- Consider the geometric series
- Suppose that
, where
is real. Employing the well-known
identity
- Finally, making use of Euler's theorem,

- Consider the geometric series
- Derive Equation (1022).
- An interference experiment employs two narrow parallel slits of separation
, and monochromatic light of wavelength
.
Estimate the minimum distance that the projection screen must be placed behind the slits in order to obtain a far-field interference
pattern.
- A double-slit of slit separation
is illuminated at normal incidence by a parallel beam from
a helium-neon laser that emits monochromatic light of wavelength
. A projection
screen is located
behind the slit. What is the separation of the central interference fringes on the screen? [From Crawford 1968.]
- Consider a double-slit interference experiment in which the slit spacing is
, and the
projection screen is located
behind the slits. Assuming monochromatic illumination at normal incidence, if the observed separation between neighboring interference
maxima at the center of the projection screen is
, what is the wavelength of the
light illuminating the slits?
- What is the mean length of the classical wavetrain (wave packet) corresponding to the light emitted by an
atom whose excited state has a mean lifetime
? In an ordinary gas-discharge source, the
excited atomic states do not decay freely, but instead have an effective lifetime
, due to
collisions and Doppler effects. What is the length of the corresponding classical wavetrain? [From Crawford 1968.]
- If a ``monochromatic'' incoherent ``line'' source of visible light is not really a line, but has a finite width of
, estimate the
minimum distance it can
be placed in front of a double-slit, of slit separation
, if the light from the slit is to generate a
clear interference pattern.
- The visible emission spectrum of a sodium atom is dominated by a yellow line which actually consists of two
closely spaced lines of wavelength
and
. Demonstrate that a diffraction
grating must have at least 328 lines in order to resolve this doublet at the third spectral order.
- Consider a diffraction grating having 5000 lines per centimeter. Find the angular
locations of the principle maxima when the grating is illuminated at normal incidence by (a) red light of
wavelength 700 nm, and (b) violet light of wavelength 400 nm.
- A soap bubble 250 nm thick is illuminated
by white light. The index of refraction of the soap film is
.
Which colors are not seen in the reflected light? Which colors
appear bright in the reflected light? What color does the soap film
appear at normal incidence?
- Suppose that a monochromatic laser of wavelength
emits a
diffraction-limited beam of initial diameter 2mm. Estimate how large a light spot
the beam would produce on the surface of the Moon (which is a mean distance
from the surface of the Earth). Neglect any effects of the Earth's atmosphere. [From Hecht and Zajac 1974.]
- Estimate how far away an automobile is when you can only just barely resolve the two headlights with your eyes. [From Crawford 1968.]
- Venus has a diameter of about 8000 miles. When it is prominently visible in the sky, in the early morning or late evening, it
is about as far away as the Sun: that is, about 93 million miles away. Venus commonly appears larger than a point to the
unaided eye. Are we seeing the true size of Venus? [From Crawford 1968.]
- The world's largest steerable radio telescope, at the National Radio Astronomy Observatory, Green Bank, West Virginia,
consists of a parabolic disk that is 300 ft in diameter. Estimate the angular resolution (in minutes of an arc) of the telescope when it is
observing the well-known 21-cm
radiation of hydrogen. [From Crawford 1968.]
- Estimate how large the lens of a camera carried by an artificial satellite orbiting the Earth at an altitude of 150 miles
would have to be in order to resolve features on the Earth's surface a foot in diameter.
- Demonstrate that the secondary maxima in the far-field interference pattern generated by three identical, equally spaced, parallel slits of
negligible width are nine times less intense than the principle maxima.
- Consider a double-slit interference/diffraction experiment in which the slit spacing is
, and the slit width
.
Show that the intensity of the far-field interference pattern, assuming normal incidence by monochromatic
light of wavelength
, is
- What is the theoretical maximum angular resolving power (in arc seconds) of a conventional reflecting telescope with a 12in main mirror?