6 1 Formulae Ray Optics

7/31/2019 6 1 Formulae Ray Optics

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=sin i

sin r

sin iab =

sin r

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

sin rba =

sin i

Denser(b)

Rarer(a)i

r

N

ab x ba = 1 or ab = 1 / ba

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

Lateral Shift:

t sin y =

cos r1

t sin(i1- r1)y =

cos r1

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

= t (i1 r1) or = t i1(1 1 /ab)

Rarer (a)

Rarer (a)

Denser(b)

N

N

r1

i1

i2

r2

M

t

y

6.1 RAY OPTICS

TIPS:

of optically rarer medium is lower and that of a denser mediumis higher.

of denser medium w.r.t. rarer medium is more than 1 and thatof rarer medium w.r.t. denser medium is less than 1.(air =vacuum = 1)

In refraction, the velocity and wavelength of light change. In refraction, the frequency and phase of light do not change.

am = ca / cm and am = a / m

Principle of Reversibility of Light:

Refraction through a Parallel Slab:sin i1

ab =sin r1

sin i2ba =

sin r2

But ab x ba = 1

sin i1

sin r1

sin i2

sin r2x = 1

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


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c >b

Apparent Depth of a Liquid:

Rarer (a)

Denser (b)

O

O

N

bhr

ha

i

r

r

i

sin iba =

sin r

sin rab =

sin ior

hrab =

ha=

Real depth

Apparent depth

Apparent Depth of a Number ofImmiscible Liquids:

ha = hi / ii = 1

n

Apparent Shift:Apparent shift = hr - ha = hr (hr / )

= hr [ 1 - 1/]TIPS:

If the observer is in rarer medium and the object is in denser medium then ha < hr.

(To a bird, the fish appears to be nearer than actual depth.) If the observer is in denser medium and the object is in rarer medium then ha > hr.

(To a fish, the bird appears to be farther than actual height.)

a

Rarer (a)

Rarer (a)

Denser(b)

N

N

b

r

i

r

r

r2

i

Denser(c)

c

N

sin i1ab =

sin r1

sin r1bc =

sin r2

ab x bc x ca= 1

sin r2ca =sin i1

ab x bc = acor

bc = ac / abor

a

Refraction through a Compound Slab:


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N N N N

O

r = 90

ic i > ic i

Rarer(air)

Denser(glass)

g

a

Conditions for TIR:

The incident ray must be in optically denser medium.

The angle of incidence in the denser medium must be greater than thecritical 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 theangle of refraction in the rarer medium is 90.

sin iga =

sin r

sin ic=

sin 90= sin ic

or1

ag =ga

1ag =

sin icor

1sin ic =

ag

gsin ic = aAlso

Redcolour hasmaximumvalue of critical angle andVioletcolour hasminimumvalue of critical angle since,

1sin ic =

ag=

1

a + (b/ 2)

Applications of T I R:

Mirage formation

Looming

Totally reflecting Prisms

Optical Fibres

Sparkling of Diamonds

Total Internal Reflection:

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


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5/14

Note:

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

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

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

4. The refractive indices 1 and2 get interchanged in the expressions.

1

- u

2

v

2 -1

R+ =

2

- u

1

v

1 -2

R+ =

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

C P

RO

Denser Medium Rarer Medium

IM

rA

vu

N

i

- u

v

1 - 2

R+ =

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

- u

v

1 - 2

R+ =

Refraction at Concave Surface:

(From Denser Medium to Rarer Medium -Virtual Image)

- u

v

1 - 2

R+ =


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R1

P1

O

2

1

i

A

vu

N1

R2

C1

I1

N2

L

C

N

P2

C2

I

1

(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,

C

(2-

1)(

CC+=

1

C

CC+ )

1

Substituting the valueswith sign conventions,

1

- u

(2 -1)

R1+ =

11

v R2- )

1(

Since 2 / 1 =

1

- u

2

R1+ =

11

v R2- )

1(

1- 1)(

or

1

- u

( 1)R1

+ =11

v R2- )

1(

When the object is kept at infinity, the image is formed at the principal focus.i.e. u = - , v = + f._So, ( 1)

R1=

11

f R2- )

1(

This equation is called Lens Makers Formula.

Also, from the above equations we get,

1

- u f+ =

11

v

Lens Makers Formula:For refraction atLP1N,

C

CI1

2 -1

CC+ =

(as if the image is formedin the denser medium)

For refraction at LP2N,

-CI1

CI

-(1 -2)

CC+ =


7/14

f

R

u

C

A

B

A

B

M

Triangles ABC and ABC are similar.

AB

AB=

CB

CB

Triangles MCF2 and ABF2 are similar.

AB

MC=

BF2

CF2

v

AB

AB

=BF2

CF2

or

2F2

F2

F1

2F1

CB

CB=

BF2

CF2

CB

CB=

CB - CF2

CF2According to new Cartesian signconventions,CB = - u, CB = + v and CF2 = + f.

1

v f- =

11

u

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

m =I

O

AB

AB

=CB

CB

+ I

- O=

+ v

- u

According to new Cartesian signconventions,AB = + I, AB = - O, CB = + v and

CB = - u.

I

O=

v

u=or

Magnification in terms of v and f:

m =

f - v

f

Magnification in terms of v and f:

m =f

f - u

Power of a Lens:

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

P=1

f

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

Linear Magnification:


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i

A

B C

e

O

P

Q

r1 r2

N1 N2D

In quadrilateral APOQ,

A + O = 180 .(1)

(since N1 and N2 are normal)

In triangle OPQ,

r1 + r2 + O = 180 .(2)

In triangle DPQ,

= (i - r1) + (e - r2)

= (i + e) (r1 + r2) .(3)

From (1) and (2),

A = r1 + r2

From (3),

= (i + e) (A)

or i + e = A +

Variation of angle of deviation with angle of incidence:

i0 i = e

m

When angle of incidence increases,the angle of deviation decreases.At a particular value of angle of incidencethe angle of deviation becomes minimumand is calledangle of minimum deviation.At m, i = e and r1 = r2 = r (say)

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

Refractive Index of Material of Prism:

A = r1 + r2A = 2rr = A / 2i + e = A + 2 i = A + m

i = (A + m) / 2

According to Snells law,sin i

=sin r1

sin i

sin r=

=

sin(A + m)

2

sinA

2

Refraction of Light through Prism:


9/14

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

sin i =

sin r1

sin e =

sin r2and

If i is assumed to be small, then r1, r2and e will also be very small. So,replacing sines of the angles by angles themselves, we get

i =

r1and

e =

r2i + e = (r1 + r2) = ABut i + e = A + So, A + = A

or = A ( 1)

Dispersion of White Light through Prism:

r

A

B C

D

Whitelight

v

Cause of Dispersion:

sin iv =

sin rv

sin ir =

sin rr

andSince v > r , rr > rv

So, the colours are refracted at different

angles and hence get separated.

R

OYGBIV

Screen

N

Dispersion can also be explained on the basis of Cauchys equation.

= a +b

2

c

4+ (where a, b and c are constants for the material)

Since v < r , v > r

But = A ( 1)

Therefore, v > r

So, the colours get separated with different angles of deviation.Violet is most deviatedandRedis least deviated.


10/14

The dispersive power of the material of a prism for any two colours is definedas the ratio of the angular dispersion for those two colours to the meandeviation produced by the prism.It may also be defined as dispersion per unit deviation.

=

where is the mean deviation and =

v + r

2

Also =v - r

or =

(v r) A(y 1) A or =

(v r)(y 1)

Scattering of Light Blue colour of the sky and Reddish appearance ofthe Sun at Sun-rise and Sun-set:

The molecules of the atmosphere and other particles that are smaller thanthe longest wavelength of visible light are more effective in scattering

light of shorter wavelengths than light of longer wavelengths. Theamount of scattering is inversely proportional to the fourth power of thewavelength. (Rayleigh Effect)

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

Angular Dispersion:

The difference in the deviations suffered by two colours in passingthrough a prism gives the angular dispersion for those colours.

The angle between the emergent rays of any two colours is called angulardispersion between those colours.

It is the rate of change of angle of deviation with wavelength. ( = d / d)

= v - r or = (v r) A

Dispersive Power:


11/14

Compound Microscope:

Fo

Fo

Fe

2Fe2Fo

fo fo

fe

EyeA

B

A

B

A

B

Objective

Eyepiece

2Fo

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

D

L

vouo

Po Pe

Angular Magnification or Magnifying Power (M):Angular magnification or magnifying power of a compound microscope is definedas the ratio of the angle subtended by the final image at the eye to the angle subtended by the object seen directly, when both are placed at the least distance ofdistinct vision.

M =

Since angles are small, = tan and = tan

M =tan

tan

M =AB

Dx

D

AA

M =AB

Dx

D

AB

M =AB

AB

M =AB

ABx

AB

AB

M = Me x Mo

Me = 1 +D

fe

and Mo =vo

- uoM =

vo

- uo( 1 +

D

fe)

Since the object is placed very close to theprincipal focus of the objective and theimage is formed very close to the eyepiece,uo fo and vo L

M = - Lfo

( 1 + D

fe)

or M - L

fox

D

fe

(Normal adjustmenti.e. image at infinity)

Me = 1 -ve

fe

or (ve = - D= - 25 cm)


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I

Image atinfinity

Fe

Fo

Objective

Eyepiece

fo fe

Po Pe

Eye

fo + fe = L

Focal length of the objectiveis much greater than that of the eyepiece.Aperture of the objective is also large to allow more light to pass through it.

Angular magnification or Magnifying power of a telescope in normaladjustment is the ratio of the angle subtended by the image at the eye asseen through the telescope to the angle subtended by the object as seendirectly, when both the object and the image are at infinity.

M =


M =tan

tan

(fo + fe = L is called the length of thetelescope in normal adjustment).

M = /Fe I

PoFe

Fe I

PeFe

M = /- I

fo

- I

- fe

M = - fo

fe

Astronomical Telescope: (Image formed at infinity Normal Adjustment)


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I

A

B

Objective

Astronomical Telescope: (Image formed at LDDV)

Po

Fo

Eye

Pe

fo

Fe

fe

Eyepiece

ue

D

Angular magnification or magnifying power of a telescope in this case isdefined as the ratio of the angle subtended at the eyeby the final imageformed at the least distance of distinct vision to the angle subtended at theeye by the object lying at infinity when seen directly.

M =


M =tan

tan

M =Fo I

PeFo/

Fo I

PoFo

M =PoFo

PeFoM =

+ fo

- ue

Multiplying by fo on both sides andrearranging, we get

M =- fo

fe( 1 +

fe

D)

-1

u

1

f

1

v=

-1

- ue

1

fe

1

- D=

or

Lens Equation

becomes

or +1

ue

1

fe

1

D=

Clearly focal length of objective must be

greater than that of the eyepiece for largermagnifying power.Also, it is to be noted that in this case M islarger than that in normal adjustmentposition.


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Newtonian Telescope: (Reflecting Type)

Concave Mirror

Plane Mirror

Eyepiece

Eye

Lightfrom star

M =fo

fe

Magnifying Power:

Resolving Power of a Microscope:

The resolving power of a microscope is defined as the reciprocal of thedistance between two objects which can be just resolved when seen throughthe microscope.

Resolving Power =1

d=

2 sin

Resolving power depends on i) wavelength , ii) refractive index of themedium between the object and the objective and iii) half angle of thecone of light from one of the objects .

Resolving Power of a Telescope:

The resolving power of a telescope is defined as the reciprocal of thesmallest angular separation between two distant objects whose images areseen separately.

Resolving Power =1

d=

a

1.22

Resolving power depends on i) wavelength ,ii) diameter of the objective a.

d

Objective

d

Objective

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6 1 Formulae Ray Optics

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