Chapter :Ray optics-I

By

SHIVAKUMAR .H

PGT PHYSICS

JNV HASSAN

RAY OPTICS-I

1. Reflection of light and reflection by spherical mirror
2. Refraction of Light
3. Laws of Refraction and Principle of Reversibility of Light
4. Refraction through a Parallel Slab
5. Refraction through a Compound Slab
6. Apparent Depth of a Liquid
7. Total Internal Reflection
8. Refraction at Spherical Surfaces - Sign Conventions
9. Refraction at Convex and Concave Surfaces
10. Lens Maker’s Formula
11. First and Second Principal Focus

​

14. Thin Lens Equation (Gaussian Form)
15. Linear Magnification
16. Refraction through a Prism
17. Expression for Refractive Index of Prism
18. Dispersion
19. Angular Dispersion and Dispersive Power
20. Blue Colour of the Sky and Red Colour of the Sun
21. Compound Microscope
22. Astronomical Telescope (Normal Adjustment)
23. Astronomical Telescope
24. Newtonian Telescope (Reflecting Type)

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Reflection of light-

• Reflection of light- The phenomenon of bouncing back of light in the same medium after striking a surface, is called reflection of light.
• Laws of reflection:
• The angle of reflection equals the angle of incidence
• The incident ray, reflected ray and the normal to the reflecting surface at the point of incidence lie in the same plane

​

Sign Conventions:-��

• All distances are measured from the pole of the mirror or the optical

centre of the lens.

• The distances measured in the same direction as the incident light are taken as positive and those measured in the direction opposite to the direction of incident light are taken as negative .
• The heights measured upwards with respect to x-axis and normal to theprincipal axis (x-axis) of the mirror/ lens are taken as positive (Fig. 9.2). The heights measured downwards aretaken as negative.

Focal length of spherical mirrors

• when a parallel beam of light is incident on a convex mirror, The reflected rays appear to be diverge from a point F on the principal axis of a convex mirror . The point F is called the principal focus of convex the mirror

• when a parallel beam of light is incident on a concave mirror, The reflected rays converge at a point F on the principal axis of a concave mirror . The point F is called the principal focusof concave the mirror

Relation between f and R

We can prove that

∠MCP = θ and ∠MFP = 2θ

​

From fig

tanθ =MD/CD and tan 2θ =MD/FD

For small angle θ, tanθ ≈ θ,tan 2θ ≈ 2θ. Therefore, above Eq. Gives

​

MD/FD = 2MD/CD

FD =CD/2

​

FD = f and CD = R. then gives

f = R/2

Path of light rays

• Mirror Formula For Concave Mirror When Real Image Is Formed:-
• When object is placed on the principal axis beyond the principal focus, a real and inverted image is formed

​

We can PT ∆ A′B′F and ∆ MPF are similar.

A’B’/MP = B’F/ FP

A’B’/AB = B’F/PF ……(i) (AB=MP) Also

We can also PT ΔA’B’P and ΔABPare similar therefore, A’B’/AB = B’P/ BP

• Or.....(ii)
• From (i) and (ii) we get
• B’F/PF = B’P/ BP …(iii)
• => (PB’-PF)/PF = B’P/BP
• => (-v-(-f))/-f =-v/(-u) (by sign convension)
• => (v/f) + (-1) = v/u (Multiple by 1/v)
• => (1/f) – 1/v = 1/u
• ie 1/f = 1/u + 1/v

Magnification(m) :

The size of the image relative to the size of the object is called linear magnification .

Theratio of the height of the image (h′) to the height of the object (h):

> m =h/h

In triangles A′B′P and ABP, we have,

> B A’/BA= B’P/BP

With the sign convention, this becomes

– h / h = -v /-u

so that

m = h’/h = - v/u

Magnification(m)

Refraction of Light:

Refraction is the phenomenon of change in the path of light as it travels from one medium to another (when the ray of light is incident obliquely).

It can also be defined as the phenomenon of change in speed of light from one medium to another.

Rarer

Rarer

Denser

N

N

Laws of Refraction:

I Law: The incident ray, the normal to the refracting surface at the point of incidence and the refracted ray all lie in the same plane.

II Law: For a given pair of media and for light of a given wavelength, the ratio of the sine of the angle of incidence to the sine of the angle of refraction is a constant for given pair of medium and for given colour of light. (Snell’s Law)

(The constant μ is called refractive index of the medium, i is the angle of incidence and r is the angle of refraction.)

μ

TIPS:

1. μ of optically rarer medium is lower and that of a denser medium is higher.
2. μ of denser medium w.r.t. rarer medium is more than 1 and that of rarer medium w.r.t. denser medium is less than 1. (μ_air = μ_vacuum = 1)
3. In refraction, the velocity and wavelength of light change.
4. In refraction, the frequency and phase of light do not change.
5. _aμ_m = c_a / c_m and _aμ_m = λ_a / λ_m

Principle of Reversibility of Light:

Rarer (a)

N

Denser (b)

or

If a ray of light, after suffering any number of reflections and/or refractions has its path reversed at any stage, it travels back to the source along the same path in the opposite direction.

A natural consequence of the principle of reversibility is that the image and object positions can be interchanged. These positions are called conjugate positions.

μ

Refraction through a Parallel Slab:

Rarer (a)

Rarer (a)

Denser (b)

N

N

r₁

M

δ

y

But _aμ_b x _bμ_a = 1

It implies that i₁ = r₂ and i₂ = r₁ since i₁ ≠ r₁ and i₂ ≠ r₂.

Lateral Shift:

or

Special Case:

If i₁ is very small, then r₁ is also very small. i.e. sin(i₁ – r₁) = i₁ – r₁ and cos r₁ = 1

or

μ

Refraction through a Compound Slab:

Rarer (a)

Rarer (a)

Denser (b)

N

N

μ_b

Denser (c)

μ_c

N

_aμ_b x _bμ_c x _cμ_a = 1

_aμ_b x _bμ_c = _aμ_c

or

_bμ_c = _aμ_c / _aμ_b

or

μ_a

μ_c > μ_b

​

Apparent Depth of a Liquid:

Rarer (a)

Denser (b)

O

O’

N

μ_b

or

Apparent Depth of a Number of Immiscible Liquids:

Apparent Shift:

Apparent shift = h_r - h_a = h_r – (h_r / μ)

= h_r [ 1 - 1/μ]

TIPS:

1. If the observer is in rarer medium and the object is in denser medium then h_a < h_r. (To a bird, the fish appears to be nearer than actual depth.)
2. If the observer is in denser medium and the object is in rarer medium then h_a > h_r. (To a fish, the bird appears to be farther than actual height.)

μ_a

Total Internal Reflection:

Total Internal Reflection (TIR) is the phenomenon of complete reflection of light back into the same medium for angles of incidence greater than the critical angle of that medium.

N

N

N

N

O

r = 90°

Rarer (air)

Denser (glass)

μ_g

μ_a

Conditions for TIR:

1. The incident ray must be in optically denser medium.
2. The angle of incidence in the denser medium must be greater than the critical angle for the pair of media in contact.

Relation between Critical Angle and Refractive Index:

Critical angle is the angle of incidence in the denser medium for which the angle of refraction in the rarer medium is 90°.

or

or

Also

Red colour has maximum value of critical angle and Violet colour has minimum value of critical angle since,

Applications of T I R:

1. Mirage formation
2. Looming
3. Totally reflecting Prisms
4. Optical Fibres
5. Sparkling of Diamonds

looming

The looming effect is the result of a superior mirage. A typical example is a mirage of a ship formed over cool water in an area where the temperature increases with height. Refraction bends light down toward the water in this case.

Mirage

A mirage is a naturally occurring optical phenomenon in which light rays bend via refraction to produce a displaced image of distant objects or the sky. 

optical fiber 

• Optical fibres are fabricated with high quality composite glass/quartz fibres. Each fibre consists of a core and cladding.
• The refractive index of the material of the core is higher than that of the cladding.

• no appreciable loss in the intensity of the light signal.
• light reflected at one side of inner surface strikes the other at an angle larger than the critical angle.
• transmitting and receiving electrical signals
• ‘light pipe’---95% transmission
• Endoscopy

Spherical Refracting Surfaces:

A spherical refracting surface is a part of a sphere of refracting material.

A refracting surface which is convex towards the rarer medium is called convex refracting surface.

A refracting surface which is concave towards the rarer medium is called concave refracting surface.

•

•

C

C

P

P

R

R

A

B

A

B

APCB – Principal Axis C – Centre of Curvature P – Pole R – Radius of Curvature

•

•

Denser Medium

Denser Medium

Rarer Medium

Rarer Medium

Prisms designed to bend rays by 90º and 180º or to invert image without changing its size make use of total internal reflection.

Assumptions:

1. Object is the point object lying on the principal axis.
2. The incident and the refracted rays make small angles with the principal axis.
3. The aperture (diameter of the curved surface) is small.

New Cartesian Sign Conventions:

1. The incident ray is taken from left to right.
2. All the distances are measured from the pole of the refracting surface.
3. The distances measured along the direction of the incident ray are taken positive and against the incident ray are taken negative.
4. The vertical distances measured from principal axis in the upward direction are taken positive and in the downward direction are taken negative.

Refraction at Convex Surface: (From Rarer Medium to Denser Medium - Real Image)

•

C

P

O

•

Denser Medium

Rarer Medium

•

•

I

M

μ₂

μ₁

α

β

γ

i

r

γ = r + β

or r = γ - β

A

According to Snell’s law,

or

or

μ₁ i = μ₂ r

Substituting for i, r, α, β and γ, replacing M by P and rearranging,

Applying sign conventions with values, PO = - u, PI = + v and PC = + R

N

i = α + γ

Refraction at Convex Surface: (From Rarer Medium to Denser Medium - Virtual Image)

Refraction at Concave Surface: (From Rarer Medium to Denser Medium - Virtual Image)

Refraction at Convex Surface: (From Denser Medium to Rarer Medium - Real Image)

Refraction at Convex Surface: (From Denser Medium to Rarer Medium - Virtual Image)

Refraction at Concave Surface: (From Denser Medium to Rarer Medium - Virtual Image)

Note:

1. Expression for ‘object in rarer medium’ is same for whether it is real or virtual image or convex or concave surface.

​

​

​

2. Expression for ‘object in denser medium’ is same for whether it is real or virtual image or convex or concave surface.

​

​

​

3. However the values of u, v, R, etc. must be taken with proper sign conventions while solving the numerical problems.

​

4. The refractive indices μ₁ and μ₂ get interchanged in the expressions.

Lens Maker’s Formula:

μ₂

μ₁

A

L

C

N

μ₁

For refraction at LP₁N,

(as if the image is formed in the denser medium)

For refraction at LP₂N,

(as if the object is in the denser medium and the image is formed in the rarer medium)

Combining the refractions at both the surfaces,

Substituting the values with sign conventions,

Since μ₂ / μ₁ = μ

or

When the object is kept at infinity, the image is formed at the principal focus.

i.e. u = - ∞, v = + f.

So,

This equation is called ‘Lens Maker’s Formula’.

Also, from the above equations we get,

First Principal Focus:

First Principal Focus is the point on the principal axis of the lens at which if an object is placed, the image would be formed at infinity.

F₁

F₂

Second Principal Focus:

Second Principal Focus is the point on the principal axis of the lens at which the image is formed when the object is kept at infinity.

F₂

F₁

Thin Lens Formula (Gaussian Form of Lens Equation): For Convex Lens:

•

C

M

Triangles ABC and A’B’C are similar.

Triangles MCF₂ and A’B’F₂ are similar.

or

According to new Cartesian sign conventions,

CB = - u, CB’ = + v and CF₂ = + f.

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

-1/u=1/f -1/v

Linear Magnification:

Linear magnification produced by a lens is defined as the ratio of the size of the image to the size of the object.

According to new Cartesian sign conventions,

A’B’ = + I, AB = - O, CB’ = + v and CB = - u.

Magnification in terms of v and f:

Magnification in terms of u and f:

Power of a Lens:

Power of a lens is its ability to bend a ray of light falling on it and is reciprocal of its focal length. When f is in metre, power is measured in Dioptre (D).

Power of a lens:-

Power of a lens is a measure of the convergence or divergence of lens,

The power P of a lens is defined as the tangent of the angle by which it converges or diverges a beam of light falling at unit distant from the optical centre

tanδ =h/f; if h=1 then tanδ=1/f

P =1/f

The SI unit for power of a lens is dioptre (D): 1D = 1m–1.

Power of a lens is positive for a converging lens and negative for a diverging lens.

Combination of thin lenses

PO = u, object distance for the first lens (A),�PI = v, final image distance and�PI1​=v1​, image distance for the first lens (A) and also object distance for second lens (B).�For the image I1​ produced by the first lens A,�1/v​1​−1/u =1/f1​​  .... (1)�For the final image I, produced by the second lens B,�1/v−1/v​1​=1/f2​​  ... (2)�Adding equations (1) and (2),�1/v​−1/u​=1/f1​​+1/f2​​   ... (3)�If the combination is replaced by a single lens of focal length F such that it forms the image of O at the same position I, then�1/v​−1/u​=1/f​ ... (4)�From equations (3) and (4),�1/f​=1/f1​​+1/f2​​...... (5 ) P=P1+P2�This F is the focal length of the equivalent lens for the combination.

Refraction of Light through Prism:

Refracting Surfaces

Prism

i

δ

e

O

P

Q

r₁

r₂

N₁

N₂

D

In quadrilateral APOQ,

A + O = 180° …….(1)

(since N₁ and N₂ are normal)

In triangle OPQ,

r₁ + r₂ + O = 180° …….(2)

In triangle DPQ,

δ = (i - r₁) + (e - r₂)

δ = (i + e) – (r₁ + r₂) …….(3)

From (1) and (2),

A = r₁ + r₂

From (3),

δ = (i + e) – (A)

or

μ

Sum of angle of incidence and angle of emergence is equal to the sum of angle of prism and angle of deviation.

Variation of angle of deviation with angle of incidence:

When angle of incidence increases, the angle of deviation decreases.

At a particular value of angle of incidence the angle of deviation becomes minimum and is called ‘angle of minimum deviation’.

At δ_m, i = e and r₁ = r₂ = r (say)

After minimum deviation, angle of deviation increases with angle of incidence.

Refractive Index of Material of Prism:

A = r₁ + r₂

A = 2r

r = A / 2

i + e = A + δ

2 i = A + δ_m

i = (A + δ_m) / 2

According to Snell’s law,

Refraction by a Small-angled Prism for Small angle of Incidence:

and

If i is assumed to be small, then r₁, r₂ and e will also be very small. So, replacing sines of the angles by angles themselves, we get

and

i + e = μ (r₁ + r₂) = μ A

But i + e = A + δ

So, A + δ = μ A

or

δ = A (μ – 1)

Dispersion of White Light through Prism:

The phenomenon of splitting a ray of white light into its constituent colours (wavelengths) is called dispersion and the band of colours from violet to red is called spectrum (VIBGYOR).

δ_r

A

B

C

D

White light

δ_v

Cause of Dispersion:

and

Since μ_v > μ_r , r_r > r_v

So, the colours are refracted at different angles and hence get separated.

ROYGB I V

Screen

N

Scattering of Light – Blue colour of the sky

The molecules of the atmosphere and other particles that are smaller than the longest wavelength of visible light are more effective in scattering light of shorter wavelengths than light of longer wavelengths. The amount of scattering is inversely proportional to the fourth power of the wavelength. (Rayleigh Effect)

Rayleighscatteringscattering is proportional to 1/λ4.

When looking at the sky in a direction away from the Sun, we receive scattered sunlight in which short wavelengths predominate giving the sky its characteristic bluish colour.

For a << λ,

Light from the Sun near the horizon passes through a greater distance in the Earth’s atmosphere than does the light received when the Sun is overhead. The correspondingly greater scattering of short wavelengths accounts for the reddish appearance of the Sun at rising and at setting.

Reddish appearance of the Sun at Sun-rise and Sun-set:

OPTICAL INSTRUMENTS

A simple magnifier or microscope:-

The linear magnification m, for the image

formed at the near point D,

m = v/u

=v x1/u

=v(1/v -1/f )

=1-v/f

=1- (-D)/f

= 1+D/f

Compound Microscope:

•

•

•

•

•

F_o

•

F_o

F_e

2F_e

2F_o

f_o

f_o

Eye

A

B

A’

B’

A’’

B’’

Objective

Eyepiece

2F_o

Objective: The converging lens nearer to the object.

Eyepiece: The converging lens through which the final image is seen.

Both are of short focal length. Focal length of eyepiece is slightly greater than that of the objective.

A’’’

α

β

D

L

v_o

u_o

P_o

P_e

Angular Magnification or Magnifying Power (M):

Angular magnification or magnifying power of a compound microscope is defined as the ratio of the angle β subtended by the final image at the eye to the angle α subtended by the object seen directly, when both are placed at the least distance of distinct vision.

Since angles are small, α = tan α and β = tan β

M = M_e x M_o

and

Since the object is placed very close to the principal focus of the objective and the image is formed very close to the eyepiece,

u_o ≈ f_o and v_o ≈ L

or

(Normal adjustment i.e. image at infinity)

or

(v_e = - D = - 25 cm)

I

Image at infinity

•

F_e

F_o

Objective

Eyepiece

Astronomical Telescope: (Image formed at infinity – Normal Adjustment)

f_o

f_e

P_o

P_e

Eye

f_o + f_e = L

Focal length of the objective is much greater than that of the eyepiece.

Aperture of the objective is also large to allow more light to pass through it.

Elescope:-Angular magnification or Magnifying power of a telescope in normal adjustment is the ratio of the angle subtended by the image at the eye as seen through the telescope to the angle subtended by the object as seen directly, when both the object and the image are at infinity.

Since angles are small, α = tan α and β = tan β

(f_o + f_e = L is called the length of the telescope in normal adjustment).

I

A

B

Objective

Astronomical Telescope: (Image formed at LDDV)

P_o

F_o

Eye

P_e

f_o

F_e

•

•

f_e

Eyepiece

u_e

D

Angular magnification or magnifying power of a telescope in this case is defined as the ratio of the angle β subtended at the eye by the final image formed at the least distance of distinct vision to the angle α subtended at the eye by the object lying at infinity when seen directly.

Since angles are small, α = tan α and β = tan β

Multiplying by f_o on both sides and rearranging, we get

or

Lens Equation

becomes

or

Clearly focal length of objective must be greater than that of the eyepiece for larger magnifying power.

Also, it is to be noted that in this case M is larger than that in normal adjustment position.

Newtonian Telescope: (Reflecting Type)

Concave Mirror

Plane Mirror

Eyepiece

Eye

Light from star

Magnifying Power:

Thank you