Show that the radius r of the mᵗʰ dark Newton’s ring, as viewed from directly above (Fig. 34–18), is given by r = √mλR where R is the radius of curvature of the curved glass surface and λ is the wavelength of light used. Assume that the thickness of the air gap is much less than R at all points and that r ≪ R . [Hint: Use the binomial expansion.]

A radio telescope, whose two antennas are separated by 55 m, is designed to receive 3.0-MHz radio waves produced by astronomical objects. The received radio waves create 3.0-MHz electronic signals in the telescope’s left and right antennas. These signals then travel by equal-length cables to a centrally located amplifier, where they are added together. The telescope can be “pointed” to a certain region of the sky by adding the instantaneous signal from the right antenna to a “time-delayed” signal received by the left antenna a time ∆t ago. (This time delay of the left signal can be easily accomplished with the proper electronic circuit.) If a radio astronomer wishes to “view” radio signals arriving from an object oriented at a 12° angle to the vertical as in Fig. 34–54, what time delay ∆t is necessary?

(II) White light passes through a 640-slit/ mm diffraction grating. First-order and second-order visible spectra (“rainbows”) appear on the wall 32 cm away as shown in Fig. 35–40. Determine the widths ℓ₁ and ℓ₂ of the two “rainbows” (400 nm to 700 nm). In which order is the “rainbow” dispersed over a larger distance?

(II) X-rays of wavelength 0.138 nm fall on a crystal whose atoms, lying in planes, are spaced 0.315 nm apart. At what angle Φ (relative to the surface, Fig. 35–28) must the X-rays be directed if the first diffraction maximum is to be observed?

(II) Suppose a thin piece of glass is placed in front of the lower slit in Fig. 34–7 so that the two waves enter the slits 180° out of phase (Fig. 34–44). Describe in detail the interference pattern on the screen.

(II) When a Newton’s ring apparatus (Fig. 34–18) is immersed in a liquid, the diameter of the tenth dark ring decreases from 2.92 cm to 2.54 cm. What is the refractive index of the liquid? [Hint: See Problem 37.]

Consider two antennas radiating 6.8-MHz radio waves in phase with each other. They are located at points S₁ and S₂, separated by a distance d = 175 m, Fig. 34–50. Determine the points on the positive y-axis where the signals from the two sources will be out of phase (crests of one meet troughs of the other).

"Two identical sources S₁ and S₂, separated by distance d, coherently emit light of wavelength λ uniformly in all directions. Defining the x axis with its origin at S₁ as shown in Fig. 34–52, find the locations (expressed as multiples of λ ) where the signals from the two sources are out of phase along this axis for x > 0 , if d = 3λ.

If you can read the bottom row of your doctor’s eye chart, your eye has a resolving power of 1 arcminute, equal to 1/60 degree. If this resolving power is diffraction-limited, to what effective diameter of your eye’s optical system does this correspond? Use Rayleigh’s criterion and assume λ = 550 nm.