Refraction: Laws, Applications and Refractive Index

Ans. If, θ is the polarising angle of incidence and μ is the refractive index

Then they are linked by,

We also know that,

Where, c = the speed of light

v = the speed of light in the medium

Which is greater than the speed of light.

Ans. The conditions for observing a rainbow are as follows,

i) The sun must appear after the rainfall

ii) The observer must be present in between the sun and the rainbow such that the sun is behind him/her.

Formation of rainbow:

• The ray of light reaches the observer through refraction which is followed by a reflection and refraction.
• The figure shows red light from drop 1 and violet light from drop 2 which are reaching the observer’s eye. 

Ans. Wavefront is an imaginary surface formed by the locus of all points of a wave having the same phase (having identical path length from the source of that wave).

BC is the distance after time that travel in medium 1 and AE is medium 2.

V₁= velocity is medium 1

V₂ = velocity in medium 2

From ΔABC, 

t/QC

sin i = V₁

ΔAEC

sin i/sinr = V₁/V₂.... (1)

n₁ = C/V₁

n₂= C/V₂

V₁/V₂= n₂/n₁.... (2)

n₁sini = n₂ sinr (from (1) and (2))

This represents the law of refraction.

Ques 2.  When it is kept in a medium of refractive index 1.4, calculate the radius of a curvature of an equi-concave lens of refractive index 1.5 to have a power of - 5D. (CBSE 2019)

Ans. As we know that the lens makers formula for the concave lens is 

1/f= (µ -1) [(1/R₁)-(1/R₂)]

Where, f = focal length

μ = μ₂/ μ₁ is the refractive index of material of the lens with respect to the surrounding medium 

R₁ and R₂ = radius of the curvature of two conceive spherical refracting surfaces

Now, as the lens is an equi-convex lens, therefore

R₁ = -R₂

Hence,

Given, refractive index of lens is μ₂ = 1.5

Refractive index of medium is μ₁= 1.4

Power of the lens = -5D

Thus, the focal length is,

f= 1/Power =1/(-5)

Or f= 100/(-5) cm =-20cm

Therefore, 

The radius of the curvature of an equi-concave lens is 2.857 cm.

Ans. 

“Refractive index” of a medium is the ratio of the speed of light in vacuum to the speed of light in a medium. 

Let us assume the medium 1 which is optically denser than the medium 2

Let the speed of light be V₁ in medium 1 and V₂in medium 2

It must be noted that v₂-v₁.

A plane wave AB propagates and hits the interface at an angle and it can be the time taken by wavefront to travel the distance BC

Now, if we want to draw the refracted wavefront, we can draw a sphere of radius V₂ with A as centre and let the surface tangent to the sphere passing through point C is the refracted wavefront.

Now, 

Let the surface be tangent at the sphere E

 In ΔABC

sin i = v₁t/AC

In ΔAEC

sin r = v₂t/AC

Dividing both the equations, we get

sin i/sin r = v₁/v₂= (c/v₂)/(c/v₁)

Thus, 

sin i/ sin r = n₂/n₁

This is verified Snell’s law of refraction.

Ans. a) 

ΔA’B’F and MFP

A’B/MP = B’F/FP OR A’B’/AB = B’F’/FP (PM=BA)

Likewise, ΔaA’B’P and ABP

B’A/BA = B’F/BP 

B’F/FP = B’P/BP

B’F = v+f

BP = u 

(v+f)/f = v/u

1 + (v/f) = v/u

Dividing by v and applying sign convention, 

1/v - 1/f = (-1)/u

1/f = 1/v + 1/u 

This is the mirror equation.
Linear magnification,

From ΔA’B’P and ABP,

B’A’/BA = B’P/BP = -v/u

Ans. (a)

Resolving the power,

\(= \frac{D}{1.22 \lambda}\)

(b) (i) Angular magnification,

=fo/fe = 20/0.01 = 2000

(ii)D/d = x/fo

d= Dfo/x = (3.5* 106 * 20)/(3.8 * 108) = 0.184m

Ans. (a) The phenomenon of reflection of light into a denser medium from an interface of this denser medium and a rare medium is called total internal reflection. The conditions for total internal reflections are as follows:-

• An incident ray should travel from a denser medium to a rarer medium
• Angle of incidence i should be greater than the critical angle C for a pair of media in contact. 

Critical angle is the angle of incidence in a denser medium to which an angle of refraction is 90° in the rarer medium.

The relation between the critical angle of incidence and refraction index of the medium:

Let us assume that an angle of incidence is equal to the critical angle,

i = c

Thus, the angle of refraction is r = 90°

Now applying the Snell’s law, we get

µ_d sin C = µ_r sin r

Where, µ_d = refractive index of denser medium

µ_r = refractive index of rarer medium

Therefore,

µ_d/µ_r = sin r/sin C = sin 90°/sin C

µ= µ_d/µ_r = 1/sin C

µ = 1/ sin C

This shows the required relation between the critical angle of incidence and refractive index of medium.
(b) Let us assume the given lenses to be 1,2 and 3 respectively as shown in the figure.

Focal length of lens 1 = +10 cm

Focal length of lens 2 = -10cm

Focal length of lens 3 = +30cm
By applying lens formula, we get

1/f = 1/v -1/u

For lens 1, we have f =+10, v = x and u = -30cm

1/10 = 1/x - 1/(-30)

=> 1/x = 1/10 - 1/30 = 2/30 

=> x = 15 cm 

For lens 2, the image of lens 1 at distance x = 15cm is the object. 

Thus, f = -10 cm

v = x

u = 10 cm

Hence, 

1/-10 = 1/x’ - 1/10

=> 1/x’ = 1/10 -1/10 =0 

=> x’ = infinity

Thus, for image 3, lens 2, which is located at infinity, is the object 

As we know, for a convex lens, rays coming from infinity converge at the focus and therefore, an image is formed at the focus of lens 3 that is 30 cm. Thus the final image formed by the combination is at 30cm.

Ans. C1 and C2 are the optical centres of the convex lenses L1 and L2

Also, I’ serves as a virtual image object for L2 from the lens formula L2,

1/v - 1/v’ = 1/f2 …(ii)

Adding equations (i) and (ii)

1/v- 1/v’ = 1/f1 + 1/f2

= 1/F

Therefore the focal length of the combination is

1/F = 1/f1 + 1/f2

Ans. (a) 

Magnifying power is defined as the ratio of angle β subtended by the final image on the eye to the angle α subtended by object on the eye.

The limitations of refracting telescope over a reflecting type telescope are,

• Due to refraction, it suffers from chromatic aberration and therefore, the image obtained is multicoloured and blurred. 
• As the lens of large apparatus can not be manufactured easily, its light gathers low power and thus, can not be used to view the faint stars.