Refraction and Reflection of Plane Waves using Huygens Principle

Wave Optics · CBSE

Theory

What this subtopic covers

NCERT 10.1 (introduction), 10.2 (Huygens principle and wavefronts) and 10.3 (reflection and refraction of plane waves). CBSE sets this section more often than any other in the chapter, almost always as "define a wavefront, state Huygens' principle, and verify the law of reflection / Snell's law".

The introduction, in the two lines CBSE uses

Descartes (1637) and Newton gave the corpuscular model, which predicted that light bending towards the normal must travel FASTER in the second medium.
Huygens (1678) gave the wave model, which predicts the opposite: bending towards the normal means the light travels SLOWER in the second medium. Foucault's 1850 measurement of the speed of light in water settled the argument in favour of the wave model.
Young's interference experiment (1801) established that light is a wave; Maxwell then showed it is an electromagnetic wave, which is why it needs no medium.
Geometrical optics is the limit in which the finiteness of $\lambda$ is neglected; a ray is the path of energy propagation as $\lambda\to0$ .

Wavefront — the definition to write

A wavefront is the continuous locus of all the particles of a medium that are vibrating in the same phase at a given instant. Equivalently, a surface of constant phase.
The speed at which the wavefront advances is the speed of the wave, and the energy of the wave travels in a direction perpendicular to the wavefront.
A ray is a line drawn normal to the wavefront in the direction of propagation. Difference asked in board questions: a wavefront is a surface (a locus of points in the same phase); a ray is a line perpendicular to that surface showing the direction of energy flow.

Source or optical element	Wavefront produced
Point source, close by	Spherical, diverging
Line source or narrow slit	Cylindrical
Distant source, e.g. a star, seen at the Earth	Plane
Point source at the focus of a convex lens	Plane, after the lens
Plane wave onto a thin convex lens	Spherical, converging to the focus $F$
Plane wave onto a thin concave mirror of radius $R$	Spherical, converging; radius of curvature $R/2$
Plane wave through a thin prism	Plane, tilted towards the base

Huygens principle — the statement to write

Every point on a given (primary) wavefront acts as a source of new secondary wavelets, which spread out in all directions with the speed of light in that medium.
The surface that touches all these secondary wavelets tangentially in the forward direction gives the new (secondary) wavefront at a later instant.
If the wavefront at $t=0$ is $F_1F_2$ , draw spheres of radius $vt$ from each of its points; their forward common tangent $G_1G_2$ is the wavefront at time $t$ .
The construction also produces a backward envelope $D_1D_2$ . Huygens removed it by assuming the amplitude of the secondary wavelets is maximum in the forward direction and zero in the backward direction. This is the ad hoc obliquity factor, and it is exactly what the 1-mark MCQ tests: the amplitude is maximum forward, zero backward.

Reflection of a plane wave — full derivation

Let a plane wavefront $AB$ be incident at angle $i$ on the plane reflecting surface $xy$ (or $MN$ ), with $A$ reaching the surface first and $B$ still to arrive.
Let $t$ be the time in which the disturbance from $B$ travels to $B'$ on the surface. Then $BB'=ct$ , where $c$ is the speed of the wave in the medium.
During the same time $t$ , the point $A$ — which reached the surface first — has been emitting secondary wavelets. Draw a sphere of radius $ct$ centred at $A$ ; then $AA'=ct$ .
Draw the tangent plane $A'B'$ from $B'$ to this sphere. $A'B'$ is the reflected wavefront.
Compare triangles $BAB'$ and $B'A'A$ : $AB'$ is common; $BB'=A'A=ct$ ; and $\angle ABB'=\angle B'A'A=90^\circ$ , since a wavefront is perpendicular to its rays.
The two right triangles are therefore congruent (RHS), so the angles they make with the surface are equal, giving $\angle i=\angle r$ . This is the first law of reflection.
Since the whole construction lies in one plane, the incident ray, the reflected ray and the normal at the point of incidence are coplanar. This is the second law of reflection.

Refraction of a plane wave, rarer to denser — full derivation

Let $PP'$ separate medium 1 (speed $v_1$ , index $\mu_1$ ) from medium 2 (speed $v_2$ , index $\mu_2$ ), with $v_2<v_1$ .
A plane wavefront $AB$ , travelling along $A'A$ , is incident on the interface at angle $i$ . $A$ touches the interface first; $B$ has still to travel to $C$ on the interface.
Let $t$ be the time for the wavefront to cover $BC$ . Then

$BC=v_1 t$

In the same time $t$ , the disturbance from $A$ has spread into medium 2 at speed $v_2$ . Draw a sphere of radius $v_2t$ about $A$ , so

$AE=v_2 t$

Draw the tangent plane $CE$ from $C$ to that sphere. $CE$ is the refracted wavefront, and $r$ is the angle it makes with the interface (the angle of refraction).
In the right triangles $ABC$ and $AEC$ , both having $AC$ as hypotenuse:

$\sin i=\dfrac{BC}{AC}=\dfrac{v_1 t}{AC},\qquad \sin r=\dfrac{AE}{AC}=\dfrac{v_2 t}{AC}$

Dividing,

$\dfrac{\sin i}{\sin r}=\dfrac{v_1 t/AC}{v_2 t/AC}=\dfrac{v_1}{v_2}$

With $\mu_1=\dfrac{c}{v_1}$ and $\mu_2=\dfrac{c}{v_2}$ , so $v_1=\dfrac{c}{\mu_1}$ and $v_2=\dfrac{c}{\mu_2}$ :

$\dfrac{\sin i}{\sin r}=\dfrac{c/\mu_1}{c/\mu_2}=\dfrac{\mu_2}{\mu_1}$

$\therefore\;\mu_1\sin i=\mu_2\sin r$

This is Snell's law of refraction — the first law. The second law (incident ray, refracted ray and normal coplanar) follows because the whole construction is planar.
Since $v_2<v_1$ here, $\sin r<\sin i$ , so $r<i$ : the ray bends towards the normal on entering the denser medium. The refracted wavefront is tilted closer to the interface normal than the incident one.

Refraction denser to rarer, and the critical angle

Now $v_2>v_1$ , so the sphere drawn in medium 2 has the LARGER radius $v_2t$ . The tangent from $C$ tilts the refracted wavefront further AWAY from the normal, and $r>i$ — the ray bends away from the normal.
The same algebra still gives $\mu_1\sin i=\mu_2\sin r$ .
Define the critical angle $i_c$ by

$\sin i_c=\dfrac{n_2}{n_1}\qquad (n_1>n_2)$

At $i=i_c$ , $\sin r=1$ so $r=90^\circ$ . For $i>i_c$ there is no refracted wavefront at all and the wave undergoes total internal reflection.

Wavelength, speed and frequency across an interface

If $BC$ in medium 1 equals one wavelength $\lambda_1$ , then $AE$ in medium 2 equals one wavelength $\lambda_2$ (the crest from $B$ reaches $C$ in the same time the crest from $A$ reaches $E$ ). Hence

$\dfrac{BC}{AE}=\dfrac{v_1}{v_2}=\dfrac{\lambda_1}{\lambda_2}\qquad\Longrightarrow\qquad \dfrac{\lambda_1}{v_1}=\dfrac{\lambda_2}{v_2}$

Both sides equal $1/\nu$ . So on entering a denser medium the wavelength and the speed both decrease by the factor $\mu$ , while the frequency $\nu=v/\lambda$ stays the same.

$\lambda_{\text{med}}=\dfrac{\lambda_{\text{vac}}}{\mu}$

Why the frequency is unchanged (a standard 2-mark reason): reflection and refraction arise from the interaction of the incident light with the atomic constituents of the medium. The atoms behave as oscillators undergoing FORCED oscillation, and a driven oscillator takes up the frequency of the driving agency. The re-radiated reflected and refracted waves therefore have the same frequency as the incident light. Frequency is a property of the source, not of the medium.
Why a reduced speed does not mean reduced energy: the energy carried by a wave depends on its amplitude (and frequency), not on the speed of propagation. Since the frequency is unchanged, the energy is not reduced.
In the photon picture, for a given frequency, intensity is set by the number of photons crossing unit area per unit time — not by an amplitude.

Behaviour of wavefronts at a prism, a lens and a mirror

Thin prism: the part of the incident plane wavefront passing near the base travels through the greatest thickness of glass, where the speed is least, and is delayed most. The emergent wavefront is still PLANE but tilted towards the base — matching the ray bending towards the base.
Thin convex lens: the central part of the plane wavefront traverses the thickest glass and is delayed the most, so the emergent wavefront has a depression at the centre. It becomes SPHERICAL and CONVERGING, concave towards the direction of propagation, and converges to the focus $F$ .
Concave mirror: a plane wave is reflected as a spherical wave converging to the focal point $F$ . Since $F$ lies at $R/2$ from the pole, the reflected wavefront has radius of curvature $R/2$ .
Fermat-style consequence CBSE quotes: the total time taken from a point on the object to the corresponding point on the image is the same along every ray. A ray through the centre of a convex lens travels a shorter distance but at a lower speed inside more glass, so its travel time matches that of a ray near the rim.

Diagram to describe if you cannot draw well

Refraction: a horizontal interface $PP'$ ; above it, a set of parallel slanted lines (the incident plane wavefront $AB$ ) with arrows $A'A$ perpendicular to them; $A$ on the interface, $B$ above; a dashed arc of radius $v_2t$ drawn below the interface centred at $A$ ; $C$ marked on the interface to the right of $A$ with $BC=v_1t$ ; the tangent $CE$ from $C$ to the arc drawn below the interface, with $AE=v_2t$ ; the normal at $A$ dotted, angle $i$ between the incident ray and the normal, angle $r$ between the refracted ray ( $\perp CE$ ) and the normal.
Reflection: reflecting surface $xy$ horizontal; incident plane wavefront $AB$ slanting down to the right, $A$ on the surface, $B$ above; $BB'=ct$ carrying $B$ down to $B'$ on the surface; an arc of radius $ct$ about $A$ above the surface; the tangent $A'B'$ from $B'$ to the arc, so $AA'=ct$ ; angles $i$ and $r$ measured from the normal at the surface.

Standard results, collected

Result	Statement
Wavefront	Locus of points vibrating in the same phase at a given instant
Huygens principle	Each point of a wavefront is a source of secondary wavelets; the forward envelope is the new wavefront
Amplitude of secondary wavelets	Maximum forward, zero backward — no backwave
Law of reflection	$\angle i=\angle r$ , from congruent right triangles
Snell's law	$\mu_1\sin i=\mu_2\sin r$ , i.e. $\dfrac{\sin i}{\sin r}=\dfrac{v_1}{v_2}=\dfrac{\mu_2}{\mu_1}$
Refractive index	$\mu=\dfrac{c}{v}$ , the ratio of the speed of light in vacuum to that in the medium
Critical angle	$\sin i_c=\dfrac{n_2}{n_1}$ ; total internal reflection for $i>i_c$
Wavelength in a medium	$\lambda_{\text{med}}=\dfrac{\lambda_{\text{vac}}}{\mu}$ ; frequency unchanged
Plane wave onto a concave mirror	Reflected wavefront spherical, radius of curvature $R/2$

Theory Questions Asked on This Subtopic

Real board questions. Answers are at the end of the page.

1
Define the wavefront of a travelling wave. Using Huygens' principle, obtain the law of refraction at a plane interface when light passes from a rarer to a denser medium.
2 marksCBSE Class 12 Physics — SA · 2020 · Section B, 2 marksAnswer 1 ↓
2
Define the term 'wavefront of light'. A plane wavefront $AB$ propagating from a denser medium (1) into a rarer medium (2) is incident on the surface $P_1P_2$ separating the two media, as shown in the figure (the incident ray labelled '1' strikes the interface at point $A$ , with $B$ marked above on the incident wavefront and the interface $P_1P_2$ horizontal through $A$ ; ray '2' shows the direction of onward propagation into medium 2). Using Huygens' principle, draw the secondary wavelets and obtain the refracted wavefront in the diagram.
2 marksCBSE Class 12 Physics — SA · 2020, An · Section B, 2 marksAnswer 2 ↓
3
A plane wave-front propagating in a medium of refractive index ' $\mu_1$ ' is incident on a plane surface making an angle of incidence ( $i$ ). It enters into a medium of refractive index $\mu_2$ ( $\mu_2>\mu_1$ ). Use Huygens' construction of secondary wavelets to trace the refracted wavefront. Hence, verify Snell's law of refraction.
3 marksCBSE Class 12 Physics — SA · 2023, Ev · Section C, 3 marksAnswer 3 ↓
4
Using Huygens' construction, show how a plane wave is reflected from a surface. Hence verify the law of reflection.
3 marksCBSE Class 12 Physics — SA · 2023 · Section C, 3 marksAnswer 4 ↓
5
A plane wavefront is incident on a surface separating two media of refractive indices $n_1$ and $n_2$ ( $n_2>n_1$ ). With the help of a suitable diagram, explain its propagation from the rarer to the denser medium. Hence, verify Snell's law.
3 marksCBSE Class 12 Physics — SA · 2022C, An · Section C, 3 marksAnswer 5 ↓
6
A plane wavefront is propagating from a rarer into a denser medium. Use Huygens' principle to show the refracted wavefront and verify Snell's law.
3 marksCBSE Class 12 Physics — SA · Term II 2021-22 · Section C, 3 marksAnswer 6 ↓
7
Define the term 'refractive index' of a medium. Verify Snell's law of refraction when a plane wavefront is propagating from a denser to a rarer medium.
3 marksCBSE Class 12 Physics — SA · Delhi 2019, Ap · Section C, 3 marksAnswer 7 ↓
8
Define the term wavefront. Using Huygens' wave theory, verify the law of reflection.
3 marksCBSE Class 12 Physics — SA · Delhi 2019 · Section C, 3 marksAnswer 8 ↓
9
Define the term wavefront. State Huygens' principle. Consider a plane wavefront incident on a thin convex lens. Draw a proper diagram to show how the incident wavefront traverses through the lens and, after refraction, focusses on the focal point of the lens, giving the shape of the emergent wavefront.
3 marksCBSE Class 12 Physics — SA · AI 2016 · Section C, 3 marksAnswer 9 ↓
10
Explain the following, giving reasons: (i) 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. (ii) When light travels from a rarer to a denser medium, the speed decreases. Does this decrease in speed imply a reduction in the energy carried by the light wave?
3 marksCBSE Class 12 Physics — SA · 2/3, AI 2016, An · Section C, 3 marksAnswer 10 ↓
11
What is a wavefront? How does it propagate? Using Huygens' principle, explain reflection of a plane wavefront from a surface and verify the laws of reflection. OR Define a wavefront. Using Huygens' principle, verify the laws of reflection at a plane surface.
5 marksCBSE Class 12 Physics — Long Answer · 2/5, 2020 · Section E, 5 marksAnswer 11 ↓
12
Define wavefront. Use Huygens' principle to verify the laws of refraction.
3 marksCBSE Class 12 Physics — SA · 3/5, AI 2017 · Section C, 3 marksAnswer 12 ↓
13
What is a wavefront? How does it propagate? Using Huygens' principle, explain the reflection of a plane wavefront from a surface and verify the laws of reflection. OR Define a wavefront. Using Huygens' principle, verify the laws of reflection at a plane surface.
5 marksCBSE Class 12 Physics — Long Answer · 2/5, 2020 · Section E, 5 marksAnswer 13 ↓
14
(a) Define a wavefront. How is it different from a ray? (b) Depict the shape of a wavefront in each of the following cases: (i) Light diverging from a point source. (ii) Light emerging out of a convex lens when a point source is placed at its focus. (iii) Using Huygens' construction of secondary wavelets, draw a diagram showing the passage of a plane wavefront from a denser into a rarer medium.
5 marksCBSE Class 12 Physics — Long Answer · AI 2015C, An · Section E, 5 marksAnswer 14 ↓

Numericals

Every numerical set on this subtopic. Full working is at the end of the page.

15
According to Huygens principle, the amplitude of secondary wavelets is
- (a)
  equal in both the forward and the backward directions.
- (b)
  maximum in the forward direction and zero in the backward direction.
- (c)
  large in the forward direction and small in the backward direction.
- (d)
  small in the forward direction and large in the backward direction.
1 markCBSE Class 12 Physics — Multiple Choice · 2023, R · Section A, 1 markAnswer 15 ↓
16
A plane wavefront is incident on a concave mirror of radius of curvature $R$ . The radius of curvature of the reflected wavefront will be
- (a)
  $2R$
- (b)
  $R$
- (c)
  $\dfrac R2$
- (d)
  $\dfrac R4$
1 markCBSE Class 12 Physics — Multiple Choice · 2023, U · Section A, 1 markAnswer 16 ↓

Answers & Solutions

Work the questions above first, then check yourself here. Each answer is set out the way it should be written in the paper.

1.↑ back to the question
A wavefront is the continuous locus of all the particles of a medium that are vibrating in the same phase at a given instant. Let $P_1P_2$ represent a surface separating medium 1 (rarer, speed $v_1$ ) from medium 2 (denser, speed $v_2<v_1$ ). A plane wavefront $AB$ , propagating in the direction $A'A$ , is incident on the interface at an angle $i$ . Let $t$ be the time taken by the wavefront to travel the distance $BC$ (from $B$ on the interface to $C$ ). $BC=v_1t$ (distance $=$ speed $\times$ time). To determine the shape of the refracted wavefront, draw a sphere of radius $v_2t$ centred at $A$ in the second medium (since the disturbance from $A$ travels at speed $v_2$ in medium 2 for the same time $t$ ). Let $CE$ be the tangent plane drawn from $C$ to this sphere; $CE$ represents the refracted wavefront, with $AE=v_2t$ . In right triangles $ABC$ and $AEC$ (both with hypotenuse $AC$ ): $\;\sin i=\dfrac{BC}{AC}=\dfrac{v_1t}{AC}\;$ and $\;\sin r=\dfrac{AE}{AC}=\dfrac{v_2t}{AC}$ , where $i$ and $r$ are the angles of incidence and refraction. $\therefore\;\dfrac{\sin i}{\sin r}=\dfrac{v_1t/AC}{v_2t/AC}=\dfrac{v_1}{v_2}$ If $c$ is the speed of light in vacuum, the refractive indices are $\mu_1=\dfrac c{v_1}$ and $\mu_2=\dfrac c{v_2}$ , so $v_1=\dfrac c{\mu_1}$ and $v_2=\dfrac c{\mu_2}$ . $\therefore\;\dfrac{\sin i}{\sin r}=\dfrac{c/\mu_1}{c/\mu_2}=\dfrac{\mu_2}{\mu_1}\;\Rightarrow\;\mu_1\sin i=\mu_2\sin r$ This is Snell's law of refraction.
2.↑ back to the question
A wavefront is the continuous locus of all the particles of a medium vibrating in the same phase at a given instant. Since the wave travels from a denser medium 1 (speed $v_1$ ) into a rarer medium 2 (speed $v_2>v_1$ ), by the same Huygens' construction as in Q3 above (with the roles of the two media swapped): while the point on the wavefront at the interface sends out a secondary wavelet of radius $v_1t$ in medium 1's remaining path, the point that has already reached the interface earlier sends out a wavelet of larger radius $v_2t$ into medium 2 (since $v_2>v_1$ ). The tangent drawn from the appropriate point on the incident wavefront to this larger sphere (radius $v_2t$ ) gives the refracted wavefront — it is tilted further away from the interface normal than the incident wavefront, consistent with the ray bending away from the normal on entering a rarer medium. This construction, exactly as derived in Q3, gives $\;\mu_1\sin i=\mu_2\sin r\;$ (Snell's law), with $r>i$ here since $v_2>v_1$ .
3.↑ back to the question
Same construction and derivation as in Q3 above (light going from the medium of refractive index $\mu_1$ , rarer since $\mu_2>\mu_1$ , into the denser medium of refractive index $\mu_2$ ): the refracted wavefront $CE$ is obtained as the tangent from $C$ to the sphere of radius $v_2t$ drawn from $A$ , giving $\;\dfrac{\sin i}{\sin r}=\dfrac{v_1}{v_2}=\dfrac{\mu_2}{\mu_1}\;\Rightarrow\;\mu_1\sin i=\mu_2\sin r$ , which is Snell's law.
4.↑ back to the question
Let $AB$ represent a plane wavefront incident on the plane reflecting surface $xy$ , with the incident rays making an angle $i$ with the normal. Let $t$ be the time taken by the secondary wavelet from $B$ to reach $B'$ on the surface, so $BB'=ct$ (where $c$ is the speed of light in the medium). By Huygens' principle, while the disturbance from $B$ travels to $B'$ , the point $A$ on the wavefront (which reaches the surface earlier) has already been sending out secondary wavelets for the same total time; draw a sphere of radius $ct$ centred at $A$ . The tangent plane $A'B'$ drawn from $B'$ to this sphere represents the reflected wavefront after time $t$ , with $AA'=ct$ as well. In triangles $BAB'$ and $B'A'A$ : $\;AB'=AB'$ (common), $\;BB'=A'A=ct$ , and $\;\angle ABB'=\angle B'A'A=90°$ (since $AB$ and $A'B'$ are wavefronts, perpendicular to the respective rays). So the two right triangles are congruent (RHS congruence), which gives $\;\angle i=\angle r$ — the angle of incidence equals the angle of reflection. Also, the incident ray, the reflected ray, and the normal to the surface at the point of incidence all lie in the same plane (the plane of the diagram) — this gives the second law of reflection.
5.↑ back to the question
Same construction and derivation as Q3 above: a plane wavefront $AB$ travelling in the rarer medium ( $n_1$ , speed $v_1$ ) is incident on the interface; secondary wavelets in the denser medium ( $n_2$ , speed $v_2<v_1$ ) spread out more slowly, so the refracted wavefront $CE$ (the tangent from $C$ to the sphere of radius $v_2t$ about $A$ ) is tilted closer to the normal than the incident wavefront — the ray bends towards the normal on entering the denser medium. This gives $\;\dfrac{\sin i}{\sin r}=\dfrac{v_1}{v_2}=\dfrac{n_2}{n_1}\;\Rightarrow\;n_1\sin i=n_2\sin r$ , which is Snell's law.
6.↑ back to the question
Same construction and derivation as Q3/Q7 above: $\;n_1\sin i=n_2\sin r$ , verifying Snell's law.
7.↑ back to the question
Refractive index of a medium is defined as the ratio of the speed of light in vacuum to the speed of light in that medium: $\;\mu=\dfrac cv=\dfrac{\text{speed of light in vacuum}}{\text{speed of light in the medium}}$ For the denser-to-rarer case: the construction is exactly as in Q4 above — the refracted wavefront (tangent from the appropriate point to the larger sphere of radius $v_2t$ , since $v_2>v_1$ in the rarer medium) gives $\;\dfrac{\sin i}{\sin r}=\dfrac{v_1}{v_2}={}^1\mu_2\;\Rightarrow\;\mu_1\sin i=\mu_2\sin r$ , verifying Snell's law.
8.↑ back to the question
A wavefront is the continuous locus of all the particles of a medium vibrating in the same phase at a given instant. Same construction and derivation as Q6 above: comparing the congruent triangles $BAB'$ and $B'A'A$ formed by the incident wavefront $AB$ , the reflected wavefront $A'B'$ , and the common side $AB'$ , we get $\angle i=\angle r$ , verifying the first law of reflection; the second law (incident ray, reflected ray and normal coplanar) follows from the planar geometry of the construction.
9.↑ back to the question
Wavefront: the continuous locus of all particles of a medium vibrating in the same phase at a given instant. Huygens' principle: every point on a given (primary) wavefront acts as a source of new secondary wavelets, spreading out in all directions with the speed of light in that medium; the surface touching these secondary wavelets tangentially, in the forward direction, gives the new (secondary) wavefront at a later instant. Diagram: a plane wavefront (a set of parallel vertical lines, each perpendicular to the horizontal incident rays) approaches a thin convex lens. The lens is thicker at the centre than at the edges, so the central part of the wavefront, travelling through more glass (where speed is lower), is delayed more than the edges. This causes the flat wavefront to progressively curve inward as it crosses the lens, emerging as a spherical wavefront that is concave towards the direction of propagation, converging to a point at the focus $F$ of the lens on the far side.
10.↑ back to the question
(i) Reflection and refraction arise from the interaction of the incident light with the atomic constituents of matter, which are set into forced oscillation at the SAME frequency as that of the incident light (like a driven oscillator following the driving frequency). Since these oscillating charges re-radiate the reflected and refracted waves, both have the same frequency as the incident light — frequency is a property of the source, unaffected by the medium. (ii) No. The energy carried by a wave depends on its frequency (and amplitude), not on its speed of propagation. Since the frequency remains unchanged on entering a denser medium (as explained in (i)), the energy of the wave is not reduced merely because its speed has decreased.
11.↑ back to the question
A wavefront is the continuous locus of all the particles of a medium vibrating in the same phase at a given instant. It propagates outward from the source, with each point on it acting (per Huygens' principle) as a source of secondary wavelets that spread out at the speed of light in the medium; the forward tangential envelope of these wavelets gives the new wavefront a little later. Reflection and verification of the laws of reflection: same construction and derivation as in Q6 above — using the congruent triangles $BAB'$ and $B'A'A$ formed by the incident wavefront $AB$ and reflected wavefront $A'B'$ (with $BB'=AA'=ct$ , $AB'$ common, and both angles at $A,A'$ equal to $90°$ ), we obtain $\angle i=\angle r$ , and the incident ray, reflected ray and normal are shown to be coplanar. This verifies both laws of reflection.
12.↑ back to the question
A wavefront is the continuous locus of all the particles of a medium vibrating in the same phase at a given instant. Same construction and derivation as Q3 above, giving $\;\mu_1\sin i=\mu_2\sin r$ (Snell's law, the first law of refraction); the incident ray, refracted ray and normal being coplanar (evident from the planar Huygens' construction) gives the second law of refraction.
13.↑ back to the question
Same content and derivation as Q13 above.
14.↑ back to the question
(a) A wavefront is the continuous locus of all points in a medium that are vibrating in the same phase at a given instant. A ray, on the other hand, is a line drawn perpendicular to the wavefront, indicating the direction of propagation of the wave at that point — a wavefront is thus a surface, while a ray is a line normal to it. (b)(i) The wavefront is spherical (increasing in radius as it moves outward), with rays radiating outward from the point source in all directions, each ray perpendicular to the spherical wavefront at that point. (ii) Since the point source is at the focus, the rays emerging from the convex lens travel parallel to each other (a standard property of a lens with a source at its focus); the corresponding wavefront is therefore PLANE, perpendicular to these parallel rays. (iii) Same construction as in Q4 above: an incident plane wavefront in the denser medium (speed $v_1$ ) meets the interface; the secondary wavelets in the rarer medium (speed $v_2>v_1$ ) spread out faster, so the refracted (tangent) wavefront on the rarer side is tilted further from the normal than the incident wavefront — showing the ray bending away from the normal as it crosses from the denser into the rarer medium.
15.Answer: (B)↑ back to the question
Huygens' construction (with the obliquity/inclination factor) predicts that secondary wavelets have maximum amplitude in the forward direction and zero amplitude in the backward direction — this is why a wavefront propagates only forward and there is no 'backwave'. Option (b).
16.Answer: (C)↑ back to the question
A plane wavefront incident on a concave mirror is reflected as a CONVERGING spherical wavefront, converging towards the focus of the mirror. Since the focus of a concave mirror lies at a distance $R/2$ from the mirror (pole), the reflected wavefront, at the instant of reflection, has its centre of curvature at the focus — i.e. its radius of curvature is $R/2$ . Option (c).