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 ( [tex]$nm$[/tex] ). 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 [tex]$532 ~ nm$[/tex]. 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 the calculator.
A. [tex]$1.8 \times 10^{14} Hz$[/tex]
B. [tex]$2.8 \times 10^{17} Hz$[/tex]
C. [tex]$5.6 \times 10^{14} Hz$[/tex]
D. [tex]$1.11 \times 10^{16} Hz$[/tex]
Answer (1)
Convert the wavelength from nanometers to meters: λ m e t ers = 5.32 × 1 0 − 7 m .
Calculate the frequency of the green laser light: ν g ree n = 5.639 × 1 0 14 Hz .
Divide the frequency of the green laser light by 2 to find the original frequency: ν b e f ore = 2.8195 × 1 0 14 Hz .
The output light frequency of the material used before doubling is 2.8 × 1 0 14 Hz .
Explanation
Understanding the Problem We are given that the wavelength of green laser light is 532 nm. We need to find the frequency of the light before it is doubled to produce this green light. The relationship between frequency ( ν ) and wavelength ( λ ) is given by the equation: ν = λ c where c is the speed of light, which is approximately 300 , 000 , 000 meters per second. We also know that 1 meter is equal to 1 , 000 , 000 , 000 nanometers.
Converting Wavelength to Meters First, we need to convert the wavelength from nanometers to meters. To do this, we divide the wavelength in nanometers by 1 , 000 , 000 , 000 : λ m e t ers = 1 , 000 , 000 , 000 nm / m 532 nm = 5.32 × 1 0 − 7 m
Calculating Frequency of Green Light Next, we calculate the frequency of the green laser light using the formula ν = λ c : ν g ree n = 5.32 × 1 0 − 7 m 300 , 000 , 000 m / s = 5.639 × 1 0 14 Hz
Finding Frequency Before Doubling Since the frequency of the emitted light is doubled to obtain the green laser light, the frequency of the light before doubling is half the frequency of the green laser light: ν b e f ore = 2 ν g ree n = 2 5.639 × 1 0 14 Hz = 2.8195 × 1 0 14 Hz
Final Answer Rounding to one decimal place, the frequency of the light before doubling is approximately 2.8 × 1 0 14 Hz .
Examples
Lasers are used in many applications, such as barcode scanners, laser printers, and medical equipment. Understanding the relationship between the wavelength and frequency of light is important in designing and using these devices. For example, in laser printers, the wavelength of the laser light determines the resolution of the printed image. In medical equipment, lasers are used for various procedures, such as eye surgery and tattoo removal. The frequency and intensity of the laser light must be carefully controlled to avoid damaging the tissue.
