Final answer:
To find the angular wave number k for a given angular frequency, one uses the formula k = ω/c. For the provided angular frequency of 6.67×10¹⁵ rad/s, the wave number is 2.223×10⁷ rad/m. For a photon with a given wave number, the frequency and wavelength can be calculated using the relationship between frequency, wavelength, and the speed of light.
Step-by-step explanation:
To calculate the angular wave number k of an electromagnetic wave with an angular frequency of 6.67×10¹⁵ rad/s, we use the relationship between wave number (k), angular frequency (ω), and the speed of light (c), where c = 3×10⁸ m/s.
The formula is k = ω/c, where ω is the angular frequency. Substituting the given angular frequency, we get k = 6.67×10¹⁵ rad/s / 3×10⁸ m/s, which simplifies to k = 2.223×10⁷ rad/m.
For a photon with a wave number of k=9π×10⁶ rad/m, the angular frequency (ω) can be found using the same relationship, ω = kc. So, ω = 9π×10⁶ rad/m × 3×10⁸ m/s, which gives us ω = 8.50×10⁴⁵ rad/s.
The frequency (f) of a photon is f = ω/2π. Using the calculated angular frequency, f = 8.50×10⁴⁵ rad/s / 2π, resulting in f = 1.35×10⁴⁵ Hz. As for the wavelength (λ), we use c = λf, which gives us λ = c/f or λ = 3×10⁸ m/s / 1.35×10⁴⁵ Hz, resulting in λ ≈ 2.22×10⁻⁷ m.