Simple Nature - Light and Matter

gives

θ= 2 tan−^1

w

2 f

for the field of view. This comes out to be 39◦and 64◦for the two lenses. (b) For small angles,

the tangent is approximately the same as the angle itself, provided we measure everything in

radians. The equation above then simplifies to

θ=

w

f

The results for the two lenses are .70 rad = 40◦, and 1.25 rad = 72◦. This is a decent approxi-
mation.
(c) With the 28-mm lens, which is closer to the film, the entire field of view we had with the
50-mm lens is now confined to a small part of the film. Using our small-angle approximation
θ=w/f, the amount of light contained within the same angular widthθis now striking a piece
of the film whose linear dimensions are smaller by the ratio 28/50. Area depends on the square
of the linear dimensions, so all other things being equal, the film would now be overexposed by
a factor of (50/28)^2 = 3.2. To compensate, we need to shorten the exposure by a factor of 3.2.
Page 838, problem 48:
You don’t want the wave properties of light to cause all kinds of funny-looking diffraction effects.
You want to see the thing you’re looking at in the same way you’d see a big object. Diffraction
effects are most pronounced when the wavelength of the light is relatively large compared to the
size of the object the light is interacting with, so red would be the worst. Blue light is near the
short-wavelength end of the visible spectrum, which would be the best.
Page 838, problem 49:
(a) You can tell it’s a single slit because of the double-width central fringe.
(b) Four fringes on the top pattern are about 23.5 mm, while five fringes on the bottom one are
about 14.5 mm. The spacings are 5.88 and 2.90 mm, with a ratio of 2.03. A smallerdleads to
larger diffraction angles, so the width of the slit used to make the bottom pattern was almost
exactly twice as wide as the one used to make the top one.
Page 839, problem 51:
For the size of the diffraction blob, we have:

λ

d

∼sinθ

≈θ

θ∼

700 nm

10 m

≈ 10 −^7 radians

For the actual angular size of the star, the small-angle approximation gives

θ∼

109 m

1017 m

= 10−^8 radians

The diffraction blob is ten times bigger than the actual disk of the star, so we can never make
an image of the star itself in this way.