Question

The wavelength of an EM wave is 800 nm. Calculate its frequency, angular frequency, wavenumber, wave vector amplitude, energy, and momentum.

Answer 1

The frequency of the EM wave is 3.75 × 10^14 Hz. The angular frequency is 7.5 × 10^14 rad/s. The wavenumber is 7.85 × 10^6 rad/m. The wave vector amplitude is also 7.85 × 10^6 rad/m. The energy of the photon is 2.48 × 10^-19 J and its momentum is 8.81 × 10^-27 kg·m/s.

Step-by-step explanation:

To calculate the frequency of an EM wave, we can use the formula c = fλ, where c is the speed of light and λ is the wavelength. Rearranging the formula, we get f = c/λ. Plugging in the values, we have:

f = (3 × 10^8 m/s) / (800 nm) = 3.75 × 10^14 Hz.

The angular frequency (ω) of an EM wave is calculated using the formula ω = 2πf. Substituting in the frequency we just calculated, we have:

ω = 2π(3.75 × 10^14 Hz) = 7.5 × 10^14 rad/s.

The wavenumber (k) of an EM wave is defined as the number of wavelengths per unit distance. It is calculated using the formula k = 2π/λ. Substituting in the wavelength, we have:

k = 2π / (800 nm) = 7.85 × 10^6 rad/m.

The wave vector amplitude (|k|) is the magnitude of the wave vector and is given by |k| = 2π/λ. Substituting in the wavelength, we have:

|k| = 2π / (800 nm) = 7.85 × 10^6 rad/m.

The energy (E) of a photon is given by the equation E = hf, where h is Planck's constant (6.63 × 10^-34 J·s) and f is the frequency. Substituting in the frequency, we have:

E = (6.63 × 10^-34 J·s) × (3.75 × 10^14 Hz) = 2.48 × 10^-19 J.

The momentum (p) of a photon is given by the equation p = hf/c, where c is the speed of light. Substituting in the frequency and the speed of light, we have:

p = (6.63 × 10^-34 J·s × 3.75 × 10^14 Hz) / (3 × 10^8 m/s) = 8.81 × 10^-27 kg·m/s.

Answer 2

Given: Wavelength (λ) = 800 nm

Frequency
`\\u=(c)/(\lambda) \Rightarrow \\u = (3*10^8)/(800* 10^(-9)) =3.75*10^(14) Hz`

Angular frequency
`\omega=2\pi \\u \Rightarrow \omega = 2\pi 3.75*10^(14) = 23.55*10^(14) rad/s`

Wavenumber,
`k=(2\pi)/(\lambda) \Rightarrow k =(2\pi)/(800* 10^(-9))=7.85* 10^6m^(-1)`

wave vector amplitude |k| = k =
`7.85* 10^6m^(-1)`

energy,
`E = h \\u \Rightarrow E = 6.63* 10^(-34)* 3.75*10^(14) = 24.86* 10^(-20)J`

momentum,
`p=(h)/(\lambda) = (6.63* 10^(-34))/(800* 10^(-9))= 8.28* 10^(-28)Ns`