Superposition：練習問題

A diffraction grating has 400 lines per mm. A beam of monochromatic light is incident normally on the grating. The third-order maximum is observed at an angle of \( 36.0^{\circ} \) to the normal.
What is the wavelength of the light?

A stationary wave is formed on a stretched string fixed at both ends. The string vibrates in its third harmonic with a frequency of \( 450 \text{ Hz} \).
What is the frequency of the fifth harmonic for this string?

A diffraction grating is used to measure the wavelength of monochromatic light. The first-order maximum occurs at an angle \( \theta \).
The grating is replaced by a different grating with double the number of lines per millimeter. The light source remains the same.
What is the new angle for the first-order maximum?

What is the primary condition for two sources of waves to be considered coherent?

A stationary wave is formed on a stretched string fixed at both ends. The string vibrates in its third harmonic with a frequency of \( 450 \text{ Hz} \).
What is the frequency of the fifth harmonic for this string?

In a ripple tank experiment, water waves with a wavelength of 3.0 cm are incident on a gap of width \( w \). Explain qualitatively how the degree of diffraction changes as the gap width \( w \) is increased from 1.0 cm to 10.0 cm.

A pipe closed at one end has a fundamental frequency of 250 Hz. If the air in the pipe is replaced by helium, where the speed of sound is 2.9 times greater than in air, calculate the new frequency of the first overtone (the second possible mode) in the helium-filled pipe.

A stationary wave is formed on a stretched string of length \( L \). The string vibrates in its second harmonic.
If the speed of the transverse waves on the string is \( v \), what is the frequency of this vibration in terms of \( v \) and \( L \)?

A laser beam of wavelength 633 nm is directed at a diffraction grating. The first-order maximum is produced at an angle of \( 18.5^\circ \).
(a) Calculate the distance between adjacent slits in the grating.
(b) Calculate the number of lines per centimeter on the grating.
(c) State and explain the effect on the position of the first-order maximum if a laser of wavelength 450 nm is used instead.

Light of wavelength \( \lambda \) is incident on a double slit with separation \( a \). The interference pattern is viewed on a screen at distance \( D \).
(a) Derive the expression \( \lambda = \frac{ax}{D} \) for the fringe width \( x \), stating any assumptions made.
(b) If the wavelength is 500 nm, \( a = 0.20 \text{ mm} \), and \( D = 1.8 \text{ m} \), calculate the value of \( x \).
(c) Explain why the central fringe is always bright regardless of the wavelength used.