5.63. Find the curvature radius of a ray of light propagating
in a horizontal direction close to the Earth's surface where the
gradient of the refractive index in air is equal to approximately
3.10-8 m-1. At what value of that gradient would the ray of light
propagate all the way round the Earth?
5.2. Interference of Light
- Width of a fringe:
Ax = —d— (5.2a)
where 1 is the distance from the sources to the screen, d is the distance between
the 'sources.- Temporal and spatial coherences. Coherence length and coherence radius:
X2
Icon = , Pcoh =7-- , (5.2b)
where is the angular dimension of the source.- Condition for interference maxima in the case of light reflected from a
thin plate of thickness 6:
26 jl r1 2 — sin?' 0 1 --= (k 1/21 X., (5.2c)
where k is an integer. - Newton's rings produced on reflection of light from the surfaces of an
air interlayer formed between a lens of radius R and a glass plate with which
the convex surface of the lens is in contact. The radii of the rings:
r = 1(X.Rk/2, (5.2d)with the rings being bright if k = 1, 3, 5,.. ., and dark if k = 2, 4, 6,...
The value k = 0 corresponds to the middle of the central dark spot.5.64. Demonstrate that when two harmonic oscillations are added,
the time-averaged energy of the resultant oscillation is equal to
the sum of the energies of the constituent oscillations, if both of
them
(a) have the same direction and are incoherent, and all the values
of the phase difference between the oscillations are equally probable;
(b) are mutually perpendicular, have the same frequency and
an arbitrary phase difference.
5.65. By means of plotting find the amplitude of the oscillation
resulting from the addition of the following three oscillations of the
same direction:gi = a cos cot, (^) Z = 2a sin cot, s = 1.5a cos (cot n/3).
5.66. A certain oscillation results from the addition of coherent
oscillations of the same direction k = a cos [cot (k — 1) T],
where k is the number of the oscillation (k = 1, 2,.. N), cp is
the phase difference between the kth and (k — 1)th oscillations.
Find the amplitude of the resultant oscillation.
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