Optics Start with a Point Source of Light

Optics Start with a point source of light

Photographic plate

object

We do not get a sharp picture of the source; we just get a diffuse blur The lens Rays get bent downwards

object image

Outer rays bend more Middle ray goes straight through

If we arrange the shape of the lens just right, the rays will focus to a point Question: If we make the lens so that an object at A focuses, then will objects at B, C, ... focus somewhere as well?

C A A'

Answer: No, they will not ... at best we can manage a good approximation .... The approximation (a) The lens in thin compared to the distances of the object and image

small

(b) The objects are small compared to the curvature radius of the lens surfaces

curvature radius of lens surface

small

All the concerned rays have angles to the horizontal that are small Why do the rays bend ? sin ✓1 n2 Snell's law = sin ✓2 n1 n1 ✓1 The ray bends closer to the normal when it enters a denser medium n2 ✓2

n1 ✓1

The path of light rays is n always reversible 2 ✓2 The ray bends away from the normal when it enters a rarer medium GR9677

38 GR9677

sin ✓ 1 = sin 90o 1.33

n1 =1 sin ✓ 0.75 ⇡ n2 =1.33 ✓

98. Suppose that a system in quantum state i has energy Ei . In thermal equilibrium, the expression

-EkTi / Â Eei i -EkT/ Â e i i represents which of the following? (A) The average energy of the system (B) The partition function (C) Unity (D) The probability to find the system with 97. A beam of light has a small wavelength spread energy Ei d l about a central wavelength l . The beam (E) The entropy of the system travels in vacuum until it enters a glass plate at an angle q relative to the normal to the plate, as 99. A photon strikes an electron of mass m that is shown in the figure above. The index of refraction initially at rest, creating an electron-positron pair. of the glass is given by n()l . The angular spread The photon is destroyed and the positron and two dq ¢ of the refracted beam is given by electrons move off at equal speeds along the initial direction of the photon. The energy of the 1 (A) dq¢ = dl photon was GR0177 n 98. Suppose that a system in quantum state i 2 (A) mc has energy Ei dn. In()l thermal equilibrium, the (B) 2mc2 (B) dq¢ = dl expression dl (C) 3mc2 2 Ee-EkTi / (D) 4mc 1 Âdl i 2 (C) dq¢= i dl (E) 5mc l dn -EkT/ Â e i sin qi dl (D) dq¢ = sin q l represents which of¢ the following?

(A) The averagetan energyq ¢ dn() lof the system (E) dq¢ = dl (B) The partitionn functiondl (C) Unity (D) The probability to find the system with 97. A beam of light has a small wavelength spread energy Ei d l about a central wavelength l . The beam (E) The entropy of the system travels in vacuum until it enters a glass plate at an angle q relative to the normal to the plate, as 99. A photon strikes an electron of mass m that is shown in the figure above. The index of refraction initially at rest, creating an electron-positron pair. of the glass is given by n()l . The angular spread The photon is destroyed and the positron and two dq ¢ of the refracted beam is given by electrons move off at equal speeds along the initial direction of the photon. The energy of the 1 (A) dq¢ = dl photon was n 2 (A) mc dn()l (B) 2mc2 (B) dq¢ = dl dl (C) 3mc2 Unauthorized(D) 4mc2 copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. 1 dl (E) 5mc2 (C) dq¢= dl l dn 66

sin q dl (D) dq¢ = sin q ¢ l

tan q ¢ dn()l (E) dq¢ = dl n dl

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98. Suppose that a system in quantum state i has energy Ei . In thermal equilibrium, the expression

-EkTi / Â Eei i -EkT/ Â e i i represents which of the following? (A) The average energy of the system (B) The partition function (C) Unity (D) The probability to find the system with 97. A beam of light has a small wavelength spread energy Ei d l about a central wavelength l . The beam (E) The entropy of the system travels in vacuum until it enters a glass plate at an angle q relative to the normal to the plate, as 99. A photon strikes an electron of mass m that is shown in the figure above. The index of refraction initially at rest, creating an electron-positron pair. of the glass is given by n()l . The angular spread The photon is destroyed and the positron and two dq ¢ of the refracted beam is given by electrons move off at equal speeds along the initial direction of the photon. The energy of the 1 GR0177 (A) dq¢ = dl photon was 98. Suppose that an system in quantum state i (A) mc2 has energy E . In thermal equilibrium, the idn ()l (B) 2mc2 expression(B) dq¢ = dl dl (C) 3mc2 Ee-EkTi / (D) 4mc2 1 Âdl i (C) dq¢= i dl (E) 5mc2 l dn -EkTi / Â e sin qi dl (D) dq¢ = represents whichsin qof¢ thel following?

(A) The average energy of the system tan q ¢ dn()l (E) dq¢ = dl (B) The partitionn functiondl (C) Unity (D) The probability to find the system with 97. A beam of light has a small wavelength spread energy Ei d l about a central wavelength l . The beam (E) The entropy of the system travels in vacuum until it enters a glass plate at an angle q relative to the normal to the plate, as 99. A photon strikes an electron of mass m that is shown in the figure above. The index of refraction initially at rest, creating an electron-positron pair. of the glass is given by n()l . The angular spread The photon is destroyed and the positron and two dq ¢ of the refracted beam is given by electrons move off at equal speeds along the initial direction of the photon. The energy of the 1 (A) dq¢ = dl photon was n 2 (A) mc dn()l (B) 2mc2 (B) dq¢ = dl dl (C) 3mc2 2 Unauthorized(D) 4mc copying or reuse of 1 dl any part of this page is illegal. (C) dq¢= dl (E) 5mc2 GO ON TO THE NEXT PAGE. l dn 66

sin q dl (D) dq¢ = sin q ¢ l

tan q ¢ dn()l (E) dq¢ = dl n dl

Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. 66 What shape of then lens will do this job ? Where do the rays focus?

small object

large

If the object if very far away, and we are looking at a small transverse region near the lens, the the rays look almost parallel

focal point

focal length f

Rays from infinity focus at the focal point f

If the object comes closer, then it is harder to focus the rays, so the image forms further away

object distance o image distance i

1 1 1 + = o i f f

1 1 1 + = o i f object distance o image distance i

f Object moves from infinity to focal point

Image moves from focal point to infinity Magniﬁcation How do we locate the image of a point off the axis ? We rotate the rays object height h image height h 0

object distance o image distance i

image height h 0 image distance i (a) Size is given by = object height h object distance o

(b) The image is inverted

h0 i We write = h o HRW 28E: The sun subtends an angle of 0.5 degrees.

You produce an image of the sun on a screen, using a convex lens of focal length 20 cm.

What is the diameter of the image ? HRW 28E: The sun subtends an angle of 0.5 degrees.

You produce an image of the sun on a screen, using a convex lens of focal length 20 cm.

What is the diameter of the image ? 1 1 ✓ 0.50 radians ⇡ ⇡ 2 57 rays from top of object Diameter f ✓ 0.18 cm ⇡ ⇡

o image distance i = f rays from bottom of object Two lenses

focal length f 1 focal length f 2 final image object height h height h 00 image height h 0

object distance o image distance i object distance o 0 image distance i 0

1 1 1 1 1 1 + = + = o i f1 o0 i0 f2

h h h Magnification M = 00 = 0 00 h h h ✓ ◆ ✓ ◆✓ 0 ◆

i i0 i i0 Final image is = = o o o o upright ✓ ◆✓ 0 ◆ ✓ ◆✓ 0 ◆ HRW A beam expander is made of two lenses as shown.

What is the ratio of the intensities of the emergent beam to the incident beam, if

f1 = 10 cm, f2 = 20 cm ?

f1 f2 HRW A beam expander is made of two lenses as shown.

What is the ratio of the intensities of the emergent beam to the incident beam, if

f1 = 10 cm, f2 = 20 cm ?

f1 f2

I f 2 1 f = 1 = I f 4 i ✓ 2 ◆ GR9677

38 GR9677

f 2 = 10 f1

f2 = 15 cm

f1 + f2 = 16.5 cm

f1 f2

38 The telescope Angular magniﬁcation

The magnification relation is

h0 i hi = h0 = h o o An object like the sun is very far away. So o is very large. But the sun is very big, so h is also very large h ↵ o h In this situation we look at the ratio o

Since all angles are assumed small, we have tan ↵ sin ↵ ↵ ⇡ ⇡

Far away objects will be described by ↵ , rather than h , and we talk about angular magnification rather than magnification The telescope objective eyepiece

f1 f2

o i o0 i0

h00 i i0 Magnification = h o o ✓ ◆✓ 0 ◆

h h00 Angular sizes ↵i = ↵f = o i0

We have i = f1 o0 = f2

↵f i f1 Angular magnification of telescope = = ↵i o0 f2 GR9277 GR9277

f2 f1 eyepiece objective

f Angular magniﬁcation = 1 = 10 f2

f1 =1

f2 = .1 Total length = 1 + .1=1.1

30. The driver of a police car hears an echo of the 34. A rod measures 1.00 m in its rest system. How car’s siren from a wall toward which the car is fast must an observer move parallel to the rod 30. The driver of a police car hears an echo of the 34. A rod measures 1.00 m in its rest system. How moving with a speed of 3.5 m/s. If the speed of to measure its length to be 0.80 m? car’s siren from a wall toward which the car is fast must an observer move parallel to the rod sound is 350 m/s and the frequency of the siren is moving with a speed of 3.5 m/s. If the speed of to(A) measure 0.50c its length to be 0.80 m? 600 Hz, the driver hears the echo at a frequency sound is 350 m/s and the frequency of the siren is (B) 0.60c nearest to which of the following? (A) 0.50c 600 Hz, the driver hears the echo at a frequency (C) 0.70c (B) 0.60c nearest(A) 588 to Hz which of the following? (D) 0.80c (C) 0.70c (B) 594 Hz (E) 0.90c (A) 588 Hz (D) 0.80c (C) 600 Hz (B) 594 Hz (E) 0.90c (D) 606 Hz 35. A particle decays in 2.0 ms in its rest frame. If (C) 600 Hz (E) 612 Hz the same particle moves at u = 0.60c in the lab (D) 606 Hz 35. A particle decays in 2.0 ms in its rest frame. If frame, how far will it travel in the lab before (E) 612 Hz the same particle moves at u = 0.60c in the lab 31. The first five harmonics produced by an organ decaying? frame, how far will it travel in the lab before pipe open at both ends are 50 Hz, 100 Hz, 150 Hz, 31. The first five harmonics produced by an organ decaying?(A) 150 m 200 Hz, and 250 Hz. Which of the harmonics, pipe open at both ends are 50 Hz, 100 Hz, 150 Hz, (B) 288 m if any, will survive once the pipe is closed at (A) 150 m 200 Hz, and 250 Hz. Which of the harmonics, (C) 360 m one end? (B) 288 m if any, will survive once the pipe is closed at (D) 450 m (C) 360 m one(A) end? 50 Hz, 150 Hz, and 250 Hz only (E) 750 m (D) 450 m (B) 100 Hz and 200 Hz only (A) 50 Hz, 150 Hz, and 250 Hz only (E) 750 m (C) 150 Hz and 250 Hz only 36. The rest mass of a particle with total energy (B) 100 Hz and 200 Hz only (D) 200 Hz only 5.0 GeV and momentum 4.9 GeV/c is (C) 150 Hz and 250 Hz only 36. The rest mass of a particle with total energy (E) None approximately (D) 200 Hz only 5.0 GeV and momentum 4.9 GeV/c is

(E) None approximately 2 32. A refracting telescope consists of two converging (A) 0.1 GeV/c 2 lenses separated by 100 cm. The eye-piece lens GRpractice(B)(A) 0.20.1 book GeV/c2 32. A refracting telescope consists of two converging 2 has a focal length of 20 cm. The angular (C) 0.5 GeV/c2 lenses separated by 100 cm. The eye-piece lens (B) 0.2 GeV/c2 magnification of the telescope is (D) 1.0 GeV/c2 has a focal length of 20 cm. The angular (C) 0.5 GeV/c (E) 1.5 GeV/c2 magnification(A) 4 of the telescope is (D) 1.0 GeV/c2

do spectroscopy over a range of visible (B)(A) 1 33. The best type of laser with which to wavelengths is Q do spectroscopy over a range of visible (C)(B) 1 wavelengths(A) a dye laser is eQ0 (B) a helium-neon laser (C) (A) a dye laser Q (C) an excimer laser (D) e0 (B) a helium-neon laser 2e (D) a ruby laser Q0 (C) an excimer laser (D) (E) a neodymium-YAG laser 2e (D) a ruby laser (E) 0 0

(E) a neodymium-YAG laser (E) 0

GO ON TO THE NEXT PAGE. -18- 24 GO ON TO THE NEXT PAGE. -18-

(B) 5 (E) 1.5 GeV/c2 (A) 4 37. If charge +Q is located in space at the point (C) 6 (B) 5 (x = 1 m, y = 10 m, z = 5 m), what is the total (D) 20 37. If charge +Q is located in space at the point (C) 6 electric flux that passes through the yz-plane? (E) 100 (x = 1 m, y = 10 m, z = 5 m), what is the total (D) 20 f1 + f2 = 100 electric(A) flux that passes through the yz-plane? (E) 100 33. The best type of laser with which to f2 = 20 do spectroscopy over a range of visible (B)(A) 1 33. The best type of laser with which to wavelengths is do spectroscopy over a range of visible (B) 1Q f1 = 80 (C) wavelengths(A) a dye laser is eQ0 (B) a helium-neon laser (C) f (A) a dye laser 1Q (C) an excimer laser Angular magniﬁcation =(D) e0 =4 (B) a helium-neon laser f2e (D) a ruby laser Q0 (C) an excimer laser (D) (E) a neodymium-YAG laser 2e (D) a ruby laser (E) 0 0

(E) a neodymium-YAG laser (E) 0

GO ON TO THE NEXT PAGE. -18- 24 GO ON TO THE NEXT PAGE. -18- HRW 47E: If the angular magniﬁcation of a telescope is 36 and the diameter of the objective is 75 mm, what is the minimum diameter of the eyepiece required to collect all the light from the objective ? HRW 47E: If the angular magniﬁcation of a telescope is 36 and the diameter of the objective is 75 mm, what is the minimum diameter of the eyepiece required to collect all the light from the objective ?

objective eyepiece

f1 f2

o i o0 i0

↵f i f1 Angular magnification of telescope = = ↵i o0 f2 f 1 = 36 f 1 2 d = 75 2.1 mm objective 36 ⇥ ⇡ Mirrors Basic law of reflection:

Angle of incidence equals ✓i angle of reflection ✓r ✓r = ✓i

We can use this to focus:

object image First we consider parallel rays, which correspond to a source that is infinitely far away

↵ ↵ R

↵

center

?? If we take the limit of small angles, then all the parallel rays do pass through a point ...

↵ R ↵

↵ = =

center focus focal length R 2 image

focal length f As the object comes closer, it becomes more difficult to converge the rays, so the image forms further away object distance o 1 1 1 image + = o i f object image i distance Object moves from infinity to focal point Image moves from focal point to infinity

object image

focal length f Magniﬁcation

Each point of object lies on an 'axis' through the center

The image of each point will lie on this axis

image

object center object distance o

image distance i

object height h image height h 0 center 2f 1 1 1 + = o i f

h0 2f i i Size is given by = = The image is inverted h 2f o o Virtual images f

1 1 1 object distance o image distance i + = o i f

If we place the object closer than the focal point, then the rays are unable to converge to a point ...

object distance o

But all is not lost, as long as the rays still behave as if they came from a point, since we can use some other lens to make them converge .... virtual image f

object distance o

We can now use another lens to finally focus the rays to a point ....

1 1 1 + = o i f Lens of a camera or lens of the eye f

object distance o

image distance i is a negative number Magniﬁcation

object distance o

image distance i is negative

h i Magnification 0 = is positive h o

So image is upright The same thing happens for a concave mirror

virtual image

image distance i object distance o is negative

1 1 1 + = o i f

i is a negative number We can rotate about the center

2f f

center image distance i object distance o is negative

1 1 1 + = o i f

i Magnification is a positive number o So the virtual image is upright

11. An object is located 40 centimeters from the first of two thin converging lenses of focal lengths 20 centimeters and 10 centimeters, respectively, 8. A positive charge Q is located at a distance L as shown in the figure above. The lenses are above an infinite grounded conducting plane, separated by 30 centimeters. The final image as shown in the figure above. What is the total formed by the two-lens system is located charge induced on the plane? (A) 5.0 cm to the right of the second lens (A) 2Q (B) 13.3 cm to the right of the second lens (B) Q (C) infinitely far to the right of the second lens (C) 0 (D) 13.3 cm to the left of the second lens (D) -Q (E) 100 cm to the left of the second lens

(E) -2Q GR0177

9. Five positive charges of magnitude q are arranged symmetrically around the circumference of a circle of radius r. What is the magnitude of the electric field at the center of the circle? ()k = 14p⑀0 (A) 0 2 (B) kq r 2 (C) 5kq r 2 (D) (/)cos/kq r af25p

(E) (/)cos/525kq r2 afp 12. A spherical, concave mirror is shown in the figure 10. A 3-microfarad capacitor is connected in series above. The focal point F and the location of the with a 6-microfarad capacitor. When a 300-volt object O are indicated. At what point will the potential difference is applied across this com- image be located? bination, the total energy stored in the two (A) I capacitors is (B) II (A) 0.09 J (C) III (B) 0.18 J (D) IV (C) 0.27 J (E) V (D) 0.41 J (E) 0.81 J

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(E) -2Q GR0177

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15E: A shaving mirror has a radius of curvature of 35 cm. It is positioned so that the (upright) image of a man's face is 2.5 times the size of the face.

How far is the mirror from the face? HRW

15E: A shaving mirror has a radius of curvature of 35 cm. It is positioned so that the (upright) image of a man's face is 2.5 times the size of the face.

How far is the mirror from the face?

35 f = 2 i = 2.5 o

1 1 1 + = o i f

o = 10.5 cm Sign changes

The standard configurations with all numbers positive and images real are

1 1 1 + = o i f object distance o image distance i

object distance

image 1 1 1 + = object o i f image i distance

Any changes bring in a negative sign, but the formulae remain the same (1) Convex lens has positive f , concave lens has negative f

(2) Concave mirror has positive f , convex mirror has negative f

(3) An object on the correct side has positive o , an object on the wrong side has negative o

(4) An image on the correct side has positive i , an object on the wrong side has negative i

GR0177

(E) 2Q -

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i2 =5 GR0177

first image

1 1 1 1 = = 11. An object is located 40 centimeters from the i1 20 40 40 first of two thin converging lenses of focal lengths i = 40 20 centimeters and 10 centimeters, respectively, 1 8. A positive charge Q is located at a distance L as shown in the figure above. The lenses are to right of first lens above an infinite grounded conducting plane, separated by 30 centimeters. The final image as shown in the figure above. What is the total formed by the two-lens system is located This is o2 = 10 charge induced on the plane? (A) 5.0 cm to the right of the second lens from the second lens (A) 2Q (B) 13.3 cm to the right of the second lens (B) Q (C) infinitely far to the right of the second lens second image (C) 0 (D) 13.3 cm to the left of the second lens 1 1 1 1 (D) -Q (E) 100 cm to the left of the second lens = + = i2 10 10 5 (E) -2Q i2 =5

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14. An 8-centimeter-diameter by 8-centimeter-long NaI(Tl) detector detects gamma rays of a specific energy from a point source of radioactivity. When the source is placed just next to the detector at (B) the center of the circular face, 50 percent of all emitted gamma rays at that energy are detected. If the detector is moved to 1 meter away, the fraction of detected gamma rays drops to (A) 10-4

(B) 2 ¥ 10-4

(D)

(E)

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GR0177 13. Two stars are separated by an angle of 15. Five classes of students measure the height of a 3 ¥ 10-5 radians. What is the diameter of the building. Each class uses a different method and smallest telescope that can resolve the two stars each measures the height many different times. using visible light ( l @ 600 nanometers) ? The data for each class are plotted below. Which (Ignore any effects due to Earth’s atmosphere.) class made the most precise measurement? (A) 1 mm (A) (B) 2.5 cm (C) 10 cm (D) 2.5 m (E) 10 m ✓ 14. An 8-centimeter-diameter by 8-centimeter-long ⇠ d NaI(Tl) detector detects gamma rays of a specific energy from a point source of radioactivity. When 5 the source is placed ✓just next to3 the10 detector at (B) the center of the circular⇠ dface,⇠ 50⇥ percent of all emitted gamma raysd at 2thatcm energy are detected. If the detector is moved⇠ to 1 meter away, the fraction of detected gamma rays drops to (A) 10-4

(B) 2 ¥ 10-4

(D)

(E)

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