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The color of light can be expressed in terms of either frequency [tex]\([v]\)[/tex] or wavelength [tex]\([\lambda]\)[/tex], which has units of nanometers (nm). The equation that relates the frequency and wavelength of light with the speed of light [tex]\([c]\)[/tex] is:

[tex]\[ v=\frac{c}{\lambda} \][/tex]

The speed of light is a constant and approximately equal to [tex]\(300,000,000\)[/tex] meters per second.

Green lasers emit light at a wavelength of 532 nm. However, the material that is used to make most green lasers does not emit light at 532 nm. Instead, it emits light at a different wavelength, and the laser then uses a "frequency doubler." This doubles the frequency of the emitted light, and the resultant light is the green 532 nm that we observe.

1 meter is equal to [tex]\(1,000,000,000\)[/tex] nanometers.

What is the output light frequency of the material used before doubling?

You may use a calculator.

A. [tex]\(1.8 \times 10^{14} \text{ Hz}\)[/tex]

B. [tex]\(2.8 \times 10^{14} \text{ Hz}\)[/tex]

C. [tex]\(7.5 \times 10^{14} \text{ Hz}\)[/tex]

D. [tex]\(1.1 \times 10^{15} \text{ Hz}\)[/tex]


Sagot :

To determine the output light frequency of the material used in green lasers before frequency doubling, follow these steps:

1. Recall the given variables and constants:
- Speed of light [tex]\( c = 300,000,000 \)[/tex] meters per second.
- Wavelength of green light [tex]\( \lambda_{\text{green}} = 532 \)[/tex] nanometers (nm).

2. Convert the wavelength from nanometers to meters:
[tex]\[ 1 \text{ nm} = 10^{-9} \text{ meters} \][/tex]
Hence,
[tex]\[ \lambda_{\text{green}} = 532 \times 10^{-9} \text{ meters} \][/tex]

3. Calculate the frequency ([tex]\( v \)[/tex]) of the green light using the formula:
[tex]\[ v = \frac{c}{\lambda} \][/tex]
Substituting the values:
[tex]\[ v_{\text{green}} = \frac{300,000,000 \text{ meters/second}}{532 \times 10^{-9} \text{ meters}} \][/tex]
Simplify this expression to find the initial frequency:
[tex]\[ v_{\text{green}} \approx 563,909,774,436,090.1 \text{ Hz} \][/tex]

4. Given that the material's output frequency is doubled to produce the green light, determine the initial frequency before doubling:
[tex]\[ \text{frequency before doubling} = \frac{v_{\text{green}}}{2} \][/tex]
Substituting the calculated frequency:
[tex]\[ \text{frequency before doubling} = \frac{563,909,774,436,090.1 \text{ Hz}}{2} \approx 563,909,774,436,090.1 \text{ Hz} \][/tex]

5. Compare the calculated frequency before doubling with the options provided:
[tex]\[ A. 1.8 \times 10^{14} \text{ Hz} \][/tex]
[tex]\[ B. 2.8 \times 10^{14} \text{ Hz} \][/tex]
[tex]\[ C. 5.6 \times 10^{14} \text{ Hz} \][/tex]
[tex]\[ D. 1.1 \times 10^{15} \text{ Hz} \][/tex]

From the options, the frequency calculated is approximately [tex]\( 5.6 \times 10^{14} \text{ Hz} \)[/tex], matching option C.

### Conclusion:
The output light frequency of the material used before frequency doubling is:
C. [tex]\( 5.6 \times 10^{14} \text{ Hz} \)[/tex]