CBSE Class 12 Physics Chapter 10: Wave Optics NCERT Solutions

The wavelength of the <math>H_a</math> line emitted by hydrogen is given as <math>\lambda = 6563 \text{ Å}</math>, which is equal to <math>6563 \times 10^{-10} \text{ m}</math>.

The observed red shift, which is the difference between the observed wavelength (<math>\lambda'</math>) and the emitted wavelength (<math>\lambda</math>), is <math>(\lambda' - \lambda) = 15 \text{ Å}</math>, or <math>15 \times 10^{-10} \text{ m}</math>.

The speed of light is a constant, <math>c = 3 \times 10^8 \text{ m/s}</math>.

Let <math>v</math> be the velocity of the star receding away from the Earth. The relationship between red shift and the recession velocity is given by the Doppler effect for light:

$$\backslash frac\{\backslash lambda\text{'}\; -\; \backslash lambda\}\{\backslash lambda\}\; =\; \backslash frac\{v\}\{c\}$$

Rearranging the formula to solve for velocity <math>v</math>:

$$v\; =\; c\; \backslash times\; \backslash frac\{(\backslash lambda\text{'}\; -\; \backslash lambda)\}\{\backslash lambda\}$$

Substituting the given values:

$$v\; =\; (3\; \backslash times\; 10^8\; \backslash text\{\; m/s\})\; \backslash times\; \backslash frac\{15\; \backslash times\; 10^\{-10\}\; \backslash text\{\; m\}\}\{6563\; \backslash times\; 10^\{-10\}\; \backslash text\{\; m\}\}$$

$$v\; =\; \backslash frac\{3\; \backslash times\; 15\}\{6563\}\; \backslash times\; 10^8\; \backslash text\{\; m/s\}$$

$$v\; \backslash approx\; 0.00687\; \backslash times\; 10^8\; \backslash text\{\; m/s\}\; =\; 6.87\; \backslash times\; 10^5\; \backslash text\{\; m/s\}$$

Therefore, the estimated speed with which the star is receding from the Earth is <math>6.87 \times 10^5</math> m/s.

Newton's corpuscular theory of light describes light as consisting of tiny particles (corpuscles). When these corpuscles strike the interface of two media, moving from a rarer medium (like air) to a denser medium (like water), the theory postulates that the surface exerts an attractive force on the corpuscles, normal to the surface. This attractive force, according to the theory, increases the component of the corpuscle's velocity perpendicular to the surface, while the component parallel to the surface remains unchanged.

Let <math>c</math> be the speed of light in vacuum (or air) and <math>v</math> be the speed of light in water. Let <math>i</math> be the angle of incidence and <math>r</math> be the angle of refraction. According to the corpuscular theory, the tangential component of velocity remains constant, so <math>c \sin i = v \sin r</math>. The relative refractive index of water with respect to air is given by <math>\mu = \frac{c}{v}</math>. Since <math>\mu > 1</math> for water, this implies <math>c > v</math> if <math>\sin i > \sin r</math> (which is true for refraction from rarer to denser medium). However, the corpuscular theory's reasoning about forces leads to the conclusion that <math>v > c</math> because the normal component of velocity increases due to attraction.

This prediction that the speed of light in a denser medium is greater than in vacuum is not confirmed by experimental determination. Experiments, such as those conducted by Foucault and Michelson, have consistently shown that the speed of light in water is less than the speed of light in vacuum (<math>v < c</math>).

The wave theory of light is consistent with the experimental results. According to wave theory, light travels slower in a denser medium because the waves interact with the particles of the medium, causing a delay in propagation. The refractive index <math>\mu</math> is related to the speeds by <math>\mu = \frac{c}{v}</math>, and since <math>\mu > 1</math> for denser media, it correctly implies <math>v < c</math>.

Let us consider a point object 'O' placed at a distance 'r' in front of a plane mirror MO'. According to Huygens' principle, every point on the object acts as a source of secondary wavelets. Let's consider two rays OA and OB originating from O, which strike the mirror at points A and B respectively.

According to Huygens' principle, when the wavefront reaches the mirror, each point on the mirror acts as a source of secondary spherical wavelets that propagate forward into the medium. The new wavefront at any instant will be the envelope of these secondary wavelets.

Let the wavefront from O reach point A at time <math>t = 0</math>. Let the distance OA be <math>r</math>. The ray OB strikes the mirror at point B after some time <math>\Delta t</math>. During this time <math>\Delta t</math>, the light travels from B to a point B' on the reflected wavefront. The distance BB' is equal to <math>c \Delta t</math>, where <math>c</math> is the speed of light.

The reflected ray from A will travel a distance AA' in the same time <math>\Delta t</math>. The reflected wavefront will be perpendicular to the reflected rays. Let the reflected rays from A and B meet at point I.

Consider the triangle OAB. Let the distance AB be <math>x</math>. The time taken for light to travel from O to A is <math>t_{OA} = r/c</math>. The time taken for light to travel from O to B is <math>t_{OB} = \sqrt{r^2 + x^2}/c</math>. The time difference is <math>\Delta t = t_{OB} - t_{OA} = (\sqrt{r^2 + x^2} - r)/c</math>.

Now, consider the reflected wavelets. The wavelet originating from A travels a distance AA' = <math>c \Delta t</math> in time <math>\Delta t</math>. The wavelet originating from B travels a distance BB' = <math>c \Delta t</math> in the same time <math>\Delta t</math>.

The reflected wavefront is the envelope of these wavelets. For a plane mirror, the reflected wavefront is also a plane. Let the reflected ray from A be AI and from B be BI. The point I is the virtual image of O.

Using the law of reflection (<math>i = r</math>) and considering the geometry of reflection from a plane mirror, it can be shown that the triangle OAB is congruent to the triangle IAB (where I is the virtual image). This congruence implies that the distance of the object from the mirror (OA) is equal to the distance of the image from the mirror (IA). Thus, the virtual image is formed at a distance equal to the object distance from the plane mirror.