REFRACTION AND REFLECTION OF PLANE WAVES USING HUYGENS PRINCIPLE -

Refraction and Reflection of plane waves using Huygens principle (Introduction)

We will see in this topic how Huygens explained the law of reflection and refraction through the wave and proved it. Here we will also see that when the light goes from one medium to another, what is the effect on its frequency, path and wavelength. Along with, we will also read how the frequency of light changes due to the relative motion between the source of light and the observer and what formulas Doppler gave it. So here we are going to discuss the topic refraction and reflection of plane waves using Huygens principle.

REFLECTION ON THE BASIS OF WAVE THEORY

Consider a plane wavefront AB is incident on the mirror XY at an angle I. All the incident rays are perpendicular to AB.

According to Huygens’ principle, every point on the wavefront AB is the source of secondary wavelets. Let the secondary wavelets from B strike the mirror XY at B’ in t second.

$REFLECTION ON THE BASIS OF WAVE THEORY$

where c is the speed of light.

In the same time t, secondary wavelets originated from A will travel a distance ( $REFLECTION ON THE BASIS OF WAVE THEORY$ ) and reaches A’.

$REFLECTION ON THE BASIS OF WAVE THEORY$

From fig, A’B’ is the secondary wavefront at time t.

from fig, $REFLECTION ON THE BASIS OF WAVE THEORY$ angle of incidence and $REFLECTION ON THE BASIS OF WAVE THEORY$ angle of reflection

In right-angled triangles ABB’ and AA’B’, AB’ is common, AA’=BB’=ct and $REFLECTION ON THE BASIS OF WAVE THEORY$ . Therefore, the two triangles are congruent.

$REFLECTION ON THE BASIS OF WAVE THEORY$

Therefore, the angle of incidence and angle of reflection are equal to each other, It is also from fig, the incident ray, normal and refracted ray which is respectively perpendicular to AB, XY, and A’B’ all lie in the same plane i.e in the plane of the paper. This is the law of reflection.

REFRACTION ON THE BASIS OF WAVE THEORY

Consider XY is a plane surface which separates rarer medium ( $REFRACTION ON THE BASIS OF WAVE THEORY$ ) to denser medium ( $REFRACTION ON THE BASIS OF WAVE THEORY$ )

Let $REFRACTION ON THE BASIS OF WAVE THEORY$ velocity of light in the rarer medium

$REFRACTION ON THE BASIS OF WAVE THEORY$ velocity of light in the denser medium

suppose a plane wavefront AB is incident on xy at an angle $\fn_cm \large i$ . All incident rays are perpendicular to AB.

According to Huygen’s principle, every point on the wavefront AB is a source of secondary wavelets.

Let the secondary wavelets originated from B strike xy at B’ in t second

$REFRACTION ON THE BASIS OF WAVE THEORY$

In the same time, secondary wavelets originated from A will travel a distance ( $V_2t$ ) in denser medium and reaches A’.

$REFRACTION ON THE BASIS OF WAVE THEORY$

from fig, A’B’ is the secondary wavefront at time t.

It is clear from fig.

$REFRACTION ON THE BASIS OF WAVE THEORY$ (Angle of the incident)

$REFRACTION ON THE BASIS OF WAVE THEORY$ (Angle of refraction)

$REFRACTION ON THE BASIS OF WAVE THEORY$

This proves Snell’s law of refraction.

It is also from fig, the incident ray, normal and refracted ray which is respectively perpendicular to AB, XY, and A’B’ all lie in the same plane i.e in the plane of the paper.

EFFECT OF SPEED

We know that

$REFRACTION ON THE BASIS OF WAVE THEORY$

we know that when light passes through rarer to denser medium, ray bends towards the normal. $REFRACTION ON THE BASIS OF WAVE THEORY$

Then from the above relation $REFRACTION ON THE BASIS OF WAVE THEORY$

i.e In a denser medium, the speed of light is smaller compared to the rarer medium and vice versa.

EFFECT OF FREQUENCY

Suppose a source of light emits two wavefronts separated by time period T. Then two wavefront would arrive at the observer separated by the same time period T. Since frequency is the characteristics of a source i.e the frequency ( $REFRACTION ON THE BASIS OF WAVE THEORY$ ) remains the same as the light goes from one medium to another.

EFFECT OF WAVELENGTH

We know that

$REFRACTION ON THE BASIS OF WAVE THEORY$

we know that, if $REFRACTION ON THE BASIS OF WAVE THEORY$ is the frequency of the source, then

$REFRACTION ON THE BASIS OF WAVE THEORY$

$\therefore$ for rarer medium $REFRACTION ON THE BASIS OF WAVE THEORY$

where $REFRACTION ON THE BASIS OF WAVE THEORY$ is the wavelength of light in the rarer medium.

and for denser medium $REFRACTION ON THE BASIS OF WAVE THEORY$

where $REFRACTION ON THE BASIS OF WAVE THEORY$ is the wavelength of light in the denser medium

$REFRACTION ON THE BASIS OF WAVE THEORY$

here $REFRACTION ON THE BASIS OF WAVE THEORY$ i.e $REFRACTION ON THE BASIS OF WAVE THEORY$

i.e when the light goes from rarer medium to denser medium wavelength is reduced and vice-versa.

THE DOPPLER EFFECT

According to the Doppler effect in light, whenever there is a relative motion between the source and the observer, the apparent frequency or (wavelength) of light received by the observer is different from the actual or true frequency (or wavelength) of the light source.

(A) WHEN SOURCE OF LIGHT IS MOVING AWAY FROM STATIONARY OBSERVER

In this case the distance between the source of light and the observer increases. The apparent frequency of light decreases and its apparent wavelength increases.

apparent frequency $Refraction and Reflection of Plane Waves using Huygens Principle$

where $Refraction and Reflection of Plane Waves using Huygens Principle$ frequency of the source

$Refraction and Reflection of Plane Waves using Huygens Principle$ velocity of source

apparent wavelength $Refraction and Reflection of Plane Waves using Huygens Principle$

where $Refraction and Reflection of Plane Waves using Huygens Principle$ wavelength of the source

$Refraction and Reflection of Plane Waves using Huygens Principle$ velocity of source

Here apparent wavelength of light increases, we say that light is red shifted i.e it is shifted to a longer wavelength.

(B) WHEN SOURCE OF LIGHT IS MOVING TOWARDS THE STATIONARY OBSERVER

In this case, the distance between the source of light and the observer decreases. The apparent frequency of light increases and its apparent wavelength decreases.

apparent frequency $Refraction and Reflection of Plane Waves using Huygens Principle$

where $Refraction and Reflection of Plane Waves using Huygens Principle$ frequency of the source

$Refraction and Reflection of Plane Waves using Huygens Principle$ velocity of source

apparent wavelength $Refraction and Reflection of Plane Waves using Huygens Principle$

where $Refraction and Reflection of Plane Waves using Huygens Principle$ wavelength of the source

$Refraction and Reflection of Plane Waves using Huygens Principle$ velocity of the source

Here apparent wavelength of light decreases, we say that light is blue-shifted i.e it is shifted to the shorter wavelength.

we know that

$Refraction and Reflection of Plane Waves using Huygens Principle$

$Refraction and Reflection of Plane Waves using Huygens Principle$ same as $Refraction and Reflection of Plane Waves using Huygens Principle$

where $Refraction and Reflection of Plane Waves using Huygens Principle$ is the speed of the star

NOTE:-

(a) This formula is valid only when the speed of the source is small compared to that of light.

(b) A more accurate formula for the Doppler effect which is valid even when the speed is close to that of light requires the use of Einstein’s special theory of relativity.

RELATED LINKS OF WAVE OPTICS
Introduction	Huygens Principle
Refraction and Reflection of Plane waves using Huygens Principle	Coherent and Incoherent Addition of Waves
Interference of Light Waves and Young’s Experiment	Diffraction
Polarisation

IMPORTANT QUESTIONS FOR PRACTICE(Refraction and Reflection of Plane waves using Huygens Principle)

1. What speed should a galaxy move with respect to us so that the sodium line at 589 nm is observed at 589.6 nm?

(a) When monochromatic light is incident on a surface separating two media, the reflected and refracted light both have the same frequency as the incident frequency. Explain why?

(b) When light travels from a rarer to a denser medium, the speed decreases. Does the reduction in speed imply a reduction in the energy carried by the light wave?

(c) In the wave picture of light, the intensity of light is determined by the square of the amplitude of the wave. What determines the intensity of light in the photon picture of light.

(3) Monochromatic light of wavelength 589 nm is incident from the air on a water surface. What are the wavelength, frequency and speed of (a) reflected and (b) refracted light? (R.I of water is 1.33).

(4)

(a) The refractive index of glass is 1.5. What is the speed of light in glass? (Speed of light in a vacuum is $\fn_jvn 3\times10^{8}m/s$ )
(b) Is the speed of light in glass independent of the colour of light? If not, which of the two colours red and violet travels slower in a
glass prism?

(5) The 6563 Å Hα line emitted by hydrogen in a star is found to be red-shifted by 15 Å. Estimate the speed with which the star is receding
from the Earth.

(6) Explain how Corpuscular theory predicts the speed of light in a medium, say, water, to be greater than the speed of light in vacuum. Is the prediction confirmed by experimental determination of the speed of light in water? If not, which alternative picture of light is consistent with the experiment?

(7) You have learnt in the text how Huygens’ principle leads to the laws of reflection and refraction. Use the same principle to deduce directly that a point object placed in front of a plane mirror produces a virtual image whose distance from the mirror is equal to the object distance from the mirror.

(8) Let us list some of the factors, which could possibly influence the speed of wave propagation

(i) nature of the source.
(ii) direction of propagation.
(iii) motion of the source and/or observer.
(iv) wavelength.
(v) intensity of the wave.
On which of these factors, if any, does
(a) the speed of light in vacuum,
(b) the speed of light in a medium (say, glass or water),
depend?

(9) For sound waves, the Doppler formula for frequency shift differs slightly between the two situations: (i) source at rest; observer moving, and (ii) source moving; observer at rest. The exact Doppler formulas for the case of light waves in vacuum are, however, strictly identical for these situations. Explain why this should be so. Would you expect the formulas to be strictly identical for the two situations in case of light travelling in a medium?