snell’s law of refraction of light state snells law of refraction of light class 10 and class 12th

find snell’s law of refraction of light state snells law of refraction of light class 10 and class 12th derivation formula ?

Reflection of Light

(i) Law of Reflection Angle of reflection = angle of incidence Aray incident along the normal to a reflecting surface retraces its path after reflection. (ii) Reflection at a plane surface (mirror)

(a) Distance of image from mirror = distance of object from mirror.

(b) Size of image = size of object.

(c) If the object moves with a certain velocity, the image moves with the same velocity but in the opposite direction. (d) Keeping the incident ray fixed, if the mirror is rotated through an angle θ, the reflected ray rotates through an angle 2θ.

(e) If two mirrors are inclined at an angle θ (in degrees), the number of images formed by the mirrors of an object is

is not placed at the same distance from the mirrors.

(f) If three mirrors are placed mutually perpendicular and adjacent to each other, the number of images of an object placed in front of them is 7.

Reflection at a Spherical Surface (Concave Mirror or Convex Mirror)

(a) Sign conventions

(i) All distances are measured from the centre (pole) of the mirror.

(ii) Distances measured in the direction of incident rays are taken as positive while those measured opposite to the direction of incident rays are taken as negative.

(iii) Distances above the principal axis are taken as positive while those below the principal axis are taken as negative.

(b) The spherical mirror formula (for both concave and convex mirrors) is

u = object distance, v = image distance, f = focal length and R = radius of curvature. For convex mirror f is positive and for an concave mirror f is negative.

For an erect image m is positive and for an inverted image m is negative.

(d) A concave mirror forms a real and inverted image of an object placed beyond its focus and a virtual and erect image if the object is placed between the pole and focus. A convex mirror forms a virtual and erect image for all positions of the object.

Snell’s Law of Refraction

If a ray of light travelling in a medium of refractive index µ₁ is incident at angle i on the boundary of a medium of refractive index µ₂, then

µ₁ sin i = µ₂ sin r

where r is the angle of refraction in medium µ₂.

Refractive Index and Speed of Light

The value of µ depends upon (i) nature of the medium and (ii) wavelength (colour) of light.

Refraction of Light through a Plane Slab When a ray of light passes through a glass slab of thickness t (Fig. 16.1),

(a) the emergent ray is parallel to the incident ray and

Real and Apparent Depth (or Height)

(a) An object in a denser medium (water) viewed by an observer in a rarer medium (air) from above [Fig. below (a)]

(b) Object in air viewed by observer under water [Fig. above (b)]
Real height OA = h, Apparent height AI = h’ = µh.
Apparent shift OI = h’ – h = µh – h = (µ – 1)h.

Total Internal Reflection

For total internal reflection,

(a) the incident ray must travel in a denser medium (µ₂) to the boundary of a rarer medium µ₁ (< µ₂) and

(b) the angle of incidence must be greater than critical angle i_c given by (Fig. 16.3) µ₂ sin i_c = µ₁ sin 90°

The value of i_c depends upon µ₁, µ₂ and wavelength of light.

Refraction through a Prism

When a ray of monochromatic light is refracted by a prism, the deviation d produced by the prism is (Fig. below)

where i₁ = angle of incidence, i₂ = angle of emergence and A = refracting angle of the prism. Also

A = r₁ + r₂

The deviation is minimum = δm, if i₁ = i₂. Then r₁ = r₂

Then

where µ₂ = refractive index of the material of the prism and µ₁ = refractive index of the medium surrounding the prism. If µ₁ = 1 (air) and µ₂ = µ, then

For a thin prism, angles i₁, i₂, r₁, r₂ and A are small and
ð = (µ – 1)A

Refraction at a Spherical Surface

If the rays from an object travelling in a medium of refractive index µ₁ are refracted at the spherical surface (convex or concave) of a medium of refractive index µ₂ forming an image, then the object and image distances u and v are related as

where R = radius of curvature of the spherical surface.

If the first medium is air, µ₁ = 1 and µ₂ = µ, then

If the incident rays from the object are in medium µ₂ and are refracted at the spherical surface of medium µ₁, then in the above formula µ₁ and µ₂ are interchanged, i.e.

Application: Image formed by a transparent sphere

Case (a): When the object O is in air outside the sphere of radius R. (Fig. below)

I’ is the angle of O due to refraction at P. For this refraction,

(u is negative, and R is positive). This image I’ serves as the virtual object for refraction at Q forming the final image I. For refraction at Q (since the incident rays are in medium µ), we have

(u’ is positive and R is negative).
Case (b): When the object O is on the surface of the sphere (Fig. below).

where u = 2R.
(u is negative and R is also negative).

Case (c): Object O inside the sphere (Fig. below)

(u is negative and R is also negative)

Refraction through a Lens

(i) Relation between u, v and f for a lens (convex or concave) is

The focal length f is positive for converging (convex) lens and negative for diverging (concave) lens.

(ii) Focal length f of a lens of refractive index µ₂ surrounded by a medium of refractive index µ₁ is given by

where R1 = radius of curvature of the surface of the lens on which the rays are incident R2 = radius of curvature of the surface of the lens from which the rays emerge after refraction through the lens. If the media on the two sides of the lens are different (e.g. a lens floating on water) as shown in Fig. below, then

This is the lens maker’s formula.

(i) If a lens of focal length f is cut along AB into two equal pieces as shown in Fig. below (a), the focal length of each piece is 2f.

But if the lens is cut along CD as shown in Fig. above (b), the focal length remains the same equal to f.

Power of a Lens

The reciprocal of the focal length (expressed in metres) is known as the power of a lens.

The SI unit of P is called diopre (D)
If the lens is placed in medium of refractive index µm, then

where f_m = focal length in medium.

Co-axial Combination of Lenses

If two thin lenses of focal lengths f1 and f2 are placed in contact co-axially (i.e. with their principal axes coinciding), the equivalent focal length F of the combination is given by

Power of combination P = P1 + P2
If the lenses are separated by a distance d,

Effect of Silvering One of the Refracting Faces of a Lens

If one surface of a convex lens is silvered, it behaves like a concave mirror. If one surface of a concave lens is silvered, it behaves like a convex mirror. In Fig. below, the rays are refracted at surface 1, reflected at surface 2 and again refracted at surface 1. The effective focal length F is given by

where f_n = focal length of lens or mirror repeated as many times as there are refractions and reflections. In the case shown in the figure, there are two refractions and one reflection. Hence

where f = focal length of the lens and f_m = focal length of the spherical mirror of radius of curvature R₂. (a) If the face of radius of curvature R₂ of a double convex lens is silvered (Fig. below),

(c) If the plane face of a plano-convex lens is silvered (Fig. below), then

(d) If the curved face of a plano-convex lens is silvered (Fig. below), then R1 = ∞ and R2 = R.

Compound Microscope

In its simplest form, a compound microscope consists of two convergent (convex) lenses, of a very short focal length called the objective and the other of a longer focal length called the eye-piece. The lenses are mounted coaxially and the separation between them can be varied. The magnifying power of a microscope is

M = magnification by objective x magnification by eye-piece

= m₀ x m_e

(a) If the final image is formed at the least distance of distinct vision (strained eye)

where u₀ = distance of object from objective, v₀ = distance (from the objective) of the image formed by objective, f_e = focal length of eye-piece and D = least distance of distinct vision.

(b) If the final image is formed at infinity (relaxed eye)

Telescope

A telescope consists of two convergent lenses called the objective and the eye-piece. The focal length of the objective is much larger than that of the eye-piece. (a) The magnifying power of a telescope in normal adjustment (i.e. when the image of a distant object is formed at infinity), i.e. for relaxed eye

f₀ = focal length of objective, f_e = focal length of eyepiece L = length of tube = f₀ + f_e.

(b) When the final image is formed at the least distance of distinct vision (i.e. for strained eye)

Dispersion of Light

Refraction of light occurs because the velocity of light changes as it travels from one medium into another. The velocity of light in a given medium depends upon its wavelength (or frequency). Light of a single wavelength or frequency is called monochromatic. White light (sunlight) is not monochromatic, it consists of many wavelengths ranging from violet (~ 300 nm) to red (~ 700 nm). When white light (or any composite light) enters a refracting medium, the different constituent colours are refracted unequally. The red is refracted the least and the violet the most. From Snell’s law

it follows that the refractive index for red is less than for violet, i.e., the speed of red light is greater than that of violet light. Thus, a medium does not have one definite refractive index; it has a range of refractive indices corresponding to the range of colours or wavelengths of the composite light. Since each colour has its own characteristic wavelength (or frequency), the refractive index of a medium will be different for different wavelengths (or frequencies). The variation of refractive index of a medium (and hence of the velocity of light in the medium) with the wavelength or frequency is referred to as dispersion. The prism disperses the colours of white light and produces its spectrum.

Angular Dispersion and Dispersive Power

If δ_v and δ_R are the deviations of violet and red lights produced by a prism of refracting angle A, the angular dispersion θ is given by

Now δ_v = (µ_v – 1)A and δ_R = (µ_R – 1)A, where µ_v and µ_R are the refractive indices of the material of the prism for violet and red lights respectively. Hence

The dispersive power of a prism is its ability to deviate the different colours of a composite light along different directions and is defined as

Dispersion without Deviation and Deviation without Dispersion

If two prisms are arranged as shown in Fig. below, then net angular dispersion

Aberrations

(i) Spherical aberration Spherical aberration occurs in a lens or a spherical mirror due to spherical nature of the surface. This defect arises because the paraxial and marginal rays do not focus at a single point (Fig. below).

Spherical aberration is minimized by

(a) using a plano-convex lens with the incident rays falling on the curved face

(b) using two plano-convex lens of focal lengths f1 and f2 separated by a distance d = f1 – f2.

(c) using a parabolic mirror.

(ii) Chromatic aberration

Chromatic aberration occurs only in lenses and not in mirrors. This defect arises because the refractive index (and hence the focal length) of a lens is different for the different colours of light. In fact f_R > f_Y > f_V. Hence rays of white light do not focus at a single point (Fig. below)

Chromatic aberration is removed by (a) using a convex lens is contact with a concave lens such that (see Fig. below)

where w and w’ are dispersive powers of the material of the lenses.

(b) using two lenses made of the same material separated by a distance d = (f + f’)/2