Question

Reflection and refraction of plane waves using huygens principle

Answer 1

Huygens’s Principle:-

Huygens proposed the wave theory of light. He suggested that light travels in the form of waves. Huygens’s principle states that each and every point of a wavefront act as a source of secondary and release secondary spherical wavelets of light. These secondary wavelets transmits with the velocity of light in the same medium. A wavefront is a real or imaginary surface where the phase of oscillation is the same. Huygens’s principle of wave theory of light is used to prove the laws of reflection and laws of refraction. 

Refraction of plane wave using Huygens Principle:-

The velocity of light changes when passes from one medium to another. This bending of light wave when it enters into other medium is called Refraction. 

As per the diagram shown below consider a plane wave front AB which is incident on the surface. Let v1 and v2 be the velocities of the incident ray and refracted ray of medium 1 and medium 2 respectively (v1>v2).The velocity of the waves depends upon the medium. From Huygens’s principle A and C forms the source of secondary spherical wavelets. Let t be the time taken from B to reach C.

So BC = v1t in medium 1

To determine the shape of the refracted wavefront, we draw a sphere of radius v2t from the point A in the second medium. It denotes the secondary spherical wavefront at time t.

AD = v2t in medium 2.

Now CD is the tangent drawn from point C to the sphere. Thus AD and CD are the refracted wavefronts.

Illustration of Refraction laws based on Huygens’s principle

Now consider ΔABC and ΔADC

Sin i / Sin r = (BC/AC) / (AD/AC)

= BC/AD

= v1t/v2t

= v1/v2

= µ which is a constant. µ is the reflective index of the medium.

Refractive Index is the ratio of velocity of light in vacuum to the velocity of light in other medium.

Hence Snell’s Law of refraction is proved using Huygens’s principle. Also the incident wavefront, the refracted wavefront and the normal lie in the same plane.

Reflection of a plane wave using Huygens Principle:-

 As discussed earlier, when light is incident on the surface it is re- emitted without any change in the frequency. This re-emitted light which is returned into the same medium from which it comes out is called Reflection of Light. Consider the figure given below:

Imagine incoming rays are incident on a surface. Here the wavefronts are plane waves. In plane wavefront, the wavefronts will be infinite parallel planes to each other with constant amplitude. Consider the plane wave AB which falls on the reflecting surface. AB is the incident wavefront and is drawn as perpendicular to the incident ray. It falls at an angle i on the surface. Now according to the Huygens’s principle every point on AB act as a source of secondary wavelets. Consider the points A and B as new sources which emits the secondary waves. The velocity of the propagation of waves is ‘v’. Let ‘t’ be the time taken. So let’s assume that vt be the distance moved by the secondary wavelets. AA1 and BE are the secondary waves. Now the new wavefront should be a tangents line which connects those two secondary waves. The reflected waves should be perpendicular to the new wavefront. A1E is the new tangential line which connects the secondary wavelets.

Illustration of Reflection laws based on Huygens’s principle

Consider ΔABE and ΔAA1E. Here AE is common.

<B = <A1 = 90° .

AA1 = BE.

These triangles are congruent triangles

 So   <i = <r   

Thus Angle of Incidence = Angle of Reflection. This is the first law of reflection.

The incident wavefront, the reflected wavefront and normal lie in the same plane which is perpendicular to the reflecting surface. This again verifies the second law of reflection. Therefore, the two Laws of Reflection are verified using Huygens’s Principle.

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Answer 2

Answer:

Reflection and refraction are the two phenomena based on the propagation of straight light, where reflection speaks about the bouncing of light rays and refraction talks about their bending.

Explanation:

Reflection using huygens principle

Diagram for reference is given in the attachment.

• We can observe that a light ray is incident on this surface, as well as a parallel ray that is incident on the same surface.
• The incident plane AB strikes the reflecting surface MN at an angle of'i '. We refer to these photons as incident rays since they are coming from the surface.
• A line which is connecting points A and B is known as a wavefront, and it is incident on the surface if we draw a perpendicular from point 'A' to this ray of light.
• Since the incidence wavefront carries the two points A and B, we may infer that light travels a distance from B to C.
• If "r" is the amount of time the wavefront took to travel from point B to point C and "v" is the speed of the wave in the medium, then the distance
• BC = vr
• We draw a sphere with a radius of vr from the point A in order to produce the reflected wavefront.
• Let CE stand in for the tangent plane formed by connecting point C with this sphere. So,
• BC = AE + vr
• The triangles EAC and BAC are congruent, hence the angles I and "r" are equivalent if we now take them into consideration.

Refraction using huygens principle

Diagram for reference is given in the attachment.

• We are aware that a light's path modifies when it passes through one transparent substance and then another.
• The angle of incidence is defined as the angle between the incident ray and the normal according to the equations of refraction.
• While the angle of refraction is defined as the angle between the refracted ray and the normal.
• Any given interface of two media has an incident ray, a reflected ray, and a normal that all reside in the same plane.
• Additionally, we are aware of the continuous relationship between the sines of the angles of incidence and refraction.
• We can observe that a light ray is incident on this surface, as well as a parallel ray that is incident on the same surface.
• We refer to these photons as incident rays since they are coming from the surface.
• Let PP' stand in for the mediums 1 and 2. V1 and V2 stand in for the speed of light in this medium.
• A line which  connecting points A and B is known as a wavefront, and it is incident on the surface if we draw a perpendicular from point 'A' to this ray of light.
• If "r" is the amount of time the wavefront took to travel from point B to point C, then the distance,
• BC = v1 r
• We so construct a sphere of radius v2r from the point A in the second medium in order to ascertain the form of the refracted wavefront.
• Let CE stand in for a tangent plane that is traced onto the sphere from point C.
• The refracted wavefront would thus be represented as CE, and AE = v2r.
• Now, if we think about the triangles ABC and AEC, we can easily
• sin i = = 1
• sin r = = 2
• where the angles of incidence and also the refraction, respectively, are 'i' and 'r'. Snell's Law is obtained by substituting the values of v1 and v2 in terms.
• n1 sin i = n2 sin (r)

Hence we have stated both refraction and reflection using huygens principle in depth.

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