Unformatted text preview: red from the middle of the central bright spot, so that the angle θspot
subtended by the entire bright spot is twice as large: θspot = 2θ (2) λ The first circular dark fringe is located by the angle θ = sin −1 1.22 (Equation 27.5), D
where D is the diameter of the spotlight and λ is the wavelength of the light.
SOLUTION Substituting Equation (2) into Equation (1) yields d = rθspot = 2rθ (3) λ Substituting θ = sin −1 1.22 (Equation 27.5) into Equation (3), we obtain D λ
694.3 × 10−9 m 3
d = 2rθ = 2r sin −1 1.22 = 2 ( 3.77 × 108 m ) sin −1 1.22 = 3.2 × 10 m D
0.20 m 41. SSM WWW REASONING Assuming that the angle θmin is small, the distance y
between the blood cells is given by
(8.1)
y = f θ min
where f is the distance between the microscope objective and
the cells (which is given as the focal length of the objective).
However, the minimum angular separation θmin of the cells is
given by the Rayleigh criterion as θmin = 1.22 λ/D (Equation
27.6), where λ is the wavelength of the light and D is the
diameter of the objective. These two relations can be used to
find an expression for y in terms of λ. Blood
cells y
f θmin
Microscope
objective D Chapter 27 Problems 1459 SOLUTION
a. Substituting Equation 27.6 into Equation 8.1 yields 1.22λ y = f θ min = f D
Since it is given that f = D, we see that y = 1.22 λ .
b. Because y is proportional to λ, the wavelength must be shorter
closer together. to resolve cells that are 42. REASONING AND SOLUTION
a. Equation 27.6 θ min = 1.22 λ / D gives the minimum angle θmin that two point objects
can subtend at an aperture of diameter D and still be resolved. The angle must be measured
in radians. For a separation s between the two circles and a distance L between the
concentric arrangement and the camera, Equation 8.1 gives the angle in radians as
θ min = s / L . Therefore, we find that c h 1.22 λ s
=
D
L or L= sD
1.22 λ Since s = 0.040 m – 0.010 m = 0.030 m, we calculate that bg
c
bg
c h
h 0.030 m 12 .5 × 10 –3 m
sD
L=
=
= 550 m
1.22 λ
1.22 555 × 10 –9 m
b. The calculation here is similar to that in part a, except that the separation s is between
one side of a diameter of the small circle and the other side, or s = 0.020 m:
sD
L=
=
1.22 λ 0
b.020 m g12.5 × 10 mh=
c
1
b.22 g555 × 10 mh
c
–3 –9 370 m 43. SSM REASONING AND SOLUTION According to Equation 27.7, the angles that
correspond to the firstorder (m = 1) maximum for the two wavelengths in question are:
a. for λ = 660 nm = 660 × 10 –9 m , 1460 INTERFERENCE AND THE WAVE NATURE OF LIGHT 660
F λ I = sin L1) F × 10
(
Gd J M G × 10
H K NH
1.1 θ = sin –1 m –9 –1 –6 m
m IO 37 °
J=
P
K
Q m
m IO 22 °
J=
P
K
Q b. for λ = 410 nm = 410 × 10 –9 m , 410
F λ I = sin L1) F × 10
(
Gd J MG × 10
H K NH
1.1 θ = sin –1 m –9 –1 –6 44. REASONING The number of lines per centimeter that a grating has is the reciprocal of the
spacing between the slits of the grating. We can determine the slit spacing by considering
the angle θ that defines the position of a principal bright fringe. This angle is related to the
order m of the fringe, the wavelength λ of the light, and the spacing d between the slits.
Thus, we can use the values given for θ, m, and λ to determine d.
SOLUTION The number of lines per centimeter that the grating has is N and is the
reciprocal of the spacing d between the slits: N= 1
d (1) where d must be expressed in centimeters. The relationship that determines the angle
defining the position of a principal bright fringe is
sin θ = m λ
d m = 0, 1, 2, 3, ... (27.7) Solving this equation for d and applying the result for a secondorder fringe (m = 2) gives ( ) 2 495 ×10−9 m
mλ
d=
=
= 6.10 × 10−6 m
sin θ
sin 9.34° or 6.10 × 10−4 cm Substituting this result into Equation (1), we find that
N= 1
1
=
= 1640 lines/cm
d 6.10 ×10−4 cm Chapter 27 Problems 1461 45. SSM REASONING AND SOLUTION The geometry of the situation is shown below. First dark
fringe y θ d Midpoint of
central bright
fringe L From the geometry, we have tan θ = y 0.60 mm
=
= 0.20
L 3.0 mm θ = 11.3° or Then, solving Equation 27.7 with m = 1 for the separation d between the slits, we have c h (1) 780 × 10 –9 m
mλ
d=
=
=
sin θ
sin 11.3° 4.0 × 10 –6 m 46. REASONING The drawing shows the angle θ
that locates a principal maximum on the screen,
along with the separation L between the grating
and the screen and the distance y from the
midpoint of the screen. It follows from the
drawing that y = L tan θ , which becomes
y = L sin θ , since we are dealing with small
angles ( tan θ ≈ sin θ ) .
For a diffraction grating, the angle θ that locates a
principal maximum can be found using
sin θ = mλ/d (Equation 27.7), where λ is the
wavelength, d is the separation between the
grating slits, and the order m is m = 0, 1, 2, 3, …. y θ
L Grating We will use the above relations to obtain an expression for the separation between adjacent
principal maxima. 1462 INTERFERENCE AND THE WAVE NATURE OF LIGHT SOLUTION The dist...
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 Spring '13
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 Physics, Pythagorean Theorem, Force, The Lottery, Right triangle, triangle

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