Answer :
Final answer:
The question involves calculating the number of allowed modes in a wavelength range for an enclosure, a problem related to physics in the field of spectroscopy and wave behavior. Wave numbers are used to represent the spatial frequency of the modes and are crucial in the calculation, but additional details are needed for a complete answer.
Explanation:
To calculate the number of allowed modes per unit volume in the wavelength range between 100 nm and 100.2 nm in a 100 cm³ enclosure, we can use the concept of spatial frequency, often expressed in terms of wave numbers. The wave number (k) is the reciprocal of the wavelength (λ) expressed in centimeters.
Thus the wave number range can be found by taking the inverse of the wavelengths (after converting them to centimeters) and finding the difference.
For a range of 100 nm (1 × 10⁻⁷ cm) to 100.2 nm (1.002 × 10⁻⁷ cm), the wave numbers are:
klow = 1 / (1 × 10⁻⁷ cm) = 107 cm⁻¹
khigh = 1 / (1.002 × 10⁻⁷ cm) = 9.98 × 106 cm⁻¹
The difference in wave numbers (Δk) is then the number of modes per centimeter. To find the number of modes per unit volume, Δk must be multiplied by the volume of the enclosure in cm³. However, the full calculation for the number of allowed modes would also need to take into account the photon polarization states and the density of states function.
Without the explicit equation for the density of states or the number of polarizations assumed, we cannot complete the calculation. In practice, for a full solution, additional information is required, which would typically be given as part of an in-depth physics problem at the college level.