( )

Problem: A fish is at a distance 2 to the left of a spherical bowl of radius filled with air. Find the location of the final image. The index of water is 1.3.

1 1First surface: 1 , 1.052

Se

R R

n n i RR i R+ = − = −

( )

( )

1 1cond surface: 1 , ' 2.073.05 '

1.05 2.07 /Magnification is 0.3562 / 3.05

1 0 1 01 ' 1 2 1 2Matrix: 1 1 1 110 1 0 1 0 11

1.6 0.6 5.8

n n i RR i R

nmn

i R Rn n

n R n RR i

R

+ = − = −−

− − = − − = +

− − − +

=2.8

0.6 2.8

5.8Then 2.072.8

R i

R

i R R

+

= − = −

60

40

N

δr r'

60i'

min

ProblemFind the exit angle ' . The glass index is 1.5.What is the minimum deviation agle ?

Snell's law: sin 40 1.5sin 25.37 (angles in degrees)' 60 ' 34.63

Snell's law: sin ' 1.5sin 34.63 ' 58

i

r rr r r

i i

δ

= → =+ = → =

= → =

min

min

.47

The minimum deviation angle from

sin sin 1.5 0.5 0.752 2

37.18

gnα δ α

δ

+ = = × =

=

Problem: A microscope has an objective lens with 1.5 cm and eyepiece with2.0 cm. An objectc is placed at 1.64 cm from the objective. Find magnification

when the nimage is formed at infinity.

The

o

e

ff

==

1 1 1 image distance from the objective: + = 17.5 cm1.64 1.5

The tube length 17.5 1.5 16.0 cm25 16Magnification is = 133. If the image is 25 cm,1.5 2

25 16 1 1 = 144 1.5 2 25

Problem: A telesco

ii

L

m

m

→ =

= − =×

= − −×

× = − + −

pe has an objective of +60 cm and an eyepiece of +5 cm. With the telescope focused at a target 3 m away and used with accommodation, what is the distance between the two lenses and the magnification?

1 1 1The first image: 75 cm300 60

1 1 1Object distance for the eyepiece: 4.17 cm25 5

75Distance = 75 + 4.17 = 79.17 cm and 18.04.17

If image is at infinity (no accommodation), the magnificati

ii

xx

m

+ = → =

− = → =

= − = −

75on is 15.5

− = −

Problem: Two polarizers are positioned with their transmission axes normal. A third polarizer is placed in between with its axix at angle relative to that of the firstplate. When unplolarized light

θ

( )

0

20 0

2 2 2 20 0

with intensity is viewed, what is the final intensity?

1 1After the fisrt plate, . After the second plate, cos . After the third plate,2 2

1 1cos cos 90 = cos sin . The maximim transmi2 2

I

I I I

I I I

θ

θ θ θ θ

=

= − 01ssion is when8

45 degrees.

Problem: Oil film of thickness 400 nm and index 1.3 covers glass of index 1.5. Findthe wavelength in the visible range (400-700 nm) which exhibits destructive interfere

I

θ =

nce.

1The condition of destructive interference is 2 because there is 2

phase change in both reflections. For 1, = 693 nm. For 2, =419 nm.

nd m

m m

λ π

λ λ

= +

= =

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

0.5

1.0

x

y

2

Problem: In an interfererence-diffraction pattern, the 5-th interferencepeak is missing. What is the ratio between the slit separtion andslip opening ?

sin sinThe pattern is given by cos

da

dπ θ αλ α

2

where

sin . If the fifth peak is suppressed by diffraction, 5 .a d aπ θαλ

= =

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

0.5

1.0

x

y

t

d

n

( )

( )

Problem: Find the thickness of glass ( 1.5) if the focal point

is to be shifted by 5 mm.

From sin sin sin ' and sin sin ' (Snell's law),

we find

1 31

g

g

gg g

g

nd

d t n

nn d t n t d d

n

θ θ θ θ θ

=

=

= − =

= − → = =−

( ) 2

Problem: In double-slit with separation , one slit is wider than the other by a factor of 2. Show that the interference pattern is given by

const. 1+8cos sin

The total electric field

d

dI πθ θλ

=

2 2 2

2 2 2 2

2is 4 4 cos sin and the light

intensity is

25 4 cos sin 1 8cos sin

dE E E

d dI E E E

π θλ

π πθ θλ λ

+ +

= + = +

Problem A mercury light source radiates, among others, a blue line of 440 nm and a green line of 550 nm. If these two lines are to have an angular separation of at least 7 degrees and if the grating a

g

vailable has 500 lines/mm, in which order must the grating be used?

1 mmGrating 2 m.500

From sin

0.44sin 0.22, 12.7 ( 1),15.96 ( 2)2

0.55sin 0.275, 26.1 ( 1),32

blue blue

green reen

d

md

m m m m

m m m

µ

λθ

θ θ

θ θ

° °

°

= =

=

= = × = = =

= = × = = ( )3.4 2

Then, 2.

m

m

° =

=

Problem: A dielectric film is /2 thick. What is the transmission through the film?What if the thickness if /4?100% because reflections at both surfaces cancel each other. Multiple reflectionsyields

λλ

Γ ( )

( ) ( )

3 2

2 3 22

2

(1 )(1 ) 1 0

where

If the film is /4 thick, total reflection is21- 1

1

The input impedance is .

If glass is coated with a dielectric film

f a

f a

in a

in a

fin

a

Z ZZ Z

Z ZZ Z

ZZ

Z

λ

− Γ + Γ − Γ − Γ − Γ − ⋅ ⋅ ⋅ =

−Γ =

+

−ΓΓ + Γ Γ − Γ − Γ + ⋅ ⋅ ⋅ = =

+ Γ +

=

2 2

/4 thick, the input impedance is

. Antireflection condition is f fin a f a g

g g

Z ZZ Z Z Z Z

Z Z

λ

= = → =

λ/2

air film air

1Γ 1+Γ

−Γ(1+Γ)

−Γ(1+Γ)(1−Γ) ΓΓ(1+Γ)−ΓΓΓ(1+Γ)

−ΓΓΓ(1+Γ)(1−Γ) ΓΓΓΓ(1+Γ)

(1+Γ)(1−Γ)

ΓΓ(1+Γ)(1−Γ)

ΓΓΓΓ(1+Γ)(1−Γ)

air film air

1Γ 1+Γ

−Γ(1+Γ)

Γ(1+Γ)(1−Γ) −ΓΓ(1+Γ)ΓΓΓ(1+Γ)

−ΓΓΓ(1+Γ)(1−Γ) ΓΓΓΓ(1+Γ)

λ/4

2

ProblemFind the resonance frequency of sound wave in a wine bottle shown.

What oscillates here is the air in the neck of the bottle which has a volumeof . The mass of air in the neck is anal r l m alπ ρ= =

2

2

1

2

2

d the equation of motion is

where the pressure perturabation can be found from

=cosnt. + =0

Then 575 / s, 91.5 Hz.s

dal F a Pdt

PPV PV P V V P aV

d P a a c fdt Vl lV

γ γ γ

ξρ δ

γδ γ δ δ ξ

ξ γ ξ ωρ

−

= =

→ = −

= − → = = =

Problem: After the switch is clsoed, 2.5 V volatge wave starts propagating on the 50 line.At the joint, the reflection coefficient is100 50 1 1. A voltage wave of 1+ 2.5 3.33 V propagat100 50 3 3

Ω

− = × = +

2 3

es toward the 50 load.

1Reflection at the load is . The voltage on the 100 line evolves as3

1 1 1 13.33 1 3.33 2.5 V (final voltage)13 3 3 13

This is consistent with the DC theory.

Ω

− Ω

× − + − + ⋅ ⋅ ⋅ = = +

Problem: Determine the resonance frequency. The tension is 20 N.

The wavelength of the standing wave in the left string is 1 m and that in the right string is 0.5 m. In the left string, the wave veloci

1

2

ty is

20= =141 m/s. Then 141 Hz. In the right string0.0005/0.5

20= =70.5 m/s. 141 Hz also. .001/0.25

c f

c f

=

=

( )

( )

1 2

2

Problem: Thick lens with 10 cm, 15 cm, 40 cm, 1.5. Object at 30 cm from the first vertex.

1 1 1 1.5 0.5First refraction 1 , 90 cm30 30 10

1.5 1 1Second refraction 1 1.5 ' 15.750 ' 15

R R dn

n n ii R i

ii

= = − ==

+ = − + = → =

+ = − → =− −

( ) ( )1 2 1 2

1 1 2

2

9 cm

90 / 1.5 15.8 0.94830 50 / 1.5

1 1 1Effective focal length

1 1 1 1 1 1where 1 , 120 cm 15 30 cm

25.71 cm

Principal plane H1 22.86 cm to the right of the first ve

m

df f f f f n

n nf R f

ffdf n

= − − = − −

= + −

= − = = − =−

=

=

1

rtex

H2 34.27 cm to the left of the second vertex.

1 1 1Equivalent thin lens formula 50.06 cm 52.86 25.71

or 15.79 cm from the vertex.

fdf n

ii

=

+ = → =

40 cm

R=10cm R=15cm

30 cm

H1H2

( )

Matrix1 01 01 1 40 1 30

1 1 11 10 1 0 1 0 11.5 1 1.51.5 10 1.515

0.333 0.0389 16.67 1.0550.00389 1.0556

0 15.79 cm0.947

1 25.71 cm

i

i i

B im A

fC

−− − − − −

= − − = → == = −

= − = 40 cm

R=10cm R=15cm

30 cm

Problem: A spherical object is placed a certain distance from a screen. A lens located between the object and the screen produces an image with radius on the screen. The lens is moved to a second poa sition (with the object fixed) such that an image with radius is formed on the screen.

Show that the radius of the object sphere is .

Let the object-screen distance be and object-lens distance

b

r ab

L

=

( ) ( )2 21 2

2 21 2

1 22 21 2

1 2

be . From1 1 1 1 14 , 4

2 2

4 4Magnification ,

4 4

Then 1 and thus 1 .

x

x L L fL x L L fLx L x f

L L fL L L fLL x L xm mx xL L fL L L fLa bm m r abr r

+ = → = + − = − −−

− − + −− −= − = − = − = −

+ − − −

= = → =

Problem The matrix of a biconcave thick lens is given by 1.1333 1.33330.1567 1.0667

The curvature radius of the first surface is 5 cm, the glass index is 1.5 and the lens thickness is 2 cm. Find

−

( ) ( )1 1 2 2

2 21 2 1 2 2

the other curvature radius and H1, H2.1 1 1 1 1Let 1 , 1 . The effective focal length is

101 / 0.1567 6.38 cm.

1 1 1 1 2 10.1 1 20 cm, 10 cm6.38 10 1.5

H1:

eff

eff

n nf R f R

f

t f Rf f f f f n f

= − = − = −

= − = −

= + − → − = − + + → = − = + ×

( )( )2

1

6.38 20.425 cm to the right of the first vertex

1.5 20

H2: 0.85 cm to the left of the second vertex

eff

eff

f tnff tnf

−= =

−

=

2

Problem: Radiation in cyclotron motionFind the radiation power emitted by 5 keV electron in a magnetic field of 0.5 T.

1Acceleration is where the velocity can be found from 5 keV2

2 2e

evBa E mvm

Evm

= = =

×= =

3 197

31

19 718 2

31

2 22 22

3 30 0

5 10 1.6 10 4.2 10 m/s9.1 10

1.6 10 4.2 10 0.5 3.7 10 m/s9.1 10

1 where cyclotron frequency6 3 2483 eV/s (initial radiation power)

cc

e

a

ee a eBP mvc c m

P

ω ωπε πε

−

−

−

−

× × ×= ×

×× × × ×

= = ××

= = = =

=

2

Problem: CLS uses electron beam of 3 GeV. (a) What is the relativity factor ?(b) Determine the wiggler spatial period if x-ray radiation of 2 nm is desired.

Energy 3 GeV(a) = 580.512 MeV

w

mc

γλ

λ

γ

=

= =

( )

( )

0 2

2

2

22 92

59

1b Electrons experience kicks at . Doppler shift shortens the wavelength by a factor 1 -2

1 1Note that 1 . Then1 2

2 2 5859 2 10 0.137 m =13.7 cm2

w

ww

cf βλ γ

βββ γ

λλ λ γ λγ

−

= =

−− = =

+

= → = = × =

0 0

Water waves increase amplitude as they approach a beach. Why?

Water wave propagates at where is the depth.

What determine the velocity of light? Sound wave?

1/ . = .

How does the

w

smol

c gh h

RTc cMγε µ

=

=

2

0 02

police radar measure the speed of a car?

1 1 cosDoppler shift ' , '1 cos 1

The frequency of electromagnetic waves in telecommunication has been steadily increasing. Why is high frequenc

f fβ β θλ λ

β θ β

− −= =

− −

y wave more beneficial? More channels (information) can be carried.

2

How much power is the earth receiving from the sun and in what form? About 1 kW/m . In the form of Poyntibg vector

Lenses of high quality optical instruments are coated with dielectric films.

S = E × H

2

Why?

Anti reflection coating. Thickness / 4.

What is the fraction of light power reflected at a glass surface?

1Power reflection = 4% if 1.5

1

Soap films, CD, DVD appear c

f f a g

gg

g

n n n

nn

n

λ =

−= = +

olored. Why? Is it due to the same mechanism as prism and rainbow?Thin fild intererence. No. Why are telescopes for astronomical observation so large?

For better resolution. =1.22 limited by diffraDλθ∆ ction

Electron microscopes can see better than optical microscopes. Why? QM wavelength of energetic electron can be much shorter than optical wavelength.In microscopes, resolution limit is .

How can CA

x λ∆ =

T (computer assisted tomography) and NMRI (nuclear magnetic resonance imaging) image internal organs? In x-ray tomography, x-rays scattered by organs are detected at different anglesand the image is numerically reconstructed by fast comupters. In optics, holographyworks in the same way.

How does the antenna work? What is the basic mechanism of radiation of electromagnetic waves?Basic mehcanism of radaition of electromagentic waves is accelerated electrons. In antenna, electrons are f

2 2

30

orced to oscillate (accelerated) for radiation. Radiation power is

(W)6

How does a laser work?Electorns are pumped (excited) to a higher energy level. When they fall to a lower energy lev

e aPcπε

=

4 4

el, lasing occurs with photon energy coresponding to the energy difference.

Why is the sky blue?Rayleigh scattering by air molecules smaller than the wavelegnths. Radiation power is 1/In Mie scat

ω λ∝ ∝tering by molecules larger than , power is insensitive to or .λ ω λ

List of common questions• Water waves increase amplitude as they approach a beach. Why?• What determine the velocity of light? Sound wave? • How does the police radar measure the speed of a car? • The frequency of electromagnetic waves in telecommunication has been steadily

increasing. Why is high frequency wave more beneficial? • How much power is the earth receiving from the sun and in what form? • Lenses of high quality optical instruments are coated with dielectric films. Why? • What is the fraction of light power reflected at a glass surface? • Soap films, CD, DVD appear colored. Why? Is it due to the same mechanism as

prism and rainbow? • Why are telescopes for astronomical observation so large?• Electron microscopes can see better than optical microscopes. Why? • How can CAT (computer assisted tomography) and NMRI (nuclear magnetic

resonance imaging) image internal organs? • How does the antenna work? What is the basic mechanism of radiation of

electromagnetic waves? • How does a laser work?• Why is the sky blue? Why is it red at sunset?• What is the basic mechanisms of synchrotron radiation?