plot a graph between refractive index corresponding to a wavelength O as a function of 1 O 2 ; and ™ calculate Cauchy’s constants for the material of the prism. Before proceeding further, we list the apparatus that you will use to perform this experiment. Apparatus Required Mercury (Hg) lamp, prism, spectrometer (collimator, prism table, telescope), magnifying glass, spirit level, torch/lamp, scale, cleansing cloth. 3.2 THEORETICAL BACKGROUND From Sec. 1.2.2, Experiment 1 of this course, you will recall that the refractive index of a prism for light of a given wavelength is given by Eq. 1.11: A  Gm sin P 2 (3.1) sin(A / 2) where A is the angle of the prism and G m is the angle of minimum deviation for the given wavelength (colour). You may also recall from Experiment 1 that, for a given wavelength, the angle of deviation is minimum when the angle of incidence is such that ray inside the prism becomes parallel to the base of the prism. For this angle of deviation, the object and the image are at the same distance from the prism, and the image is the brightest. Refer to Sec. 1.4 to quickly revise the steps used to set up the experiment to determine the angle of the prism and the angle of minimum deviation using the sodium light. In this experiment, you can use a particular colour or white light to determine the angle of prism. Cauchy’s equation is an empirical relation between the refractive index of a material and the wavelength of light: B C PO A  ...... (3.2) O2 O4 where constants A, B, C are referred to as Cauchy’s constants. As you can see, this relation predicts that as wavelength of light increases, the refractive index decreases. Typically, it is sufficient to use only the first two terms of this equation: B PO A (3.3) O2 In this experiment, you will determine Cauchy’s constants A and B using the form of Cauchy’s equation expressed in Eq. (3.3). Cauchy’s constants will be 36 Experiment 3 Cauchy’s Constants of the Material of a Prism 1 determined by plotting a graph of P O versus once refractive index O2 corresponding to different wavelengths (colors) has been determined. In Experiment 1, you worked with sodium vapor lamp which emits a doublet with wavelengths ~ 589.0 nm and 589.6 nm. It means that with sodium vapor lamp, you can determine the refractive indices of the prism for two wavelengths only. (However, in view of very small difference between the wavelengths of the sodium doublet, we considered the light emitted by the sodium source as of single wavelength and carried out calculations accordingly in Experiment 1.) But the spectrum of a mercury vapour lamp consists of several (seven) wavelengths and can be used to study the variation of refractive index with wavelength. The spectrum of a mercury vapor lamp is shown in Fig. 3.1.. Fig. 3.1: A representative emission spectrum of a mercury-vapour lamp [Picture credit: D-Kuru, CC BY-SA 2.0 AT , via Wikimedia Commons] You may now like to know as to what happens when a composite light like that from a mercury lamp enters a prism. You may recall from your earlier classes that an interesting phenomenon of dispersion is observed in which light splits into its constituent colours (of different wavelengths) as shown in Fig. 3.2. This is because the refractive index of the material of the prism is different for different wavelengths. The refractive index increases from red to violet, so the angle of deviation is greater for violet than for red, as you can note from Eq. 3.2. Gr Fig. 3.2: Dispersion of light from a prism. The angle of deviation is G r for the red colour. 37 BPHCL-138 Waves and Optics: Laboratory 3.3 MEASUREMENT OF THE ANGLE OF PRISM AND ANGLE OF MINIMUM DEVIATION First, you must follow the steps outlined in Sec. 1.4.1, Experiment 1 to set up the spectrometer and focus the collimator for parallel light. Then, follow the instructions given here for taking the measurements required to calculate Cauchy’s constants of the prism. 3.3.1 Angle of Prism To determine the angle of prism, follow the steps outlined in Sec. 1.4.2 using one particular wavelength emitted by mercury vapour lamp. Record your readings in Observation Table 3.1. Take at least two sets of readings. Least Count (LC) of vernier of the spectrometer =.......... Wavelength of light used =.......... Observation Table 3.1: Angle of Prism (A) st nd No. of Vernier 1 Position of 2 Position of Difference Angle Observ Telescope Telescope = 2A of ation Prism MSR VSR Total MSR VSR Total A I V1 A1= 1 II V 2 A2= I V1 A3= 2 II V 2 A4= A1  A2  A3  A4 Angle of the Prism A..........q 4 3.3.2 Angle of Minimum Deviation for Different Colours of Light After determining the angle of the prism, you have to determine the angle of minimum deviation for each of the prominent colours of mercury light, namely violet, indigo, blue, green, yellow, orange and red following the steps outlined in Sec. 1.4.2, Experiment 1. Record your readings in Observation Table 3.2. Take at least two sets of readings for each colour. Calculate the mean value of the angle of minimum deviation for each colour. 38 Experiment 3 Cauchy’s Constants of the Material of a Prism Observation Table 3.2: Angle of minimum deviation G m for light of different colours Sl. Color No. Vernier Minimum Deviation Direct Ray Difference Mean* No of Light of Ray = Gm Angle of Obser MSR VSR Total MSR VSR Total Minimum vation Deviation s 1. Violet I V1 G m1 = 1 II V 2 G m2 = I V1 G m3 = 2 II V 2 Gm4 = 2. Indigo I V1 G m1 = 1 II V 2 G m2 = I V1 G m3 = 2 II V 2 Gm4 = 3. Blue I V1 G m1 = 1 II V 2 G m2 = I V1 G m3 = 2 II V 2 Gm4 = 4. Green I V1 G m1 = 1 II V 2 G m2 = I V1 G m3 = 2 II V 2 Gm4 = 5. Yellow I V1 G m1 = 1 II V 2 G m2 = I V1 G m3 = 2 II V 2 Gm4 = 6. Orange I V1 G m1 = 1 II V 2 G m2 = I V1 G m3 = 2 II V 2 Gm4 = 7. Red I V1 G m1 = 1 II V 2 G m2 = I V1 G m3 = 2 II V 2 Gm4 = * Mean Angle of Minimum Deviation G m1  G m2  G m3  G m 4 Gm...........q 4 39 BPHCL-138 Waves and Optics: Laboratory We now calculate the refractive index for each color of light and the Cauchy’s Constants A and B. 3.4 CALCULATIONS 3.4.1 Refractive Index for Different Colours of Light You can calculate the refractive index P O for each colour (with a typical wavelength as given in the margin remark) using Eq. (3.1). Enter the results of The standard values of your calculations in Observation Table 3.3. wavelength ( O ) for the different colours are: Table 3.3: Refractive indices for different colours Violet : 400 nm Indigo : 420 nm Sl. Colour of Wavelength Angle of Minimum Refractive Blue : 450 nm No. Light O Deviation Index Green : 550 nm PO Yellow : 580 nm Orange : 600 nm 1. Red : 650 nm (Source: brittanica.com/science/color/ The-visible-spectrum) 2. 3. 4. 5. 6. 7. 3.4.2 Calculation of Cauchy’s Constants We now calculate the Cauchy’s Constants using Eq. (3.3). To do this, you have to plot a graph of the inverse of the square of the § 1 · wavelength ¨ ¸ for each wavelength O along the x-axis and the © O2 ¹ corresponding refractive index, PO for that wavelength along the y-axis. Do the needed calculations and enter the data required for plotting this graph in Observation Table 3.4. The values of PO and O are to be taken from 40 Observation Table 3.3. Experiment 3 Cauchy’s Constants of the Material of a Prism 1 Observation Table 3.4: Values of PO and O2 Sl. No. Wavelength 1 Refractive Index O 2 PO O 1. 2. 3. 4. 5. 6. 7. You should get a graph as the one shown in Fig. 3.3. While plotting the graph, the scales should be so chosen that full span of the graph is utilised. By doing so, you will minimise error in your calculations. PO 'P O B Slope '§¨ ·¸ 1 © O2 ¹ A § 1 · ¨ 2¸ ©O ¹ 1 Fig 3.3: Plot of PO with O2 41 BPHCL-138 Waves and Optics: Laboratory The values of A and B are calculated as the intercept on the y-axis and the slope of the graph, respectively. To calculate the slope, you should use the maximum possible intercept of the straight line. Result: The values of the Cauchy’s constants A and B for the material of the prism are: A =.............................................. B =............................................. 42 Experiment 4 Wavelength of Sodium Light using Fresnel’s Biprism EXPERIMENT 4 WAVELENGTH OF SODIUM LIGHT USING FRESNEL’S BIPRISM Structure 4.1 Introduction 4.4 Determination of Wavelength of Expected Skills Sodium Light 4.2 Interference of Light Adjusting the Apparatus Measurement of Fringe Width 4.3 Fresnel’s Biprism and Coherent Sources 4.1 INTRODUCTION As a child, you may have enjoyed blowing soap bubbles and seeing bright rainbow colours reflected from them. On a rainy day, you must have also observed brilliant, though irregular, colour patterns on the wet road surface due to a thin layer of oil spilt by a motor vehicle. You may have also realised that colour patterns change if you look at them from different angles. Have you ever looked at a fairly transparent piece of silk or polyester cloth from a distance? If you do so, you would observe patterns of bright and dark bands. The bright and dark bands so produced are known as interference fringes. All the phenomena described above arise due to interference of light waves. You have learnt about interference of light in Unit 6 of the fourth semester course entitled Waves and Optics (BPHCT-137). The simplest demonstration of interference of light waves was devised by Thomas Young. You have learnt about Young’s double slit experiment in Two sources are said to be coherent if light Unit 6 of BPHCT-137. You may recall that in this experimental setup, waves originating from monochromatic light from a point source is made to give rise to two coherent them are of the same sources by placing two closely spaced narrow slits in its path. The frequency and have a superposition of waves from these two coherent sources produces a clear constant phase interference pattern comprising bright and dark fringes on a screen placed difference between some distance away. Do you know why we need coherent sources to observe them. interference pattern? 43 BPHCL-138 Waves and Optics: Laboratory The coherent sources can be produced by a variety of experimental set-ups. In the theory course BPHCT-137, you have learnt that unlike Young, Fresnel used a biprism to produce two coherent sources. In this experiment, you will learn to obtain interference pattern using a biprism and determine fringe width. (It is the distance between two consecutive dark (or bright) fringes.) This will enable you to determine the wavelength of the incident monochromatic light. Expected Skills After performing this experiment, you should be able to: ™ set up an optical bench to observe interference pattern; ™ use a biprism to obtain interference fringes; ™ determine the distance between two virtual coherent sources; ™ determine the factors on which fringe width depends; and ™ determine the wavelength of sodium light. You will require the following apparatus for this experiment. Apparatus Required A biprism, optical bench with uprights, sodium vapour lamp, slit, micrometer eye-piece and a convex lens of short focal length. 4.2 INTERFERENCE OF LIGHT Interference is a phenomenon in which waves, under certain circumstances, reinforce (intensify or weaken) each other. The phenomenon is understood on the basis of the principle of superposition of waves. You have learnt this principle in your school physics course as well as in Unit 2 of the course BPHCT-137. You know that two identical progressive mechanical waves travelling along a wire, fixed at the ends, in opposite directions give rise to stationary waves. The stationary waves are characterised by succession of nodes and anti-nodes. The nodes are positions of minimum intensity whereas anti-nodes are positions of maximum intensity. In other words, there is a redistribution of energy carried by the two superposing waves. This redistribution of energy is one of the most significant characteristics of the interference phenomenon. In Unit 6 of the BPHCT-137 course, you learnt that the interference phenomenon is also observed with light when light from two coherent sources superpose. When two or more light waves of same frequency and having constant phase relation between them are superposed, the intensity of the resultant light in the region of superposition is found to vary from point to point. At some points, the intensity equals the sum of the intensities of individual waves while at some other points it is almost zero. (These are known as points of maxima and minima, respectively.) This is termed as the phenomenon of interference. The interference pattern comprises a series of regularly spaced maxima and minima. If the resultant intensity is zero, or in 44 Experiment 4 Wavelength of Sodium Light using Fresnel’s Biprism general, less than what we expect from individual waves, we have destructive interference (seen as dark fringes/bands). On the other hand, if resultant intensity is greater than the intensities of individual waves, we have constructive interference (seen as bright fringes/bands in the pattern). In the present experiment, you will obtain interference pattern produced by light form two coherent sources and make certain measurements to determine the wavelength of the light used. For this purpose, you need to have an expression which relates the experimentally measured quantities (fringe width, in the instant case) with the wavelength of light used. Let us now obtain the expression relating fringe width and wavelength of light used to obtain the interference pattern. (You have derived this expression in Unit 6 of BPHCT-137. But, we are giving it here for the sake of completeness.) Suppose that a narrow slit S is illuminated by a monochromatic source of light. This, in turn illuminates two other narrow equidistant slits S1 and S2 (called double slit) separated through a distance d from each other, as shown in Fig. 4.1. The interference pattern is obtained on a screen placed at a distance, say D, from the double-slit and parallel to the plane containing these slits. Fig. 4.1: Schematic diagram of the double slit arrangement used to observe Note that we are interference of light. discussing here the interference of light Let us consider a point P on the screen which is the nearest maxima or caused due to minima from the origin. Suppose that the two waves emanating from slits S1 superposition of two and S2 respectively are given as light waves (electromagnetic y1 a sin(Zt  kx) (4.1a) waves). Thus, you should keep in mind y2 a sin(Zt  kx  I) (4.1b) that the displacements y 1 and y 2 used in Note that we have chosen the amplitude of both the waves equal to a because Eqs. (4.1a) and (4.1b) the two slits are very close to each other. Further, the phase difference I actually represent the arises because the wave originating from slit S2 travels an extra distance as magnitude of the electric fields compared to the wave originating at slit S1. Now, let us know as to what associated with the light happens when these waves reach point P on the screen. For simplicity, let us waves emanating from take point P to be the origin so that the kx term in Eqs. (4.1a) and (4.1b) can slits S 1 and S 2 be dropped. Thus, at point P, we can write the displacements due to the two respectively. waves as 45 BPHCL-138 Waves and Optics: Laboratory y1 a sin Zt y2 a sin(Zt  I) Note that the slits S1 and S2 are essentially coherent sources. Therefore, the phase difference I between the two waves is constant. Since a path difference of one wavelength corresponds to a phase difference of 2S radians, we can write 2S I u ( path difference) (4.2) O Refer to Fig. 4.1. We can write the path difference between S1P and S2P as S 2 P  S1P d sin T (4.3) where d is the separation between the slits S1 and S2. Thus we can write 2S I d sin T (4.4) O y y1  y 2 Now, according to the superposition principle, the resultant displacement, y at P is given by (see the margin remark) a sin Zt  a sin(Zt  I) I I Using the identity y y1  y 2 2a cos §¨ ·¸. sin(Zt  ) (4.5) © 2¹ 2 sin A  sin B AB AB 2 sin( ) cos( ) Eq. (4.5) shows that the expression for displacement y of the resultant wave at 2 2 I we can write point P corresponds to a harmonic wave with amplitude 2a cos §¨ ·¸. Further, © ¹ 2 I I y a[2 sin(Zt  ) cos( )] you know that intensity of wave is proportional to the square of the amplitude. 2 2 I I Thus, we can write the intensity of light at point P as 2a cos( ) sin(Zt  ) 2 2 I I 4a 2 cos 2 §¨ ·¸ (4.6) ©2¹ Eq. (4.6) shows that the intensity is maximum ( 4a 2 or four times the intensity of either wave) if I nS n = 0, 1, 2, … (4.7a) 2 and minimum (in fact, zero) if I § n  1 ·S ¨ ¸ n = 0, 1, 2, … (4.7b) 2 © 2¹ Thus, by substituting the value of (I/2) from Eq. (4.7a) in Eq. (4.4), we can write the condition for constructive interference as nO sin T n n = 0, 1, 2, 3, … (4.8a) d and from Eq. (4.7b) for destructive interference as sin T n §n  1 · O ¨ ¸ n = 0, 1, 2, 3, … (4.8b) 46 © 2¹ d Experiment 4 Wavelength of Sodium Light using Fresnel’s Biprism Now, to obtain an expression for the fringe width, refer to Fig. 4.1 again. Let OP = x. Then, we can write x D tan T (4.9a) Thus, the positions of the maxima and minima are given by xn D tan T n (4.9b) where Tn is given by Eqs. (4.8a) and (4.8b) for bright (constructive The wavelength, interference) and dark (destructive interference) fringes, respectively. O ~ 6000 A֯ ~ 0.6 Pm. And, the typical value Finally, if the slit separation is much greater than the wavelength of light used of slit separation, nO d ~ 1 mm. (d !! O), then for non-zero values of n, the value of will be very small. d So, d >> O. Therefore, it readily follows from Eq. (4.8a) that Tn will be very small. Then in the small angle approximation, we can take sinTn | tan Tn | Tn Hence, Eq. (4.9b) can be written as xn D Tn (4.10) So, Eq. (4.8a), in small angle approximation, reduces to nO Tn (4.11) d Thus, from Eqs. (4.10) and (4.11), we get the position of the nth bright fringe on the screen as nOD xn n = 0, 1, 2, 3, … (4.12) d Similarly, Eq. (4.8b), in small angle approximation, takes the form § 1· O Tn ¨n  ¸ (4.13a) © 2¹ d Thus, from Eqs. (4.10) and (4.13), we get the position of the nth dark fringe on You can create double slits by cutting the screen as very fine slits in a § n  1 · OD black art paper using ¨ ¸ a shaving blade. xn © 2¹ n = 0, 1, 2, 3, … (4.13b) Then using an d ordinary lamp, you Note that by using Eqs. (4.12) and (4.13.b), you can calculate the fringe should be able to obtain interference width (that is, distance between two consecutive bright fringes or the distance pattern. Discuss your between two consecutive dark fringes): findings with your peers as well as your (n  1)OD nOD OD E x n 1  x n  (4.14) academic counsellor. d d d So, once we know the wavelength O of the light, slit separation, d and the distance, D between the double slit and the screen, we can easily calculate the fringe width. However, in the present experiment, you will measure fringe width, d and D to determine the wavelength of light using Eq. (4.14). 47 BPHCL-138 Waves and Optics: Laboratory In the above discussion on the phenomenon of interference of light, we confined to the double slit arrangement in which two coherent sources were obtained from a given source of light. But, the double slit arrangement (Fig. 4.1) has some inherent limitations which impact the quality of the interference pattern. If slits S1 and S2 are very narrow, the amount of light available for forming the fringes will be very small and the (bright) fringes will be of feeble intensity. Also, you can argue that these slits may diffract light and the observed pattern will not be interference pattern. To overcome such limitations, Fresnel designed an experimental set up to obtain interference pattern wherein the double slit arrangement was replaced by a biprism to create virtual coherent sources of light. He demonstrated that the light from such virtual sources gives rise to interference pattern. We will now briefly discuss Fresnel biprism arrangement. 4.3 FRESNEL’S BIPRISM AND COHERENT SOURCES A biprism is made up of two identical prisms of very small (~ 0.5 ) refracting angles placed base to base. To understand how a biprism can be used for creating two coherent sources of light, refer to Fig. 4.2. S is a narrow vertical slit illuminated by monochromatic source of light. The light from S is made to fall symmetrically on the biprism having its refracting edges parallel to the slit. The light incident on each half of the prism is refracted by the corresponding refracting edge. This gives rise to virtual images S1 and S2 of the slit S located on its either side. The distance between S1 and S2 is d. You may recall from Experiment 1 that the prism equation is given as (Eq. (1.11)): $  GP VLQ ˩  VLQ $ Since the biprism is very thin, the angle, A (which, in case of biprism, we have denoted by D) of the prism is very small and Fig. 4.2: Formation of virtual images S1 and S 2 of slit S by a biprism. we can write DG DG These two virtual images S1 and S2 act as two coherent sources. S1 and S2 sin 2 2 are fairly close to the source S. ( S1 , S and S2 are in the same plane) as the sin(D 2) D 2 angles of deviation are small. You can verify from Fig. 4.2 that SS1 SS2 aG, Thus, we can write where a is the distance between the source S and the biprism and G is the ( D  G) 2 angle of deviation. We know that angular deviation produced by a biprism is P given by (see the margin remark) D2 PD (D  G) G (P  1) D G (P  1)D where P is the refractive index of the material of the prism and D is base 48 angle. Thus, we can write Experiment 4 Wavelength of Sodium Light using Fresnel’s Biprism S1S2 S1S  SS 2 d 2a (P  1) D (4.15) If the eye-piece is held at a distance b from the biprism anywhere in the region of overlap of the two refracted beams, the distance of the pair of sources from the plane of interference will be D (a  b ) (4.16) On substituting for d and D from Eqs. (4.15) and (4.16), respectively in Eq. (4.14), we get the expression for the fringe width as (a  b ) O E (4.17) 2a (P  1) D This result shows that we can calculate the wavelength of light once we have measured a, b, D and E, for a biprism of given refractive index. Since biprism is very thin, the angle D is very small ( | 6 u 10 3 rad) and it is not convenient to measure it. So, Eq. (4.15) is not very useful for determining d in this experiment. We, therefore, resort to an alternative method called method of displacement, wherein d is connected to the separation between the images of the two virtual sources rather than D. You may like to know how this is achieved. The answer to this question is given in the following paragraphs. A convex lens of short focal length (f ~ 15 cm to 20 cm) is introduced between the biprism and the eye-piece (Fig. 4.3). The eye-piece is kept at a large distance from the slit (5f > D > 4f). This condition on D minimises the error in the measurement of d (see the margin remark on the next page). The convex lens converges the two refracted beams. We can adjust its position to obtain clear well-defined images in the plane of the cross wires in the eye-piece. In fact, while performing the experiment, you will observe that once positions of slit, biprism and the eye-piece are fixed, there are two positions of the lens, shown as L1 and L2 in Fig. 4.3, for which clear images of S1 and S2 are obtained in the eye-piece. When the lens is at one of these positions (say at L1 ), we obtain magnified images while in the other position (say L2 ), we obtain diminished images of the sources. Fig. 4.3: Displacement method to determine the distance between the coherent virtual sources in Fresnel biprism experiment. Two positions of the lens between the biprism and the eye-piece correspond to enlarged images I1 , I 2 and diminished images I 1c , I 2c of S1 and S2. 49 BPHCL-138 Waves and Optics: Laboratory Suppose that the separation between the two magnified images as seen in the The condition D > 4f is a theoretical eye-piece is d1. If the actual distance between the virtual sources S1 and S2 consideration (see is d, the expression for magnification by the lens is given by Eq. (4.19a)) but the d1 condition 5f > D arises m1 (4.18a) from the practical d consideration of And, if d 2 is the distance between diminished images of S1 and S2 , as seen minimising the error in in the eye-piece, the magnification is given by the measurement of d, which is geometrical d2 m2 (4.18b) mean of d1 and d 2. d The error consideration for the condition can be Now, if u and v are distances of the object and the image, respectively, we obtained as follows: can write from Fig. 4.3 that d d1 d 2 u v D Taking logarithm of or, u D v both sides, we get From the lens formula, we know that 1 1 ln d ln d1  ln d 2 1 1 1 2 2  v u f On differentiation, we get On substituting for u, we can write 'd 'd1 'd 2 1 1 1   d 2d1 2d 2 v D v f so that if we denote or ' d /d by e, then we D v v 1 D 1 can write Ÿ v (D  v ) f v (D  v ) f 1 e (e1  e2 ) 2 This can be rewritten as and v 2  Dv  fD 0 'd1 'd 2 e1e2. For real roots of the this quadratic equation, we must have d1 d 2 D 2  4fD ! 0 'd 1 ' d 2 d2 or D ! 4f (4.19a) = constant Further, if the real roots are v1 and v2, then the sum of the roots is Since v1  v 2 D (4.19b) (e1  e2 )2 (e1  e2 )2  But, from Fig. 4.3, we have 4e1e2, e1  e2 will be minimum when e1 e2. u1  v 1 u2  v 2 D (4.20) This condition implies where u1, v1 are the object and image distances when lens is in position L1 that and u2, v2 are object and image distances when the lens is in the position L2. 'd1 'd 2 On substituting for v 1 D  v 2 from Eq. (4.19b) in Eq. (4.20), we can write d1 d2 u1  D  v 2 D Ÿ u1 v2 But 'd1 'd 2 ? d1 d2 On eliminating v 2 by combining Eqs. (4.19b) and (4.20), you can prove that That is, d1 and d 2 u2 v1. Since should be almost equal. v1 v1 m1 Ÿ m1 50 u1 v2 Experiment 4 Wavelength of Sodium Light using Fresnel’s Biprism Similarly, you can show that The sum of the roots of a quadratic equation v2 v2 m2 u2 v1 ax 2  bx  c 0 Hence, m1 u m2 1 is equal to  b / a. Here b = D and a = 1. 1 Therefore, sum of roots or m1 (4.21) m2 v1  v 2 D On combining Eqs. (4.18a) (4.18b) and (4.21), we can write d d1 d2 (4.22) That is, d is geometric mean of d 1 and d 2. By combining Eqs. (4.16a) and (4.22), we can write the expression for the fringe width as DO (a  b ) O E (4.23) d d1 d2 This expression for fringe width constitutes the working formula for this experiment. Note that all quantities appearing on the right hand side (a, b, d , d1 , d 2 and E) can be measured to determine the wavelength, O of the light used in the experiment. In the next Section, we outline the procedure for determination of wavelength of light using this working formula. 4.4 DETERMINATIOIN OF WAVELENGTH OF SODIUM LIGHT In this experiment, you have to first obtain coherent sources using a biprism and then get interference fringes in the plane of the cross wires to measure the fringe width and determine the distance between the coherent sources. We now give the steps that you have to follow to adjust the apparatus to obtain interference fringes. 4.4.1 Adjusting the Apparatus 1. Refer to Fig. 4.4. It shows a sodium lamp, an optical bench with four uprights and an eye-piece. Fig. 4.4: The experimental setup for observing interference pattern due to Fresnel’s biprism. 51 BPHCL-138 Waves and Optics: Laboratory 2. Arrange the sodium lamp at one end of the optical bench. The sodium lamp is normally kept in a rectangular box having a small rectangular opening on one side to allow light to pass. 3. Mount a slit of adjustable width on the first upright and the biprism on the second upright. You must note that the slit is provided with a screw to rotate it in its own plane. Using this screw, ensure that the slit is vertical. Keep the width of the slit very small. 4. Just like the slit, the biprism can also be rotated in its own plane. Also make sure that the edge of the biprism is parallel to the slit. 5. Now view the slit (illuminated by sodium light) through the biprism. Move your eye sideways. What do you observe? Does one of the bright vertical lines appear and disappear suddenly? If it is so, then you can be sure that the edge of the biprism is exactly parallel to the slit. If the bright line appears or disappears gradually from top to bottom, then the edge of the biprism is not parallel to the slit. Rotate the biprism in its own plane till it is exactly parallel to the slit. In doing so, remember to keep the slit and the biprism as close as possible 9about 15 cm apart). 6. Now, put the micrometer eye-piece at about 15 to 20 cm from the biprism. Keep your eye just above the eye-piece and make sure that you see two images of the slit. If you do not, adjust the position of the biprism or the eye-piece by moving either of them laterally. However, you should not disturb the vertical alignment of the biprism while moving it. 7. Next, look through the eye-piece. You should see a number of vertical bright and dark fringes. The fringes can be seen only if the slit and the edge of the biprism are exactly parallel to each other. If you do not see sharp fringes in the field of view, narrow down the slit S and slightly rotate the biprism in its plane. These two adjustments should enable you to obtain sharp fringes in the field of view. 8. The next step is to align the biprism and the eye-piece. For this, you should move the eye-piece away from the biprism along the optical bench. While you move the eye-piece, keep looking through it to check whether or not the fringes shift to one side as a whole. If you observe a lateral shift of the fringes, it means that the line joining the slit and the central edge of the biprism is not parallel to the length of the optical bench. To remove this lateral shift, move the biprism (using the screw on the side of the upright) through a small distance transversely to the bench in a direction opposite to the direction of the shift till this lateral shift vanishes. 9. Now move the eye-piece forward and check whether the fringes become narrow without showing lateral shift. The above adjustments should be done alternately and repeatedly till a longitudinal movement of the eye- piece on the optical bench does not give rise to a side-ways shift of the whole fringe pattern. With the above adjustments, your experimental set up is ready for making measurements of fringe width. Let us now learn to do so. 4.4.2 Measurement of Fringe Width 1. Note the pitch and calculate the least count of the micrometer of the eye- piece. Record it in Observation Table 4.1. Consider the left extreme line