M A M EL-Morsy Optics II Nature of light and wave
NATURE OF LIGHT
1-NEWTON'S CORPUSCULAR THEORY
The branch of optics that deals with the production, emission andpropagation of light, its nature and the study of the phenomena of interference, diffraction and polarization is called physical optics. The basic principles regarding the nature of light were formulated in the latter half of the seventeenth century. Until aboutthis time, the general belief was that light consisted of a stream of particles called corpuscles. These corpuscles were given out by a light source (an electric lamp, a candle, sunetc.) and they travelled in straight lines with large velocities. The originatorof theemission or corpuscular theory was Sir Isaac Newton. Accordingto this theory, a luminous body continuously emits tiny, light and elasticparticles called corpuscles in all directions. These particles or corpusclesare so small that they can readily travel through the interstices of the particlesof matter with the velocity of light and they possess the propertyof reflection from a polished surface or transmission through a transparentmedium. When these particles fall on the retina of the eye, they producethe sensation of vision. On the basis of this theory, phenomena like rectilinear propagation, reflection and refraction could be accounted for, satisfactorily. Since the particles are emitted with high speed from a luminousbody, they, in the absence of other forces, travel in straight lines accordingto Newton's second law of motion. This explains rectilinear propagationof light.
Fig. 1
2-REFLECTION OF LIGHT ON CORPUSCULAR THEORY
LetSS' be a reflecting surface andIM the path of a light corpuscleapproaching the surfaceSS'. When the corpuscle comes within a very smalldistance from the surface (indicated by the dotted lineAB) it, accordingto the theory, begins to experience a force of repulsion due to the surface(Fig. 1)
The velocityv of the corpuscle atMcan be resolved into two components x andy parallel and perpendicular to the reflecting surface. Theforce of repulsion acts perpendicular to the surface SS' and consequentlythe component y decreases up to Oand becomes zero at O the point ofincidence on the surfaceSS' .Beyond O, the perpendicular component of the velocity increases up toN. its magnitude will be againy atN but in the opposite direction. Theparallel componentx remains thesame throughout. Thus atN, thecorpuscle again possesses two components of velocity x andv and theresultant direction of the corpuscleis alongNR. The velocity of thecorpuscle will bev. Between thesurfaces AB andSS', the path ofthe corpuscle is convex to the reflecting surface. Beyond the point N, theparticle moves unaffected by the presence of the surfaceSS'.
x = v sin i = v sin r, theni = r
Further, the angles between the incident and the reflected paths ofthe corpuscles with the normalsat Al andN are equal. Also, the incidentand the reflected path of the corpuscle.and the normal lie in the sameplane viz. the plane of the paper.
3-REFRACTION OF LIGHT ON CORPUSCULAR THEORY
Newton assumed that when a light corpuscle comes within a verysmall limiting distance from the refracting surface, it begins to experiencea force of attraction towards the surface. Consequently the component ofthe velocity perpendicular to the surface increases gradually fromAB to A' S' is the surface separating the two media (Fig. 2).IM is theincident path of the corpuscle travelling in the first medium with a velocityv and incident at an anglei. AB toA' B' is a narrow region within whichthe corpuscle experiences a force of attraction.NR is the refracted pathof the corpuscle. Letv siniand v cosibe the components of the velocityof the corpuscle atM parallel and perpendicular to the surface. The velocity parallel to the surface increases by an amount which is independentof the angle of incidence, but which is different for different materials.Letv and v' be the velocity of the corpuscle in the two media and r theangle of refraction in the second medium.
As the parallel component of the velocity remains the same,
Fig. 2
Thus, the sine of the angle of incidence bears a constant ratio to the sine of the angle of refraction. This is the well known Snell's law of refraction. If i > r, then v\ > v. i.e., the velocity of light in a denser medium like water or glass is greater than that in a rarer medium such as air.
But the results of Foucault and Michelson on the velocity of light show that the velocity of light in a denser medium is less than that in a rarer medium. Newton's corpuscular theory is thus untenable. This is not the only ground on which Newton's theory is invalid. In the year 1800, Young discovered the phenomenon of interference of light. He experimentally demonstrated that under certain conditions, light when added to light produces darkness. The phenomena belonging to this class cannot be explained, if following Newton, it is supposed that light consists of material -particles. Two corpuscles coming together cannot destroy each other.
Another case considered by Newton was that of simultaneous reflection and refraction. To Explain this he assumed that the particles had fits.so that some were in a state favourable to reflection and others were ina condition suitable for transmission. No explanation of interference, diffraction and polarization was attempted because very little was knownabout these phenomena at the time of Newton. Further, the corpusculartheory has not given any plausible explanation about the origin of the forceof repulsion or attraction in a direction normal to the surface.
4-ORIGIN OF WAVE THEORY
The test and completeness of any theory consists in its ability to explain the known experimental facts, with minimum number of hypotheses.From this point of view, the corpuscular theory is above all prejudices andwith its help rectilinear propagation, reflection and refraction could he explained.
By about the middle of the seventeenth century, while the corpusculartheory was accepted, the idea that light might be some sort of wave motionhad begun to gain ground. In 1679, Christian Huygens proposed the wave theory of light. According to this, a luminous body is a source of disturbance in a hypothetical medium called ether. This medium pervades all space. The disturbance from the source is propagated in the form of wavesthrough space and the energy is distributed equally, in all directions. Whenthese waves carrying, energy are incident on the eye, the optic nerves areexcited and the sensation of vision is produced. These vibrations in thehypothetical ether medium according to Huygens are similar to those produced in solids and liquids. They are of a mechanical nature. The hypothetical ether medium is attributed the property of transmitting elasticwaves, which we perceive as light. Huygens assumed these waves to belongitudinal, in which the vibration of the particles is parallel to the direction of propagation of the wave.
Assuming that energy is transmitted in the form of waves, Huygenscould satisfactorily explain reflection, refraction and double refraction noticed in crystals like quartz or calcite. However, the phenomenon of polarization discovered by him could not he explained. It was difficult toconceive unsymmetrical behaviour of longitudinal waves about the axis ofpropagation. Rectilinear propagation of light also could not he explainedon the basis of wave theory, which otherwise seems to be obvious according to corpuscular theory. The difficulties mentioned above were overcome, when Fresnel and Young suggested that light waves are transverseand not longitudinal as suggested by Huygens. In a transverse wave, thevibrations of the ether particles take place in a direction perpendicular to.the direction of propagation. Fresnel could also explain successfully therectilinear propagation of light by combining the effect of all the secondarywaves starting from the different points of a primary wave front.
5-WAVE MOTION
Before proceeding to study the various optical phenomena on thebasis of Huygens wave theory, the characteristics of simple harmonicmotion (the simplest form of wave motion) and the composition orsuperposition of two or more simple harmonic motions are discussed. Thepropagation of a simple harmonic wave through a medium can be transverse or longitudinal. In a transverse wave, the particles of the mediumvibrate perpendicular to the direction of propagation and in a longitudinalwave, the particles of the medium- vibrate parallel to the direction of propagation. When a stone is dropped on the surface of still water, transversewaves are produced. Propagation of sound through atmospheric air is in theform of longitudinal waves. When a wave is propagated through a medium,the particles of the medium are displaced from their mean positions of restand restoring forces come into play. These restoring forces are due to theelasticity of the medium, gravity and surface tension. Due to the periodicmotion of the particles of the medium, a wave motion is produced. At any instantthe contour of all the particles of the medium constitutes a wave.
LetP be a particle moving on the circumference ofa circle of radiusa with a uniform velocity v (Fig. 3),Let be the uniform angularvelocity of the particle(v = a ).The circle along whichP moves is called the circleof reference. As the particleP movesround the circle continuously with uniformvelocity, the foot of the perpendicularM, vibrates along the diameter YY' or (XX'). Ifthe motion ofP is uniform, then the motionofM is periodic i.e., it takes the same timeto vibrate once between the points Y andY\. At any instant, the distance ofM fromthe centre O of the circle is called the displacement. If the particle moves fromX toP in time t, then POX = MPO = = t
Fig. 3
From the MPO
OMis calledthe displacement of the vibrating particle. The displacement of a vibrating particle at any instant can be defined as its distancefrom the mean position of rest. The maximum displacement of a vibratingparticle is called its amplitude.
Displacement =y = a sin t (i)
The rate of change of displacement is called the velocity of the vibrating particle:
The rate of change of velocity of a vibrating particle is called its acceleration.
Acceleration = Rate of change of velocity
The changes in the displacement, velocity and acceleration of a vibrating particle in one complete vibration are given in the following table
Thus, the velocity of the vibrating particle is maximum (in the direction OY or OY\ ) at the mean position of rest and zero at the maximum position of vibration. The acceleration of the vibrating particle is zero at the mean position of rest and minimum at the maximum position of vibration. The acceleration is always directed towards the mean position of rest and is directly proportional to the displacement of the vibrating particle. This type of motion, where the acceleration is directed towards a fixed position (the mean position of rest) and is proportional to the displacement of the vibrating particle, is called simple harmonic motion.
Further,
Thus, in general, the time period of a particle vibrating simple harmonically is given by T = where K is the displacement per unit acceleration.
If the particle P revolves round the circle, n times per second, then the angular velocity is given by
Fig. 4
On the other hand, if the time is counted [Fig. 4 (i)]from the instant P is at S (SOX = ) then the displacement
If the time is from the instant P is at Su[Fig. 7.4 (ii)], then
Phase of the vibrating particle. (i) The phase of the vibrating particle is defined as the ratio of the displacement of the vibrating particleat any instant to the amplitude of the vibrating particle , or(ii) it is also defined as the fraction of the time interval Mat has elapsed since the particle crossed the mean position of rest in the positive direction,or(iii) it is also equal to the angle swept by the radius vector since the vibrating particle last crossed its mean position of rest, e.g., in the above equations t , (t + ) or (t -) are called phase angles. The initial phase angle when t = 0, is called the epoch. Thus is called the epoch in the above expression.
Representation of S.H.M. by a wave.
Fig. 5
Let P be a particle movingon the circumference of a circle of radius a. The foot of the perpendicular vibrates on the diameter YY
The displacement graph is a sine curve represented by ABCDE (Fig. 5).The motion of the particle M is simple harmonic. This is the type of motion that can be expected in the case of elastic media, where the deforming forces obey Hook's law. The distance AE, after which the curve repeats itself, is called the wavelength and it is denoted by .
6-EQUATION OF A SIMPLE HARMONIC WAVE
The equation y = a sin t represents the displacement of a single particle vibrating simple harmonically. Let O, A, B, C etc, be different particles in the medium. Let the distance of the particle A, B, C etc. from the particle Obex1, x2' x3etc. Let t1, t2' t3 etc., be the time intervals taken by the wave to travel from the point Oto the points A, B, C etc. The displacement of the particle O at any instant is given by
wherev is the velocity of the wave. Thus the displacement of the particles A, B etc., will be given by the equation
Substitution
In equation (iii) , is called the phase and is the phase difference between the vibrating particle at Oand A. The distance travelled by the disturbance in time T is and in time t1is x1.
Thus equation (iii) can also be written as
If the distance x1 = then the phase difference =
and
i.e., the phase difference between the particles Oand A will be zero or the, two particles vibrate in phase. Similarly, all the particles distant 2, 3 etc., from Owill be vibrating in phase.
7-KINETIC ENERGY Vr A VIBRATING PARTICLE
The displacement of vibration particle is given by
If m is the mass of the vibrating particle, then the kinetic energy at any instant given by
The average kinetic energy of the particle in one complete vibration
wheremis the mass of the vibrating particle, ais the amplitude of vibration and n is the frequency of vibration. Also, the average kinetic energy of a vibrating particle is directly proportional to the square of the amplitude.
8-TOTAL ENERGY OF A VIBRATING PARTICLE
But
The kinetic energy at the instant of the displacement is y
Potential energy of the vibrating particle is the amount of work done in overcoming the force through a distance y
Acceleration = -2 y
Force = - m 2 y
The – ve sign show that the direction of the acceleration and force are opposite to the direction of motion of the vibrating particle
Total energy of the particle at the instant the displacement is y
T .E = K.E +P.E.
But
As the average kinetic energy of the vibrating particle = , the average potential energy = . the total energy at any instant is constant.
9-COMPOSITION OF TWO SIMPLE HARMONIC MOTIONS IN A STRAIGHT LINE
(a) Analytical method.
Let the two simple harmonic vibrations be represented by the equations
where y1 and y2 are the displacements of a particle due to the two vibrations, a1 and a2are the amplitudes of the two vibrations x1and x2are the epoch angles. Here, the two vibration are assumed to be of the same frequency and hence is the same for both. The resultant displacement y of the particle is given by
Since the amplitude a1 and a2 and the angles 1 and 2are constant, the coefficients of in equation (iii) can be substituted by
Squaring and adding equations (iv) and (v)
Dividing equation (v) and (iv)
Equations (vi) and (vii) give the values of A and in terms of a1, a2, x1and x2,
Equation (viii) is similar to the original equations (i) and (ii). The amplitude of the resultant vibration is A and epoch angle is . The time period of the resultant vibration is the same as the original vibrations. The values of A and are given by equations (vi).and (vii). Thus, the resultant of two simple harmonic vibrations of the same period and acting in the same line is also a simple harmonic vibration with a resultant amplitude A and epoch angle .
Case (i) if
(b) Graphical method.
Let OP and OQ represent the radius vectors at any instant ( Fig. 6)
From the OPB andOQC
But the projection of OQ on the Y axis is equal to the projection of PR on the y axis.
Similarly the projection of OP and OQ on the X axis will be
Thus, the diagonal OR represents completely the resultant of two collinear simple harmonic motions. The resultant amplitude Aand the epoch angle are given by the equations (i) and (iii). The resultant displacement is represented by the equation
9- COMPOSITION OF TWO SIMEPLE HARMONICVIBRATIONS ACTING AT RIGHT ANGLES
Let
represent the displacement of a particle along the x and y axis due to the influence of two simple harmonic vibrations acting simultaneously on a particle in perpendicular directions. Here, the two vibrations arc of the same time period but are of different amplitudes and different phase angles.
From equation (ii),
From equation (i)
Substituting the value of in equation (iii)
Squaring
This represents the general equation of an ellipse. Thus, due to the superimposition of two simple harmonic vibrations, the displacement of the particle will be along a curve given by equation (iv).
Special cases :
10- NATURE OF LIGHT
(i) Corpuscular theory. Rectilinear propagation of light is a natural deduction on the basis of corpuscular theory. This theory can also explain reflection and refraction, though the theory does not clearly envisage why, how and when the force of attraction or repulsion is experienced perpendicular to the reflecting or refracting surface by a corpuscle. Newton assumed that the corpuscles possess fits which allow them easy reflection at one stage and easy transmission at the other. According to Newton'scorpuscular theory the velocity of light in a denser medium is higher thanthe velocity in a rarer medium. But the experimental results of Foucaultand Michelson show that the velocity of light in a rarer medium is higherthan that in a denser medium. Interference could riot be explained on thebasis of corpuscular theory because two material particles cannot cancelone another's effect. The phenomenon of diffraction viz., bending of lightround corners or illumination of geometrical shadow cannot be conceivedaccording to corpuscular theory, because a corpuscle travelling at highspeed will not be deviated from its straight line path. Certain crystalslike quartz, calcite etc. exhibit the phenomenon of double refraction. Explanation of this has not been possible with the corpuscle concept. Theunsymmetrical behaviour of light about the axis of propagation (viz. polarization of light) cannot be accounted for by the corpuscular theory.
(ii)Wave theory. Huygens wave theory could explain satisfactorilythe phenomena of reflection and refraction. Applying the principle of secondary wave points, rectilinear propagation of light can be correlated. Thephenomenon of interference can also be understood considering that lightenergy is propagated in the form of waves. Two wave trains of equal frequency and amplitude and differing inphase can annul one another's effectandproduce darkness. Similar to sound waves, bending of waves round obstacles is possible, thus enabling the understanding of the phenomenon ofdiffraction. Double refraction can also be explained on the basis of wavetheory. According to Huygens, propagation of lightis in the form of longitudinal waves. But in the caseof longitudinal waves, one cannot expectthe unsymmetrical behaviour of a beam of light about the axis of propagation. This difficulty was overcome when Fresnel suggested that the lightwaves are transverse and not longitudinal. On the basis of this concept,the phenomenon of polarization can also be understood. Finally, on thebasis ofwave theory it can be shown mathematically, that the velocity oflight in a rarer medium is higher than the velocity of light in a denser medium.This is in accordance with the experimental results -on the velocity of light.