ICSE Class 10 Physics Chapter 04 Refraction of Light at Plane Surfaces

Refraction Of Light At Plane Surfaces

Syllabus

Refraction of light through a glass block and a triangular prism, qualitative treatment of simple applications such as real and apparent depth of objects in water and apparent bending of sticks in water. Application of refraction of light.

Scope of syllabus: Partial reflection and refraction due to change in medium. Laws of refraction, the effect on speed (V), wavelength (λ) and frequency (f) due to refraction of light; conditions for a light ray to pass undeviated. Values of speed of light (c) in vacuum, air, water and glass; refractive index μ = c/V, V = fλ. Values of μ for common substances such as water, glass and diamond, experimental verification; refraction through glass block; lateral displacement; multiple images in thick glass plate/mirror; refraction through a glass prism; simple applications: real and apparent depths of object in water; apparent bending of a stick under water. Simple numerical problems and approximate ray diagrams required.

Total internal reflection; Critical angle; examples in triangular glass prisms; comparison with reflection from a plane mirror (qualitative only). Application of total internal reflection.

Scope of syllabus: Transmission of light from a denser medium (glass/water) to a rarer medium (air) at different angles of incidence; critical angle C, μ = 1/sin C, essential conditions for total internal reflection. Total internal reflection in a triangular glass prism; ray diagram, different cases - angles of prism (60°, 60°, 60°), (60°, 30°, 90°), (45°, 45°, 90°); use of right angle prism to obtain δ = 90° and 180° (ray diagram); comparison of total internal reflection from a prism and reflection from a plane mirror.

Refraction, Laws Of Refraction And Refractive Index

In class IX, we have read the reflection of light from the plane and spherical mirrors. The return of light in the same medium after striking a surface is called reflection of light. The reflection of a light ray obeys two laws: (i) the angle of reflection is equal to the angle of incidence, and (ii) the incident ray, the normal at the point of incidence and the reflected ray, all lie in one plane. Here we shall study the refraction of light through the plane and spherical surfaces.

Light has the maximum speed in vacuum and it travels with different speeds in different media. It travels faster in air than in water and faster in water than in glass. The speed of light is 3 × 10⁸ m s⁻¹ in air, 2.25 × 10⁸ m s⁻¹ in water and 2 × 10⁸ m s⁻¹ in glass. The speed of light is constant in a transparent homogeneous medium.

While passing from one medium to the other, if light slows down, the second medium is said to be optically denser than the first medium and if light speeds up, the second medium is said to be optically rarer than the first medium. Thus water and glass are optically denser than air (or air is optically rarer than water and glass). Similarly, glass is optically denser than water (or water is optically rarer than glass).

Refraction Of Light

Partial reflection and refraction at the boundary of two different medium: In a transparent medium although light travels in a straight line path, but when a ray of light travelling in one transparent medium strikes obliquely at the surface of another transparent medium, a part of light comes back to the same medium obeying the laws of reflection and is called the reflected light. The remaining part of light passes into the other medium and travels in a straight path different from its initial direction and is called the refracted light.

Thus, at the boundary separating the two media, light suffers a partial reflection and partial refraction. Thus the change in direction of the path of light, when it passes from one transparent medium to another transparent medium, is called refraction. The refraction of light is essentially a surface phenomenon.

In Fig. 4.1 and Fig. 4.2, SS' is the surface separating the two media (say, air and glass). When a light travelling on one media falls on the surface SS', a small part of it is reflected back in the same medium obeying the laws of reflection and the rest of it is refracted through the other medium i.e., there is a partial reflection and partial refraction at the boundary surface. The intensity (or the amplitude) of the refracted light will obviously be less than that of the incident light because a part of the incident light has suffered reflection.

In Fig. 4.1 and 4.2, for the incident ray AO, the refracted ray is OB, the reflected ray is OC and the normal at the point of incidence O is NOM. The angle of incidence i is angle AON and the angle of refraction r is angle BOM. Note that the angle r is not equal to the angle i (i.e., OB is not in direction of OA).

It has been experimentally observed that

(1) When a ray of light travels from a rarer medium to a denser medium (say, from air to glass), it bends towards the normal (i.e., angle r is less than angle i) as shown in Fig. 4.1. The deviation of the ray (from its initial path) is δ = i - r.

(2) When a ray of light travels from a denser medium to a rarer medium (say, from glass to air), it bends away from the normal (i.e., angle r is greater than angle i) as shown in Fig. 4.2. The deviation of the ray is then δ = r - i.

(3) The ray of light incident normally on the surface separating the two media, passes undeviated (i.e., such a ray suffers no bending at the surface). Thus if angle of incidence angle i = 0°, then angle of refraction angle r = 0° as shown in Fig. 4.3. The deviation of the ray is zero (i.e., δ = 0°).

Note: In discussing refraction now onward, the reflected ray from the boundary surface will not be shown although it is always there.

Cause of Refraction (or Cause of Change in Direction)

When a ray of light passes from one medium to another medium, its direction (or path) changes because of change in speed of light in going from one medium to another. In passing from one medium to other, if light slows down, it bends towards the normal and if light speeds up, it bends away from the normal. For normal incidence (angle i = 0°), the speed of light changes but the direction of light does not change.

Laws Of Refraction

The refraction of light obeys two laws of refraction which were given by the Dutch scientist Willebrod Snell, so they are known as Snell's laws after his name. They are:

(1) The incident ray, the refracted ray and the normal at the point of incidence, all lie in the same plane.

(2) The ratio of the sine of the angle of incidence i to the sine of the angle of refraction r is constant for the pair of given media. i.e., mathematically

\[\frac{\sin i}{\sin r} = \text{constant}\,\mu_2\]

The constant is called the refractive index of the second medium with respect to the first medium. It is generally represented by the Greek letter μ₂ (mew).

Refractive Index

The refractive index of second medium with respect to the first medium is defined as the ratio of the sine of the angle of incidence in the first medium to the sine of the angle of refraction in the second medium.

Unit: The refractive index has no unit as it is the ratio of two similar quantities.

Speed Of Light In Different Media; Relationship Between Refractive Index And Speed Of Light (μ = c/V)

The speed of light is maximum in vacuum and is equal to 3 × 10⁸ m s⁻¹. The speed of light in air is nearly same as in vacuum. It is denoted by the symbol c. In any other transparent media, the speed of light is less than that in air (or vacuum).

The refractive index of a medium is generally defined with respect to vacuum (or air), and it is called the absolute refractive index (or simply the refractive index) of the medium. It is denoted by the letter μ.

The refractive index of a medium is defined as the ratio of the speed of light in vacuum (or air) to the speed of light in that medium, i.e.,

\[\mu = \frac{\text{Speed of light in vacuum or air (c)}}{\text{Speed of light in that medium (V)}}\]

The refractive index of a transparent medium is always greater than 1 (it can not be less than 1), because speed of light in any medium is always less than that in vacuum (i.e., V is less than c).

Examples: (1) The speed of light in air is 3 × 10⁸ m s⁻¹ and in glass it is 2 × 10⁸ m s⁻¹, therefore the refractive index of glass is

\[\mu_{\text{glass}} = \frac{3 \times 10^8}{2 \times 10^8} = 1.5\]

(2) The speed of light in water is 2.25 × 10⁸ m s⁻¹, so the refractive index of water is

\[\mu_{\text{water}} = \frac{3 \times 10^8}{2.25 \times 10^8} = \frac{4}{3} = 1.33\]

(3) The refractive index of diamond is 2.41, it means that light travels in air 2.41 times faster than in diamond.

The refractive index of some common transparent substances are given in the table ahead.

Note: If the refractive indices of medium 1 and medium 2 are the same, the speed of light will be same in both the media, so a ray of light will pass from medium 1 to medium 2 without any change in its path even when the angle of incidence in medium 1 is not zero.

In general, the refractive index of second medium with respect to first medium is related to the speed of light in the two media as follows:

\[\mu_{12} = \frac{\text{Speed of light in medium 1}}{\text{Speed of light in medium 2}}\]

where μ₁₂ represents the refractive index of medium 2 with respect to medium 1.

If V₁ is the speed of light in medium 1 and V₂ is the speed of light in medium 2, then from eqn. (4.3),

\[\mu_{12} = \frac{V_1}{V_2} = \frac{c/\mu_1}{c/\mu_2} = \frac{\mu_2}{\mu_1}\]

Here μ₁ and μ₂ are the absolute refractive indices of the medium 1 and 2 respectively.

Examples: (1) Refractive index of glass with respect to water is

\[\text{water}^{\mu}_{\text{glass}} = \frac{\text{Speed of light in water}}{\text{Speed of light in glass}} = \frac{2.25 \times 10^8}{2.0 \times 10^8} = 1.125\]

or

\[\text{water}^{\mu}_{\text{glass}} = \frac{\mu_{\text{glass}}}{\mu_{\text{water}}} = \frac{3/2}{4/3} = \frac{9}{8} = 1.125\]

(2) Refractive index of water with respect to glass is

\[\text{glass}^{\mu}_{\text{water}} = \frac{\text{Speed of light in glass}}{\text{Speed of light in water}} = \frac{2.0 \times 10^8}{2.25 \times 10^8} = 0.89\]

or

\[\text{glass}^{\mu}_{\text{water}} = \frac{\mu_{\text{water}}}{\mu_{\text{glass}}} = \frac{4/3}{3/2} = \frac{8}{9} = 0.89\]

Conditions For A Light Ray To Pass Undeviated On Refraction

A ray of light passes undeviated from medium 1 to medium 2 in either of the following two conditions:

(1) When the angle of incidence at the boundary of two media is zero (i.e., angle i = 0°) as shown in Fig. 4.3.

(2) When the refractive index of medium 2 is same as that of medium 1 (Fig. 4.4) i.e., i = r.

Effect On Speed (V), Wavelength (λ) And Frequency (f) Due To Refraction Of Light

1. When a ray of light gets refracted from a rarer to a denser medium, the speed of light decreases; while if it is refracted from a denser to a rarer medium, the speed of light increases.

2. The frequency of light depends on the source of light, so it does not change on refraction. If V is the speed of light in a medium and λ is the wavelength of light in that medium, the frequency of light is given as \[f = \frac{V}{\lambda}\] or \[V = f\lambda\]

3. When light passes from a rarer to a denser medium, the wavelength decreases (since speed of light decreases, but its frequency remains unchanged). When light passes from a denser medium to a rarer medium, the speed of light and hence its wavelength increases.

If a ray of light of frequency f and wavelength λ suffers refraction from air (speed of light = c) to a medium in which the speed of light is V, then the frequency of light in the medium remains unchanged (equal to f), but the wavelength of light changes to λ' such that in air \[f = \frac{c}{\lambda}\] and in medium \[f = \frac{V}{\lambda'}\]

\[\therefore \frac{c}{\lambda} = \frac{V}{\lambda'} \text{ or } \lambda' = \frac{c}{V}\lambda\]

But \[\frac{c}{V} = \mu\] the refractive index of the medium.

\[\therefore \lambda' = \frac{\lambda}{\mu}\]

Obviously when light passes from a rarer to a denser medium (μ greater than 1), its wavelength decreases (λ' is less than λ), but if light passes from a denser to a rarer medium (μ less than 1), its wavelength increases (λ' is greater than λ).

Note: Due to change in speed of light in refraction from one medium to other, the direction of ray of light changes except for angle i = 0°.

3. The colour or wavelength of light: The speed of light of all colours is same in air (or vacuum), but in any other transparent medium, the speed of light is different for different colours. In a given medium, the speed of red light is maximum and that of the violet light is least, therefore the refractive index of that medium is maximum for violet light and least for red light (i.e., μᵥ is greater than μᵣ). The wavelength of red light is more than that of violet light, so refractive index of a medium decreases with the increase in wavelength.

Principle Of Reversibility Of The Path Of Light

According to this principle, the path of a light ray is reversible.

In Fig. 4.5, a ray of light AO is incident at an angle i on a plane surface SS' separating the two media 1 and 2. It is refracted along OB at an angle of refraction r. The refractive index of medium 2 with respect to medium 1 is

\[\mu_{1}\mu_2 = \frac{\mu_2}{\mu_1} = \frac{\sin i}{\sin r} = \frac{V_1}{V_2}\]

Now if the refraction takes place from the medium 2 to 1, the principle of reversibility requires that the ray of light incident along BO at O at an angle of incidence r in medium 2 will get refracted only along OA at an angle of refraction i in medium 1 and in no other direction than OA. The refractive index of medium 1 with respect to medium 2 is then

\[\mu_2\mu_1 = \frac{\mu_1}{\mu_2} = \frac{\sin r}{\sin i} = \frac{V_2}{V_1}\]

From eqns. (i) and (ii),

\[\mu_{1}\mu_2 \times \mu_2\mu_1 = 1\]

or

\[\mu_{12} \text{ or } \mu_2 = \frac{1}{\mu_{1}2}\]

Thus, if refractive index of glass with respect to air is 3/2, the refractive index of air with respect to glass will be μₐ = 3/2 ÷ 2/3 = 3.

Note: From relation \[\mu_{12} = \frac{\mu_2}{\mu_1}\], the refractive index of glass with respect to water is

\[\mu_g = \frac{\mu_g}{\mu_w} = \frac{3/2}{4/3} = \frac{9}{8}\]

and refractive index of water with respect to glass is

\[\mu_{g}^w = \frac{\mu_w}{\mu_g} = \frac{4/3}{3/2} = \frac{8}{9} = 0.89\]

Factors Affecting the Refractive Index of a Medium

The refractive index of a medium depends on the following three factors:

(1) Nature of the medium i.e. its optical density: As smaller the speed of light in a medium relative to air, higher is the refractive index of that medium. For example V_glass = 2 × 10⁸ m s⁻¹, μ_glass = 1.5 and V_water = 2.25 × 10⁸ m s⁻¹, μ_water = 1.33.

(2) Physical condition such as temperature: With increase in temperature, the speed of light in medium increases, so the refractive index of medium decreases.

Teacher's Note: When you look at a stick partially immersed in water, it appears bent due to refraction. This happens because light from the underwater part bends as it travels from water to air, making the stick appear at a different position than it actually is.

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