Question

A 9 × 1014 Hz laser emits a 8.8 μs pulse, 5.0 mm in diameter, with a beam energy density of 0.8 J/m3. The number of wavelengths in the length of the beam is closest to?

Answer 1

When a 9 × 1014 Hz laser emits a 8.8 μs pulse, 5.0 mm in diameter, the number of wavelengths is approximately 47,094.

Step-by-step explanation:

To calculate the number of wavelengths in the length of the beam, we need to determine the length of the beam first.

The diameter of the beam is given as 5.0 mm, so the radius would be half of that, or 2.5 mm (0.0025 m).

Since the beam is circular, we can find the length of the beam using the formula for the circumference of a circle:

C = 2πr = 2π(0.0025) = 0.0157 m

Next, we need to calculate the wavelength of the laser beam. The speed of light is approximately 3.00 x 10⁸ m/s, so we can use the formula:

λ = c/f = (3.00 x 10⁸)/(9 x 10¹⁴) ≈ 3.33 x 10⁻⁷ m

Now we can calculate the number of wavelengths in the length of the beam:

Number of wavelengths = Length of beam / Wavelength = 0.0157 / 3.33 x 10⁻⁷ ≈ 47,093.98

Therefore, the number of wavelengths in the length of the beam is closest to 47,094.

Answer 2

To solve this problem we will apply the concepts related to Frequency (Reverse to Period) and the description of the wavelength as a function of the speed of light at the rate of frequency.

Our Laser frequency is given as

`f = 9*10^(14)Hz`

Therefore the laser wavelength would be

`\lambda = (c)/(f)`

Where,

c = Speed of light

f = Frequency

`\lambda = (3*10^8)/( 9*10^(14))`

`\lambda = 3.33*10^(-7)`

The laser pulse is emitted at a period (T) of
`8.8*10^(-6)s`

Therefore the pulse wavelength would be

`\lambda' = (c)/(f)`

`\lambda' = c (1)/(f) \rightarrow (1)/(f) = T`

`\lambda' = c *T`

`\lambda' = (3*10^8)(8.8*10^(-6))`

`\lambda' = 2640m`

Finally the number of wavelengths is the ratio between the two wavelengths, then

`n = (\lambda')/(\lambda)`

`n = (2640)/(3.33*10^(-7) )`

`n = 7.927*10^9`

The number of wavelengths in the beam length is closer to
`7.927*10^9`