Wave Nature of Light

aNature of LightOptics is a branch of physics which deals with study of phenomena like reflection, refraction, dispersion, interference, diffraction etc.This note provides us an information on nature and sources of light. Nature and Sources of LightOptics is a branch of physics which deals with study of phenomena like reflection, refraction, dispersion, interference, diffraction etc. The subject of optics has been divided into ray optics and wave optics. The branch of optics that deals with the production, emission and propagation of light, its nature and the study of the phenomenon of interference, diffraction and polarization is called wave optics.Nature and Sources of LightScientists have made discussion on true nature of light and have postulated various theories. These are discussed below: Newton’s Corpuscular TheoryNewton's theory of corpuscles, which are tiny particles shot out by luminous objects, states that they travel in straight lines and bounce off or pass into an object striking on it. However, this theory cannot explain interference, diffraction and polarization of light, making it incomplete and inaccurate.Huygens’ Wave TheoryAccording to this theory, light propagates from the source in the form of a wave. For the propagation of wave a medium is necessary. So, it was assumed that space is filled with medium called ether, which has the property of both elasticity and inertia. This theory says that light wave is longitudinal but actually light wave is transverse.Electromagnetic TheoryTheoretically light gets propagated in the form of electromagnetic waves, consisting of electric and magnetic fields mutually perpendicular as well as transverse to the direction of propagation of light. The electromagnetic waves propagate in free space with the velocity of light. Hertz demonstrated experimentally that electromagnetic waves propagate with velocity equal to that of light.Quantum Theoryin 1905 AD Einstein proposed a new theory of light called a quantum theory, in order to explain the photoelectric effect. According to this theory, light is transmitted as tiny packets of energy called photons and energy of each photon is given by E=hfwhere f is the frequency of light and h is Planck’s constant.Duel Nature of LightLight can exist in both particle and wave form. Using quantum theory, we can explain photoelectric effect. But this theory could not explain interference, diffraction and polarization. These phenomena can be explained using wave theory but wave theory could not explain photoelectric effect. Wave front According to wave theory, a source of light sends out disturbance or waves in all directions. A wave front as the locus of all adjacent points at which the phase of vibrations of a physical quantity associated with the wave is the same.

Types of Wave frontDepending on the nature of the source of light, there are three types of wave fronts:

Spherical Wave frontIt is produced by the point source of light. This is all because such points which are equidistant from the point source will lie on a sphere. A spherical wave front is shown in figure.Cylindrical Wave frontWhen the source of light is linear in shape (e.g a slit), all points equidistant from the linear source lie on the surface of a cylinder. Such a wave front is called a cylindrical wave front as shown in the figure.Plane Wave frontA small portion of cylindrical wave front or spherical wave front originating from a distant source will appear a plane and hence termed as plane wave front. Huygens‘ PrincipleHuygens ’ Principle enables us to determine what its position will be at some later time. In other words, the principle gives a method to know as to how light spread out in the medium. A source of light sends out wave front is propagated forwards through a homogeneous isotropic medium, Christian Huygens made the following assumptions. 1. Each point on a wave front acts as a new source of the disturbance. The disturbances from these points are secondary wave lets. These wavelets spread out in all directions in the medium with the velocity of light.2. The new wave front is then obtained by constructing a tangential plane to all the secondary wavelets. The new wave front is the envelope of to secondary wavelets at that instant.Laws of Reflection on the Basis of Wave TheoryConclusion of laws of reflection on the basis of wave theory:1. The angle of reflection r$r$ is equal to the angle of incidence i for all wavelengths and for any pairs of materials.r=i$$r=i$$2. The incident ray, reflected ray and normal to the reflecting surface all lie on the same plane.

If you look at the figure, AA’ is the wavefront incident on a reflecting surface XY having an angle of incidence i. Following Huygen’s principle, every point on AA’ will act as a source of secondary wavelets. Time taken from A’ to D = time taken from B’ to C A’D / v = B’C / v A’D = B’C A’C sin(i) = A’C sin(r) Thus, i = r Hence, the angle of incidence and angle of reflection are both equal. This is also stated by the first law of reflection. Additionally, as the incident wavefront AB, the normal and reflected wavefront are on the same plane, we can also verify the second law of reflection.Laws of Refraction on the Basis of Wave Theory

Let the angle of incidence be i$i$ and the refraction is r$r$.From the 𝛥A1B′B1$𝛥{\mathrm{A}}_1{\mathrm{B}}^{\prime }{\mathrm{B}}_1$, we getSin∠B′ A1B1=Sini≡B′B

AB1$\mathrm{S}\mathrm{i}\mathrm{n}\angle B^{\prime }{A}_{1}B1=\mathrm{S}\mathrm{i}\mathrm{n}\mathrm{i}\equiv \frac{B^{\prime }B}{A{B}_1}$From the △A1CB1$△{\mathrm{A}}_1\mathrm{C}{\mathrm{B}}_1$, we getSin∠A1B1C=Sinr=A1C

AB1$\mathrm{S}\mathrm{i}\mathrm{n}\angle A_{1}B1C=\mathrm{S}\mathrm{i}\mathrm{n}r=\frac{A_{1}C}{A{B}_1}$aThus Sini

Sinr=B′B

A1C$\begin{array}{l}Thus\\ \frac{\mathrm{S}\mathrm{i}\mathrm{n}i}{\mathrm{S}\mathrm{i}\mathrm{n}r}=\frac{B^{\prime }B}{A_{1}C}\end{array}$ =v1t

v2t$=\frac{{\mathrm{v}}_1\mathrm{t}}{{\mathrm{v}}_2\mathrm{t}}$or, Sini

Sinr=v1

v2=1 𝜇2$\frac{\mathrm{S}\mathrm{i}\mathrm{n}i}{\mathrm{S}\mathrm{i}\mathrm{n}r}=\frac{{\mathrm{v}}_1}{{\mathrm{v}}_2}{=}^1{𝜇}_2$ (constant)This proves Snell's law of refraction. The constant 1𝜇2${}^1{𝜇}_2$ is called the refractive index of the second medium w.r.t the first medium. Questions: Discuss about merits and demerits of Huygens' theory. Ans: Merits of Huygens' wave theory of light 1. It gives satisfactory explanation for laws of reflection, refraction and double refraction.2. It also explains the theory of interference and diffraction. 3. It experimentally proved that velocity of light in rarer medium is greater than that in a denser medium. De-merits of Huygens wave theory1. It could not explain Compton Effect, photoelectric effect, Raman Effect, rectilinear propagation of light etc. 2. It could not explain properly the propagation of light through vacuum (as it assumes the presence of material medium i.e. ether for the propagation of light wave in the whole universe).3. This theory is silent about the backward envelop of secondary wavelets. NumericalThe velocity of light in air is 3 x 108 m/s Find the velocity and wavelength of sodium light (𝜆${\displaystyle 𝜆}$= 5893Å${\displaystyle Å}$) in glass of refractive index 1.658. Given, Velocity of light in air (c) = 3 x 108 m/s Wavelength in air ( 𝜆${\displaystyle 𝜆}$) = 5893Å${\displaystyle Å}$ Refractive index of glass (µ) = 1.658 Velocity of light in glass (vg) = ? Wavelength of light in glass (𝜆g${\displaystyle {𝜆}_g}$) = ? We know that, 𝜇 = c

vg${\displaystyle 𝜇=\frac{c}{v_g}}$ vg = c

𝜇${\displaystyle v_g=\frac{c}{𝜇}}$ vg = 3 ×108

1.658=1.809×108 m/s${\displaystyle v_g=\frac{3\times {10}^8}{1.658}=1.809\times {10}^{8}m/s}$ We know For air c= f ×𝜆${\displaystyle c=f\times 𝜆}$ f=c

𝜆=3×108

5893 ×10-10${\displaystyle f=\frac{c}{𝜆}=\frac{3\times {10}^8}{5893\times {10}^{-10}}}$When light travels from one medium to another, its frequency remains unchanged. So, We can use the following relation For glassvg= f ×𝜆g${\displaystyle v_g=f\times {𝜆}_g}$ 𝜆g = vg

f= 1.809×108×5893×10-10

3×108${\displaystyle {𝜆}_g=\frac{v_g}{f}=1.809\times {10}^8\times \frac{5893\times {10}^{-10}}{3\times {10}^8}}$ = 3554.28 Å${\displaystyle =3554.28Å}$

Math 80 03

 

Math 80 2