Chapter 2 Michelson Morley Experiment and Velocity of Light

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1 Chapter 2 Mihelson Morley Experiment and Veloity of Light 2. Attempts to Loate Speial Privileged Frame Sientists tried to determine relative veloity of light with respet to earth. For this purpose, the veloity of earth relative to ether frame of speial privileged frame was essential. Therefore, sientists performed various experiments involving eletromagneti waves. In this regard, the best and most important experiment was first performed by Mihelson in 88 and repeated the experiment more arefully in ollaboration with Morley in 887 and also reiterated many times thereafter. Atually, Mihelson and Morley experiments laid the experimental foundations of speial relativity. Mihelson was awarded the Nobel prize for his experiment in The Mihelson Morley Experiment (M M The experiment of M M was an attempt to measure the veloity of earth through the ether. The experiment failed in the sense that it showed onlusively that ether does not exist. More generally, it demonstrated that there are no privileged observer that the veloity of light is irrespetively of the state of motion of the observer who measures it. The apparatus of the experiment are: Monohromati soure S of light, Interferometer A, one semi-silvered plate B, two mirrors, M, M 2 (see Fig. 2.. The light from a soure at S falls on a semi-silvered plate B inlined at 45 to the diretion of propagation. The plate B, due to semi-silvered, divides the beam of light into two parts namely refleted and transmitted beams. The two beams, after refleting at mirrors M and M 2 are brought together at A, where interferene fringes are formed due to the small differene between the path lengths of two beams. Let us assume that earth (arrying the apparatus is moving with veloity v along BM = l relative to the privileged frame. The time (t required for the transmitted light ray Springer India 204 F. Rahaman, The Speial Theory of Relativity, DOI 0.007/ _2

2 2 2 Mihelson Morley Experiment and Veloity of Light Fig. 2. Experimental set-up of Mihelson Morley Experiment A v B l S M 2 l 2 2 M to traverse the length l both ways, i.e. along BM B is t = l ( v + l ( + v = ( 2l (2. [ is the veloity of light in the ether. So, downstream speed is ( + v and upstream speed is ( v] Now, onsider the path of the refleted light ray, i.e. the path BM 2 B: As the light travels from B to M 2 ( BM 2 = l 2, the whole apparatus has moved a distane x along BM. The light has therefore travelled a distane (l2 2 + x2. We have seen that earth moves some distane during the same time. Aording to Fig. 2.2, the mirror moves the distane x = vt in time t and light moves in time t is t = (l2 2 + x2. Fig. 2.2 Mirror moves some distane in time t t=0 t= 2 t 2 M2 t=t 2 l 2 B vt 2 B

3 2.2 The Mihelson Morley Experiment (M M 3 Hene, Or, t = x v = (l x2 = (l x2 l 2 ( v2 (2.2 On the bak, it travels an equal distane, so the time for to and fro travel of the light beam ( i.e. the path BM 2 B is t 2 = ( 2l 2 (2.3 Interferene fringes are reated by the differene between t and t 2. The whole apparatus is now swung through 90, so the roles of the two arms are interhanged. Therefore, l takes the plae of l 2 and vie versa. We have, t = ( 2l t 2 = ( 2l 2 ( v2 (2.4 (2.5 The time differenes between the two paths, before and after the apparatus is swung round, are t = t t 2 = ( 2 l l 2 (2.6 t = t t 2 = ( 2 l The hange in t brought about by rotating the apparatus is t t = ( 2 (l + l 2 l 2 ( v2 (2.7 (2.8

4 4 2 Mihelson Morley Experiment and Veloity of Light We expand the right-hand side in a Binomial series, retaining terms up to the seond power of (v/ to yield t t (l + l 2 v2 (2.9 Hene, the shift in the number of fringes is N = t t T = γ (l + l 2 v2 (2.0 [Period of one vibration is T = ( γ = ( λ, where γ and λ are frequeny and wave length of the light] Earth s veloity is presumed to be 30 km/s = m/s, ( v 2 = 0 8 and ( γ 0 4 m for visible light, so N 2(l + l (2. Sine l and l 2 were several metres, suh a shift would be easily detetable and measurable. However, no fringe shift was observed. This experiment was repeated with multiple mirrors to inrease the lengths l and l 2, however, the experimental onlusion was that there was no fringe shift at all. In the year 904, Trouton and Noble done again the experiment using eletromagneti waves instead of visible light, however, no fringe shift was observed. Sine the veloity of earth relative to ether annot always be zero, therefore experiment does not depend solely on an absolute veloity of earth through an ether, but also depends on hanging veloity of earth with respet to ether. Suh a hanging motion through the ether would be easily deteted and measured by the preision of experiments, if there was an ether frame. The null result seems to rule out an ether (absolute frame. Thus, no preferred inertial frame exists. Also, one way to interpret the null result of M M experiment is to onlude simply that the measured speed of light is the same, i.e. for all diretions in every inertial system. In the experiment, the downstream speed and upstream speed being rather than + v or v in any frame. Note : In 882, Fitzgerald and Lorentz independently proposed a hypothesis to explain null results of Mihelson Morley experiment. Their hypothesis is Every body moving with a veloity v through the ether has its length in the diretion of motion ontrated by a fator v2, while the dimensions in diretions perpendiular to motion remain unhanged. Let us onsider in Mihelson Morley experiment, the arms are equal, i.e. l = l 2 = l at rest relative to ether and aording to Fitzgerald and Lorentz hypothesis its length l when it is in motion with veloity v relative to ether will be l ( v2.

5 2.2 The Mihelson Morley Experiment (M M 5 By this, they interpreted the null result of Mihelson Morley experiment as follows: From, Eqs. (2. and (2.3, we have t = ( 2l ( v2 ( 2l ( + v2 ( 2l, t 2 = ( 2l ( + v2 Using Fitzgerald and Lorentz hypothesis, we get by replaing l by l, t = ( 2l ( + v2 = ( 2l v2 Expanding binomially and negleting v4 4, we get, t = ( 2l ( + v2 = t 2 ( + v2 Therefore, aording to Fitzgerald and Lorentz hypothesis, the times taken by refleted and transmitted beams are the same and hene no shift of fringe is observed. However, Fitzgerald and Lorentz hypothesis fails to give orret explanation of null results of Mihelson Morley experiment when the two ears of the interferometer are not equal. Note 2: If some instant, the veloity of earth were zero with respet to ether, no fringe shift would be expeted. From, Eqs. (2. and (2.3, we have, by putting v = 0, t = ( 2l = t 2 Then, there is no relative veloity between earth and the ether, i.e. earth drags the ether with the same veloity as earth during its motion. However, this explanation is in ontradition with various experiments like Bradley s result of aberration, et. 2.3 Phenomena of Aberration: Bradley s Observation In the year 727, Bradley observed that the star, γ Draonis revolves in nearly irular orbit. He found that angular diameter of this irular orbit is about 4 s of ar and other stars appeared to move in elliptial orbit have almost same period. This phenomena of apparent displaement is known as aberration. This an be explained as follows. Let a light ray omes from a star at zenith. An observer together with earth has a veloity v will see the diretion of light ray will not be vertial. This is

6 6 2 Mihelson Morley Experiment and Veloity of Light Fig. 2.3 Telesope will have to tilt to see the star A Atual Diretion A B Apparent Diretion t A vt B A vt B due to relative motion of the observer and light ray. To see the star, the telesope will have to tilt, i.e. diretion slightly away from the vertial as shown in the Fig The angle α between the atual and apparent diretions of the star is alled aberration. Let v be the veloity of earth in the diretion AB. The light ray enters the telesope tube at A and reahes the observer s eye at B in time t. As the light reahes from A to B in the mean time eyepiee moves from A to B. Hene, A B = t, AB = v t. The tilt of the telesope α is given by (see Fig. 2.3 or tan α = v α = tan v Using the veloities of earth and light as 30 km/s and km/s, respetively, one gets, α = tan v = tan = tan (0 4 = 20.5 ars Sine earth s motion is nearly irular, therefore every six months, the diretion of aberration will reverse and telesope axis is traing out a one of aberration during the year. Hene the angular diameter of the one would be 2α = 4 ar s. In 727, Bradley observed this motion and found the angular diameter was 4 ar s for the star γ Draonis. This experiment apparently showed the absolute veloity of earth is 30 km/s. The outome of this experiment is that ether is not dragged around with earth. If it were, then the phenomena of aberration would not be happened.

7 2.4 Fizeau s Experiment Fizeau s Experiment In 87, Fresnel proposed that light would be partially dragged along by a moving medium. Using ether hypothesis, he provided the formula for the effet of moving medium on the veloity of light. In 95, Fizeau onfirmed this effet experimentally. Figure 2.4 shows the Fizeau s experimental arrangement. Fizeau used water as the medium inside the tubes. Water flows through the tubes with veloity u. The S is the soure of light. The light ray from S falls on a semi-silver plate inlined 45 from the horizontal is divided into two parts. One is refleted in the diretion M M 2 and other part is transmitted in the diretion M D. Here, one beam of light is entering towards the veloity of water and other opposite to the motion of water. One an note that refleted beam travels the path M M 2 CDM in the diretion of motion of water and the transmitted beam travels the path M EDCM 2 M opposite to the motion of water. Both beams finally enter the telesope. Differene in time taken by both the beams travel equal distane d due to the motion of the water and veloity v of earth. It was assumed that spae is filled uniformly with the ether and ether to be partially dragged. Now, the time differene is t = d μ + ( μ 2 v + u ( μ 2 d μ + ( μ 2 v u ( μ 2 It was assumed that ( ether to be partially dragged and as a result omponent of the veloity in the tube v. It follows also that if a material body is moving with μ 2 ( μ 2. Here, veloity u through ether, the veloity of ether will be u onstant. Negleting higher order terms, we get the time differene as t = 2udμ2 ( μ 2 ( μ 2 = Fig. 2.4 Experimental set-up of Fizeau s experiment B C M2 S M E D F T

8 8 2 Mihelson Morley Experiment and Veloity of Light The phase differene is given by f = 2udμ 2 ( μ 2 T = 2udnμ2 ( μ 2 where, n be the frequeny of the osillation and T is time period of one osillation (nt =. Fizeau observed the shift in the fringes and found exatly the same as given by the above equation. This result also onfirms the result that ( the veloity of the light in any medium of refrative index μ is not simply μ but μ ±, u is μ 2 the veloity of the medium. The fator ( μ is alled Fresnel s ether dragging 2 oeffiient. Thus, Fresnel s formula for ether drag was verified by Fizeau. 2.5 The Relativisti Conept of Spae and Time The null results of Mihelson Morley Experiment ruled out the absolute onept of spae and laid Einstein to formulate the new onept of spae and time. Consider the following two ontraditory statements: [] In lassial mehanis, the veloity of any motion has different values for observers moving relative to eah other. [2] The null results of Mihelson Morley experiment onludes that the veloity of light is not affeted by the motion of the frame of referene. The genius Einstein applied his intuition and realized that this ontradition omes from the imperfetion of the lassial ideas about measuring spae and time. He ruled out the Newton s onept of absolute time by ritiizing our preoneived ideas about simultaneity. He ommented time is indeed doubtful. His intuition may be supported as follows: X O Y Suppose an observer sitting at O where O is the middle point of the straight line XY. Suppose that two events ourring at X and Y whih are fixed in an inertial frame S and the two loks giving absolute time are kept at X and Y. In the inertial frame S, the events ourring at X and Y are said to be simultaneous, if the loks plaed at X and Y indiate the same time when the events our. Let a light signal is giving at X and Y simultaneously. Suppose XOY is moving with v veloity in the diretion XY. Then, one an show that these light signals will not reah simultaneously the observer at O whih is moving with veloity v. Here, the moving O seems to lie in OY ( v another inertial frame. The light signal takes t time to reah from Y to O as and the time t 2 taken by the light signal from X to O is (+v OX. [ is the veloity of light in all diretion. So, aording to Newtonian mehanis, downstream speed is ( + v and upstream speed is ( v]

9 2.5 The Relativisti Conept of Spae and Time 9 One an note that t = OY OX = ( v ( + v = t 2, as + v = v, OX = OY Therefore, time is not absolute. It varies from one inertial frame to another inertial frame, i.e. t = t 2. In other words, every referene frame has its own partiular time. Therefore, onept of absolute time was ruled out by Einstein. The spae oordinate is transformed due to Galilean Transformation as x = x vt. Sine, time depends on frame of referene, therefore, distane between two points will also depend on the frame of referene. In other words, distane between two points is not absolute. Hene, Einstein also ruled out the onept of absolute spae. Einstein proposed his new relativisti onepts of spae and time based on the fat that observers who are moving relative to eah other measure the speed of light to be the same. This new onept of spae and time is not only valid for mehanial phenomenon but also for all optial and eletromagneti phenomenon and known as theory of relativity. To develop these new onepts of spae and time, Einstein replaed Galilean Transformation equations by a new type of transformation equations alled Lorentz transformation, whih was based on the invariant harater of the speed of light. The theory of relativity having new onept of spae and time is divided into two parts. [] Speial theory of relativity (STR [2] General theory of relativity (GTR. The STR deals with inertial systems, i.e. the systems whih move in uniform retilinear motion relative to one another. GTR deals with non-inertial system, i.e. it is an extension of inertial system to aelerated systems, i.e. the systems moving with aelerated veloity relative to one another. The GTR is applied to the area of gravitational field and based on whih laws of gravitation an be explained in a more perfet manner than given by Newton.

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