Huygens’ Principle and Wave-fronts Questions in English

(B) In the case of refraction, if $CD$ is the refracted wavefront and $v_1$ and $v_2$ are the speeds of light in air and glass respectively, then in the time the wavelet from $B$ reaches $C$, the wavelet from $A$ reaches $D$.
Thus, the time taken is $t = \frac{BC}{v_a} = \frac{AD}{v_g}$, which implies $\frac{BC}{AD} = \frac{v_a}{v_g}$. .....$(i)$
From the geometry of the diagram, in $\Delta ACB$, $BC = AC \sin \theta$. .....$(ii)$
In $\Delta ACD$, $AD = AC \sin \phi '$. .....$(iii)$
Substituting $(ii)$ and $(iii)$ into $(i)$, we get $\frac{v_a}{v_g} = \frac{\sin \theta}{\sin \phi '}$.
Since the refractive index $\mu = \frac{v_a}{v_g}$, we have $\mu_g = \frac{\sin \theta}{\sin \phi '}$.

(B) Huygens' principle states that every point on a primary wavefront acts as a source of secondary spherical wavelets,which spread out in all directions with the speed of light in the medium. The new wavefront at a later time is the forward envelope of these secondary wavelets. Therefore,it is a geometrical method to find the shape and position of a wavefront at any subsequent time. Thus,option $B$ is correct.

(C) Huygen's wave theory treats light as a wave propagating through a medium.
It successfully explains wave phenomena such as interference,diffraction,and polarization.
However,it fails to explain the particle nature of light,such as the photoelectric effect,which requires the quantum theory of light (photons).

(B) The concept of secondary wavelets for the propagation of a wave was proposed by Christiaan Huygens in his wave theory of light. According to Huygens' principle,every point on a primary wavefront acts as a source of secondary spherical wavelets,which spread out in all directions with the speed of light in the medium. The new wavefront at a later time is the forward envelope of these secondary wavelets. Therefore,the correct option is $B$.

(A) wavefront is defined as the locus of all points (particles) in a medium that are vibrating in the same phase at a given instant of time. Therefore,all particles on a wavefront have the same phase of vibration. Hence,the correct option is $A$.

(B) By definition,a wavefront is the locus of all points having the same phase of oscillation.
In any wave propagation,the direction of wave motion (the ray direction) is always perpendicular to the wavefront at every point.
Therefore,the correct relationship is that the wavefront is perpendicular to the direction of wave motion.

(D) Huygens' principle is a geometric method used to determine the shape of a wavefront at any time $t'$ if its shape at time $t$ is known.
Using this principle,one can successfully explain the laws of reflection,refraction,and the phenomenon of diffraction.
However,Huygens' wave theory does not provide an explanation for the origin of spectra,which is related to the quantum nature of light and atomic energy levels.
Therefore,the correct option is $D$.

(B) wave front is defined as the locus of all points that are in the same phase of oscillation. For a point source of light,the light waves travel in all directions with the same speed in a homogeneous medium. Therefore,the distance of all points in the same phase from the source is equal,forming a sphere. Thus,the wave front originating from a point source is spherical.

(C) Newton's corpuscular theory of light was primarily based on the observation that light travels in straight lines,which is known as the rectilinear propagation of light.
According to Newton,light consists of tiny,massless particles called corpuscles that travel at very high speeds.
Since these particles move in straight lines in a vacuum or a uniform medium,this theory successfully explained the rectilinear propagation of light.
Therefore,the correct option is $C$.

(C) Huygens' principle states that every point on a wavefront acts as a source of secondary spherical wavelets,which spread out in the forward direction at the speed of light.
By drawing a common tangent surface to all these secondary wavelets at a later time $t$,we can determine the new position of the wavefront.
Therefore,the principle is primarily used to find the new position of the wavefront.

(D) Huygens' principle is a geometrical construction that allows us to determine the position of a wavefront at any later time. According to this principle,every point on a primary wavefront acts as a source of secondary wavelets,and the new wavefront is the forward envelope of these secondary wavelets. Therefore,it primarily explains the propagation of wave fronts.

(C) The wave theory of light was proposed by Christiaan Huygens in $1678$.
According to this theory,light travels in the form of waves,and every point on a wavefront acts as a source of secondary wavelets.

(B) In Huygens' principle, the wave theory was established by assuming that light waves were longitudinal, not transverse. So, option $(A)$ is incorrect.
Maxwell's electromagnetic theory of light provided the compelling theoretical evidence that light is an electromagnetic wave, which is transverse in nature. So, option $(B)$ is correct.
Thomas Young conducted the double-slit experiment to demonstrate the interference of light, which confirmed the wave nature of light. However, he did not specifically prove Huygens' assumption regarding the nature of the wave (longitudinal vs transverse). So, option $(C)$ is incorrect.
These statements highlight specific aspects of the wave nature of light but do not provide a complete definition of light, such as its dual nature (particle-wave duality). Therefore, they do not collectively answer the question "what is light" in its entirety. So, option $(D)$ is incorrect.

(A) parallel cylindrical beam of light consists of rays that are parallel to each other.
By definition,the wavefront is the locus of points having the same phase.
For a parallel beam of light,the wavefronts are planes perpendicular to the direction of propagation.
Therefore,the initial shape of the wavefront is plane.

(D) Huygens' wave theory successfully explains the laws of reflection,refraction,and diffraction by considering the propagation of wavefronts.
However,Huygens' original wave theory could not explain the origin of spectral lines or the emission of a spectrum,as it did not account for the quantum nature of light or the interaction of light with atomic energy levels.

(B) Huygens' wave theory successfully explained reflection,refraction,and the rectilinear propagation of light by assuming the existence of a hypothetical medium called 'ether'.
However,the theory faced significant criticism because the existence of the 'ether' medium could not be experimentally verified (notably by the Michelson-Morley experiment).
While Huygens' theory could explain many phenomena,the primary historical drawback cited in textbooks regarding the fundamental premise of his theory was the requirement of the 'ether' medium,which was later proven non-existent.
Therefore,the failure of experimental verification of the ether medium is considered a major drawback.

(D) Huygens' principle is a geometric method used to determine the position of a new wavefront at a later time based on the knowledge of the wavefront at an earlier time.
It is based on the concept that every point on a wavefront acts as a source of secondary wavelets.
This principle is a general wave theory and is applicable to all types of waves,including light waves (electromagnetic waves),sound waves (mechanical waves),and water waves.
Therefore,it is applicable to all the waves mentioned above.

(D) Huygens' principle states that every point on a wavefront acts as a source of secondary wavelets,and the new wavefront is the envelope of these secondary wavelets at a later time.
This principle is a fundamental tool in wave optics that successfully explains the laws of reflection and refraction of light.
Furthermore,it provides a theoretical basis for understanding the phenomenon of diffraction,where light bends around corners or obstacles.
Since it explains propagation,reflection,and diffraction,the correct answer is $D$.

(C) Huygens' principle states that every point on a wave front acts as a source of secondary wavelets.
These secondary wavelets spread out in all directions with the speed of the wave in the medium.
The new wave front at any later time is the forward envelope of these secondary wavelets.
Therefore,Huygens' principle is primarily used to determine the geometric propagation and position of a wave front at a subsequent time.

(C) The refractive index is given by $\mu(I) = \mu_0 + \mu_2 I$. Since the intensity $I$ is maximum at the axis and decreases as we move away from the axis (as shown by $x$ in the figure),the refractive index $\mu$ is also maximum at the axis and decreases as we move away from it.
Since the speed of light $v = c/\mu$,the speed is minimum at the axis and increases as we move away from the axis.
Consequently,the part of the wavefront near the axis travels slower than the parts further away from the axis.
This causes the wavefront to bend and converge towards the axis,resulting in a convex shape as shown in the figure.

(D) Huygens' principle is a geometric construction used to determine the position of a new wavefront at a later time based on the current wavefront.
It successfully explains the laws of reflection and refraction.
It also explains the phenomenon of diffraction by considering each point on the wavefront as a source of secondary wavelets.
However,Huygens' principle (in its original form) does not explain the origin of spectral patterns (like the photoelectric effect or the discrete nature of atomic spectra),as these are quantum mechanical phenomena that require the concept of photons and energy quantization,which are beyond the scope of classical wave theory.

(C) According to Huygens's principle,the time taken by the wavefront to travel the distance $b$ in air is equal to the time taken to travel the distance $d$ in water.
Let $v_a$ be the speed of light in air and $v_w$ be the speed of light in water.
Then,the time $t = \frac{b}{v_a} = \frac{d}{v_w}$.
Rearranging the terms,we get $\frac{v_a}{v_w} = \frac{b}{d}$.
By definition,the refractive index of water with respect to air is $\mu = \frac{v_a}{v_w}$.
Therefore,$\mu = \frac{b}{d}$.

(A) The refractive index of the medium is given by $\mu(I) = \mu_0 + \mu_2I$. Since the intensity $I$ of the beam decreases as the radius increases,the refractive index $\mu$ also decreases from the axis towards the periphery.
Because the beam is initially parallel and cylindrical,the rays of light are parallel to each other. By definition,the wavefront of a parallel beam of light is a plane perpendicular to the direction of propagation.
Therefore,the initial shape of the wavefront is planar.

(C) According to the problem,the refractive index $\mu$ of air increases with height from the ground.
Since the speed of light $v$ in a medium is given by $v = c/\mu$,where $c$ is the speed of light in vacuum,the speed of light $v$ decreases as the height from the ground increases.
Consider a plane wavefront moving horizontally. The part of the wavefront at a greater height travels slower than the part of the wavefront closer to the ground.
According to Huygens' principle,each point on the wavefront acts as a source of secondary wavelets.
Because the lower part of the wavefront travels faster,the wavefront tilts,causing the direction of propagation (which is perpendicular to the wavefront) to bend upwards.
Therefore,the light beam bends upwards.

(C) When a plane wavefront is incident on a thin convex lens,the lens causes the light rays to converge towards the focus.
In diagram $(i)$,the wavefront after passing through the lens becomes spherical and converges towards the focus,which is the correct behavior for a convex lens.
In diagram $(iv)$,the wavefront after passing through the lens also becomes spherical and converges towards the focus,representing the same correct physical phenomenon.
Diagrams $(ii)$ and $(iii)$ show incorrect wavefront curvatures after refraction.
Therefore,$(i)$ and $(iv)$ are the correct representations.

(A) When spherical wavefronts strike a plane mirror,each point on the wavefront acts as a source of secondary wavelets according to Huygens' principle.
The reflected wavefronts will appear to originate from a virtual source located behind the mirror,which is the mirror image of the original point source.
Since the incident wavefronts are spherical and converging towards the mirror,the reflected wavefronts will also be spherical,appearing to diverge from the virtual image of the source point.
Based on the geometry of reflection,the reflected wavefronts will maintain their spherical shape and curvature,consistent with the virtual image formed by the mirror. This corresponds to the pattern shown in option $A$.

(B) When a plane wavefront enters a medium of refractive index $\mu,$ the speed of light decreases from $c$ to $\frac{c}{\mu}.$
Because the speed is lower in the medium,the part of the wavefront passing through the medium travels a shorter distance in the same time interval compared to the part traveling in vacuum.
This causes the wavefront to lag behind in the region where the medium is present.
As a result,the wavefront becomes discontinuous,with the portion inside the medium lagging behind the portion in the vacuum,forming a broken line shape at point $P$ as shown in option $(B).$

(C) wavefront is defined as the locus of all points having the same phase of oscillation.
Since $cd$ is a wavefront in medium-$1$,all points on the line $cd$ have the same phase. Therefore,$\phi_c = \phi_d$.
Similarly,$ef$ is a wavefront in medium-$2$,so all points on the line $ef$ have the same phase. Therefore,$\phi_e = \phi_f$.
Now,consider the expression $(\phi_d - \phi_f) = (\phi_c - \phi_e)$.
Since $\phi_c = \phi_d$ and $\phi_e = \phi_f$,we can substitute these into the expression:
$(\phi_d - \phi_f) = (\phi_c - \phi_e)$
$\phi_d - \phi_f = \phi_c - \phi_e$
This is a consistent identity based on the properties of wavefronts.

(B) wavefront is defined as the locus of all points that are in the same phase of oscillation. Thus,statement $A$ is correct.
Light rays are defined as lines perpendicular to the wavefront,indicating the direction of wave propagation. Thus,statement $C$ is correct.
Huygens proposed the existence of an 'ether' medium to explain the propagation of light waves,but the Michelson-Morley experiment later proved that no such medium exists. Thus,statement $D$ is correct.
Wavefronts can take various shapes depending on the source and the medium,such as cylindrical,spherical,or plane. Since wavefronts are not restricted to only spherical or plane shapes (e.g.,a line source produces a cylindrical wavefront),statement $B$ is incorrect.

(D) When a light source is at a very large distance from the observer,the radius of curvature of the wavefront becomes extremely large.
As the distance $r \to \infty$,the curvature $1/r \to 0$.
Consequently,a small portion of a spherical or cylindrical wavefront appears to be flat or planar.
Therefore,the wavefront of a distant source is approximately a plane wavefront.

(A) wave front is defined as the locus of all points in a medium that are in the same phase of vibration. This means that at any given instant,all points on the wave front have reached the same stage of their oscillatory cycle.

(D) For a point source of light, the light waves spread out in all directions, forming spherical wave fronts.
As the distance $r$ from the source increases, the surface area of the spherical wave front increases as $A = 4\pi r^2$.
Since the total power $P$ emitted by the source is constant, the intensity $I$ is given by $I = P / A = P / (4\pi r^2)$.
Therefore, the intensity $I$ is inversely proportional to the square of the distance $r$, i.e., $I \propto 1/r^2$.
This means the intensity decreases as the distance from the source increases.

(A) Light propagates rectilinearly due to its wave nature. According to the Huygens-Fresnel principle,every point on a wavefront acts as a source of secondary wavelets. In a homogeneous medium,these wavelets interfere constructively along the direction of propagation,resulting in rectilinear propagation.

(N/A) The shape of the wavefront for light diverging from a point source is spherical.
$(b)$ The shape of the wavefront for light emerging from a convex lens when a point source is placed at its focus is plane (planar).
$(c)$ The portion of the wavefront of light from a distant star intercepted by the Earth is plane (planar) because the source is at an effectively infinite distance.

(N/A) No,the prediction is not confirmed. According to Newton's corpuscular theory,when light corpuscles travel from a rarer medium (air) to a denser medium (water),they experience an attractive force normal to the interface. This force increases the normal component of the velocity,while the tangential component remains constant. This leads to the relation $v = \mu c$,where $v$ is the speed in the medium and $c$ is the speed in vacuum. Since the refractive index $\mu > 1$,the theory predicts $v > c$.
This prediction contradicts experimental results,which show that the speed of light in a denser medium is less than in a vacuum $(v < c)$. The wave theory of light,proposed by Huygens,correctly predicts that $v = c / \mu$,which is consistent with experimental observations.

(N/A) Let an object at $O$ be placed in front of a plane mirror at a distance $r$ (as shown in the figure).
$A$ spherical wavefront is emitted from the point object $O$. The part of the wavefront $XY$ reaches the plane mirror at point $O'$.
According to Huygens' Principle,each point on the wavefront acts as a source of secondary wavelets. When the wavefront $XY$ strikes the mirror,the mirror reflects these wavelets.
If the mirror were absent,the wavefront would continue to propagate and form a similar wavefront $X'Y'$ at a distance $r$ behind the mirror.
Since the mirror reflects the light,the secondary wavelets appear to originate from a point $I$ behind the mirror,which is the virtual image of $O$. By symmetry,the distance of this virtual image $I$ from the mirror is equal to the distance $r$ of the object $O$ from the mirror.

(N/A) The portion of the electromagnetic spectrum with wavelengths ranging from $4000 \ \mathring{A}$ to $8000 \ \mathring{A}$ is known as visible light. Light itself is invisible,but it allows us to see objects.
Various views on the nature of light are as follows:
$(1)$ Corpuscular model of light (Newton's particle theory of light):
- The corpuscular theory was proposed by Descartes in $1637$. It derived Snell's law and explained the laws of reflection and refraction at the interface of two media.
- The model predicted that if a light ray bends towards the normal during refraction,its speed must be greater in the second medium. Thus,it suggested that the speed of light is lower in a rarer medium and higher in a denser medium.
- This theory is famously associated with Isaac Newton,who considered light to be composed of tiny particles called corpuscles.
$(2)$ Huygens' wave theory:
- In $1678$,Christiaan Huygens proposed the wave theory of light.
- This theory successfully explained reflection and refraction. It predicted that if a wave bends towards the normal during refraction,its speed would be less in the second medium. This contradicted the prediction of the corpuscular model.
- In $1850$,Foucault experimentally proved that the speed of light in water is less than in air,which supported the wave theory over the corpuscular model.

(N/A) The wave theory was initially not accepted due to Newton's authority and the belief that light,unlike sound,could travel through a vacuum,whereas waves were thought to require a medium to propagate.
In $1801$,Thomas Young performed the interference experiment,which firmly established that light is a wave phenomenon.
Since the wavelength of visible light is much smaller than the dimensions of typical mirrors and lenses,light can be assumed to travel approximately in straight lines.
The branch of optics in which the finiteness of the wavelength is completely neglected is called geometrical optics.
$A$ ray is defined as the path of energy propagation in the limit where the wavelength tends to zero.
Numerous experiments involving the interference and diffraction of light waves were carried out and satisfactorily explained. Thus,by the middle of the $19^{th}$ century,the wave theory of light was well established.

(N/A) The primary theories explaining the propagation of light are:
$1$. $Corpuscular$ $Theory$: Proposed by $Isaac$ $Newton$,it suggests that light consists of tiny particles called corpuscles.
$2$. $Wave$ $Theory$: Proposed by $Christiaan$ $Huygens$,it suggests that light travels in the form of waves through a hypothetical medium called $luminiferous$ $ether$.
$3$. $Electromagnetic$ $Wave$ $Theory$: Proposed by $James$ $Clerk$ $Maxwell$,it states that light consists of oscillating electric and magnetic fields propagating through space.
$4$. $Quantum$ $Theory$ ($Photon$ $Theory$): Proposed by $Max$ $Planck$ and $Albert$ $Einstein$,it explains that light consists of discrete packets of energy called $photons$.

(B) According to Newton's Corpuscular theory,light travels faster in a denser medium (like water) than in a rarer medium (like air).
However,experimental measurements by Foucault and Fizeau showed that the speed of light is lower in water than in air.
This experimental result contradicted the Corpuscular theory and provided strong evidence in favor of the Wave theory of light,which correctly predicts that light slows down when entering a denser medium.

(N/A) The main difficulty in establishing the wave theory of light was the assumption that light waves are mechanical in nature.
According to the classical physics of that time,mechanical waves require a material medium for propagation.
Since light travels through the vacuum of space (e.g.,from the Sun to the Earth),it was difficult to explain how light could propagate without a medium.
This led to the hypothetical concept of the 'luminiferous ether',a pervasive,massless,and transparent medium,which was later disproven by the Michelson-Morley experiment.

(N/A) When we drop a small stone into a calm pool of water,waves spread out from the point of impact. Every point on the surface starts oscillating with time; hence,at any instant,the surface shows circular rings where the disturbance is maximum.
All points on such a circle are oscillating in phase because they are at the same distance from the source. Such a locus of points,which oscillate in phase,is called a wavefront. Thus,a wavefront is defined as a surface of constant phase.
The speed with which the wavefront moves outwards from the source is called the speed of the wave.
The energy of the wave travels in a direction perpendicular to the wavefront.
$A$ line perpendicular to the wavefront,indicating the direction of propagation of the wave,is called a ray. Hence,the wavefront and the ray are perpendicular.
Types of wavefronts:
$1$. Spherical Wavefront: If a point source emits waves uniformly in all directions,the locus of points having the same amplitude and vibrating in the same phase are spheres (in three dimensions). This is known as a spherical wavefront,as shown in figure $(a)$. Such waves are diverging.
$2$. Plane Wavefront: At a large distance from the source,a small portion of the sphere can be considered as a plane. It is known as a plane wavefront,as shown in figure $(b)$.
$3$. Cylindrical Wavefront: Wavefronts originating from a linear source and propagating in a three-dimensional homogeneous and isotropic medium are cylindrical wavefronts. For example,waves emanating from a tubelight,as shown in figure $(c)$.

Principle: Every point or particle of a wavefront behaves as an independent secondary source and emits secondary spherical waves by itself. After a very small time interval,the surface tangential to all such secondary spherical wavelets gives the position and shape of the new wavefront.
Basically,the Huygens' principle is a geometric construction.
Suppose that $F_{1} F_{2}$ represents a part of a spherical wavefront at $t=0$,which is a wave propagating outwards.
According to Huygens' principle,all points of this wavefront $(F_{1} F_{2})$ (e.g.,$A, B, C, \ldots$) behave as secondary sources. If the velocity of the wave is $v$,then the distance covered in time $\tau$ is $v \tau$.
To determine the shape of the wavefront at $t=\tau$,draw spheres of radius $v \tau$ from each point on the spherical wavefront and draw a common tangent to all these spheres. The surface tangent to these spheres at time $\tau$ gives the position and shape of the new wavefront,which is $G_{1} G_{2}$ in the forward direction. This is a spherical wavefront with center $O$. $A$ backward spherical wavefront $D_{1} D_{2}$ is also formed. The points $A^{\prime}, B^{\prime}, C^{\prime}$ on $G_{1} G_{2}$ act as new secondary sources.

(N/A) Principle: Every point on a wavefront acts as an independent source of secondary spherical wavelets,which spread out in all directions with the speed of light in that medium. The new wavefront at any later time is the forward envelope (tangential surface) of these secondary wavelets.
Explanation:
$1$. Huygens' principle is a geometric method to determine the shape of a wavefront at a future time if its current shape is known.
$2$. Consider a spherical wavefront $F_1 F_2$ at time $t=0$ originating from a point source $O$.
$3$. According to the principle,every point on $F_1 F_2$ (such as $A, B, C, \dots$) acts as a secondary source. If the wave speed is $v$,then in a time interval $\tau$,each secondary wavelet travels a distance $v\tau$.
$4$. To find the new wavefront at time $t = \tau$,draw spheres of radius $v\tau$ centered at each point on the original wavefront. The forward common tangent surface $G_1 G_2$ to these spheres represents the new wavefront.
$5$. The backward tangent surface $D_1 D_2$ is also formed,but it is generally ignored as the wave propagates in the forward direction. The points $A', B', C'$ on $G_1 G_2$ then act as new secondary sources for further propagation.

(N/A) The geometric representation of a plane wavefront propagating to the right at time $t=0$ is shown in the figure,and after time $t=\tau$,the new wavefront $G_{1} G_{2}$ is shown in the forward direction.
Here,if the wave velocity is $v$,then the distance covered by the wave in time $\tau$ is $v \tau$.
According to Huygens' principle,all particles like $A_{1}, B_{1}, C_{1}, D_{1}, \ldots$ on the wavefront $F_{1} F_{2}$ act as independent secondary sources and emit secondary spherical waves having a radius of $v \tau$.
After the time interval $\tau$,the surface tangential to all such secondary wavelets gives the position and shape of the new wavefront,shown as $G_{1} G_{2}$.
Thus,a new wavefront is formed at time $\tau$,and the wave propagates forward in the medium.
Lines $A_{1} A_{2}, B_{1} B_{2}, C_{1} C_{2}, D_{1} D_{2}, \ldots$ are perpendicular to both wavefronts $F_{1} F_{2}$ and $G_{1} G_{2}$,which are known as light rays.
$A$ line perpendicular to the wavefront that indicates the direction of propagation of the wave is called a ray.
The most important point of Huygens' wave theory is that it can be applied to all types of spherical or plane waves.

(B) Huygens' principle states that every point on a primary wavefront acts as a source of secondary wavelets,which spread out in all directions with the speed of light in the medium. The new wavefront at any later time is the forward envelope of these secondary wavelets.
The fundamental principle on which Huygens' principle is based is the principle of superposition of waves. This principle states that when two or more waves overlap in space,the resultant displacement at any point is the algebraic sum of the displacements of the individual waves.

(N/A) According to Huygen's principle,every point on a wavefront acts as an independent secondary source and emits small secondary wavelets.
Huygen's principle assumes that the amplitude of these secondary wavelets is maximum in the forward direction and zero in the backward direction.
However,Huygen's could not provide a theoretical justification for why the wave does not propagate backwards; this was an ad-hoc assumption.
Later,scientists such as Voigt and Kirchhoff provided a mathematical explanation,stating that the intensity of the secondary wave is proportional to the factor $(1 + \cos \theta)^2$,where $\theta$ is the angle made by the wavefront with the direction of propagation.
In the forward direction,$\theta = 0^{\circ}$,so the intensity factor is $(1 + \cos 0^{\circ})^2 = (1 + 1)^2 = 4$ (maximum).
In the backward direction,$\theta = 180^{\circ}$ (or $\pi$ radians),so the intensity factor is $(1 + \cos 180^{\circ})^2 = (1 - 1)^2 = 0$. Thus,there is no propagation of waves in the backward direction.

(A) The most fundamental and important point of Huygens' wave theory is the principle of secondary wavelets.
According to this,every point on a given wavefront acts as a fresh source of new disturbance,which sends out secondary wavelets in all directions.
The new wavefront at any later time is the forward envelope of these secondary wavelets.

(N/A) Let $PP'$ represent the surface separating medium-$1$ and medium-$2$. Let $v_1$ and $v_2$ be the speeds of light in medium-$1$ and medium-$2$ respectively.
$A$ plane wavefront $AB$ propagating in the direction $AA'$ is incident on the interface at an angle $i$. Let $\tau$ be the time taken by the wavefront to travel the distance $BC$. Thus,$BC = v_1 \tau$.
To determine the shape of the refracted wavefront,draw a sphere of radius $v_2 \tau$ from point $A$ in the second medium. Let $CE$ be the tangent plane drawn from point $C$ to this sphere. Then $AE = v_2 \tau$,and $CE$ represents the refracted wavefront.
In $\triangle ABC$,$\sin i = \frac{BC}{AC} = \frac{v_1 \tau}{AC}$.
In $\triangle AEC$,$\sin r = \frac{AE}{AC} = \frac{v_2 \tau}{AC}$.
Dividing the two equations:
$\frac{\sin i}{\sin r} = \frac{v_1 \tau / AC}{v_2 \tau / AC} = \frac{v_1}{v_2} = n_{21}$.
This is Snell's law of refraction. Since the incident ray,the refracted ray,and the normal all lie in the same plane,the laws of refraction are derived.