Wave Theory of Light

According to Newton’s corpuscular theory of light, the speed of light is more in denser medium than in rarer medium. But it was experimentally found wrong in Foucault’s experiment based on Newton’s corpuscular theory. Huygens proposed wave theory of light in which the velocity of light was found more in rarer medium than in denser medium which agreed with Foucault’s experiment. 

The assumptions of wave theory of light are; 
1. A source of light emits light waves in all directions. 
2. The light wave travels in the universe through a hypothetical medium call ether medium. 
3. Different colours of light are due to different wavelengths. 
4. The light wave is a longitudinal wave

All the phenomena such as reflection and refraction were explained on the basis of this theory. Beside these phenomena, other three phenomena where also discovered which are; Interference, Diffraction and Polarization. The phenomenon of polarization cannot be explained on the basis of this theory because in this theory, light is supposed to be longitudinal wave but actually it is transverse wave.

Wavefront and Wavelet

According to wave theory of light, a source of light emits light wave in all directions. The locus of points on which ether particles are in the same state of vibration is called wavefront. The propagation of light in a medium in a particular direction means the propagation of wavefront. The direction in which the wave travels is at right angle to the wavefront and a ray of light represents this direction. Hence, ray and wavefront are at right angle to each other. Each point on a wave surface can act like a new source of smaller spherical waves which are called wavelets.

The shape of the wavefront depends on the shape of the source and on the properties of the medium. Depending upon the shape of the source of light, wavefront is of three types;

Spherical wavefront: When the source of light is a point, then the wavefront will be spherical in shape.

Spherical Wavefront

Cylindrical wavefront: When the source of light is linear in shape, the wave front will be cylindrical in shape.

Cylindral wavefront

Plane wavefront: When a source of any shape is at a very large distance, then the wavefront is plane in shape.

Plane Wavefront

Huygens’ Principle

Huygens gave the following postulates to describe the propagation of wavefront; 
1. Every point of the primary wavefront is the origin of the secondary wavelets. 
2. The wavelet travels with the speed of light. The backward wavefront is not taken in consideration. 

Consider a point source $S$ from where the primary wavefront $AB$ is drawn. After certain time $t$, a secondary wavefront $A’B’$ is constructed. Then, every point of $AB$ is a new source of secondary wavefront. As shown in the figure given below, spheres of radius $ct$ can be formed in different points of the primary wavefront $AB$. Another surface ${A}^{”}{B}^{”}$ touching all these spheres is formed. Then, $A’B’$ is the forward wavefront and $A^{”}B^{”}$ is the backward wavefront. Energy cannot be transmitted backward, therefore, $A’B’$ is the secondary wavefront.

Huygens' Principle

Laws of Reflection on the basis of Wave Theory of Light

Laws of reflection are; 
1. The incident ray, the reflected ray and the normal drawn at the point of incidence all lie in the same plane. 
2. Angle of incidence is equal to angle of reflection.
Explanation:

Laws of Reflection on the basis of wave theory of light

​Let $MN$ be a reflecting surface in which a primary wavefront $AB$ is incident at an angle of incidence $i$. Let $1$, $2$ and $3$ light rays of the wavefront $AB$ are incident which are reflected in the form of waves $1’$, $2’$ and $3’$. If the point $B$ takes time $t$ to come to the point $C$, then the distance $BC$ is $ct$. $AD=ct$ is cut, then, $CD$ is the secondary reflected wavefront. Let the angle of reflection made by $CD$ on reflecting surface be $r$.

\[\text{In ΔABC,}\] \[\sin i=\frac{BC}{AC}\text{ __(1)}\] \[\text{In ΔADC,}\] \[\sin r=\frac{AD}{AC}\text{ __(2)}\] \[\text{Dividing (1) by (2),}\] \[\frac{\sin i}{\sin r}=\frac{BC}{AD}=\frac{ct}{ct}=1\] \[\sin i=\sin r\] \[i=r\] Hence, angle of incidence is equal to the angle of reflection.

Laws of Refraction on the basis of Wave Theory of Light

Laws of refraction are;
1. The incident ray, the refracted ray and the normal drawn at the point of incidence all lie in the same plane.
2. The ratio of sine of angle of incidence in the first medium to the sine of angle of refraction in the second medium is a constant which is called refractive index of the second medium with respect to the first medium. \[\frac{\sin i}{\sin r}=\text{constant}\] \[\frac{\sin i}{\sin r}={}_1μ_2\] \[\frac{\sin i}{\sin r}=\frac{μ_2}{μ_1}\] This is known as Snell’s law
Explanation:

Laws of Refraction on the basis of wave theory of light

​Let $MN$ be a refracting surface which separates two mediums. Above $MN$, there is a rarer medium of refractive index $μ_1$ in which velocity of light is $v_1$. If $c$ is the velocity of light in vacuum, then, \[μ_1=\frac{c}{v_1}\] \[v_1=\frac{c}{μ_1}\text{ __(a)}\]

Below $MN$, there is a denser medium of refractive index $μ_2$ in which velocity of light is $v_2$. \[μ_2=\frac{c}{v_2}\] \[v_2=\frac{c}{μ_2}\text{ __(b)}\] \[\text{Dividing (a) by (b),}\] \[\frac{v_1}{v_2}=\frac{μ_2}{μ_1}\]

Let a primary wavefront $AB$ is incident to the refracting surface $MN$ at an angle of incidence $i$. Let $1$, $2$ and $3$ light rays of the wavefront $AB$ are incident which are refracted in the form of waves $1’$, $2’$ and $3’$. If the point $B$ takes time $t$ to come to the point $C$, then the distance $BC$ is $v_1t$. $AD=v_2t$ is cut, then, $CD$ is the secondary refracted wavefront. Let the angle of refraction made by $CD$ on refracting surface be $r$.

\[\text{In ΔABC,}\] \[\sin i=\frac{BC}{AC}\text{ __(1)}\] \[\text{In ΔADC,}\] \[\sin r=\frac{AD}{AC}\text{ __(2)}\] \[\text{Dividing (1) by (2),}\] \[\frac{\sin i}{\sin r}=\frac{BC}{AD}=\frac{v_1t}{v_2t}=\frac{μ_2}{μ_1}\] \[\frac{\sin i}{\sin r}={}_1μ_2\] This proves the Snell’s law.


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