Calculate the frequency of each of the following wavelengths of electromagnetic radiation.
Part A 632.8 nm (wavelength of red light from a helium-neon laser)
Part B 12.24 cm (wavelength of a microwave oven)
Express your answer using three significant figures.
Part C 337.1 nm (wavelength of a nitrogen laser)
Express your answer using three significant figure
The correct answer and explanation is :
To calculate the frequency of each wavelength of electromagnetic radiation, we can use the formula that relates the speed of light ((c)), wavelength ((\lambda)), and frequency ((f)):
[
c = \lambda \times f
]
Where:
- (c) is the speed of light, approximately (3.00 \times 10^8 \, \text{m/s}),
- (\lambda) is the wavelength (in meters), and
- (f) is the frequency (in Hz).
We can rearrange this equation to solve for frequency ((f)):
[
f = \frac{c}{\lambda}
]
We will first need to convert the given wavelengths to meters and then use the formula to calculate the frequencies.
Part A: 632.8 nm (wavelength of red light from a helium-neon laser)
- Convert the wavelength to meters:
[
632.8 \, \text{nm} = 632.8 \times 10^{-9} \, \text{m} = 6.328 \times 10^{-7} \, \text{m}
] - Now calculate the frequency:
[
f = \frac{3.00 \times 10^8 \, \text{m/s}}{6.328 \times 10^{-7} \, \text{m}} = 4.74 \times 10^{14} \, \text{Hz}
]
Thus, the frequency of the red light is approximately (4.74 \times 10^{14} \, \text{Hz}).
Part B: 12.24 cm (wavelength of a microwave oven)
- Convert the wavelength to meters:
[
12.24 \, \text{cm} = 12.24 \times 10^{-2} \, \text{m} = 0.1224 \, \text{m}
] - Now calculate the frequency:
[
f = \frac{3.00 \times 10^8 \, \text{m/s}}{0.1224 \, \text{m}} = 2.45 \times 10^9 \, \text{Hz}
]
Thus, the frequency of the microwave radiation is approximately (2.45 \times 10^9 \, \text{Hz}).
Part C: 337.1 nm (wavelength of a nitrogen laser)
- Convert the wavelength to meters:
[
337.1 \, \text{nm} = 337.1 \times 10^{-9} \, \text{m} = 3.371 \times 10^{-7} \, \text{m}
] - Now calculate the frequency:
[
f = \frac{3.00 \times 10^8 \, \text{m/s}}{3.371 \times 10^{-7} \, \text{m}} = 8.90 \times 10^{14} \, \text{Hz}
]
Thus, the frequency of the nitrogen laser light is approximately (8.90 \times 10^{14} \, \text{Hz}).
Summary:
- Part A: The frequency of the red light from a helium-neon laser (632.8 nm) is (4.74 \times 10^{14} \, \text{Hz}).
- Part B: The frequency of the microwave radiation (12.24 cm) is (2.45 \times 10^9 \, \text{Hz}).
- Part C: The frequency of the nitrogen laser light (337.1 nm) is (8.90 \times 10^{14} \, \text{Hz}).
Explanation:
The frequency of electromagnetic radiation is inversely proportional to its wavelength. This means that shorter wavelengths correspond to higher frequencies, and longer wavelengths correspond to lower frequencies. In each part of this problem, we used the relationship between speed, wavelength, and frequency to calculate the frequency for each given wavelength. Since the speed of light is constant in a vacuum, once the wavelength is known, the frequency can be directly calculated using the formula.