An infrared wave traveling through a vacuum has a frequency of $4.0 \times 10^{14} Hz$. What is the wave's wavelength?
A. $7.5 \times 10^7 m$
B. $7.5 \times 10^{-7} m$
C. $1.2 \times 10^{14} m$
D. $1.2 \times 10^{-14} m
Answer (2)
The problem provides the frequency of an infrared wave and asks for its wavelength.
Recall the formula relating speed of light, wavelength, and frequency: c = λ f .
Rearrange the formula to solve for wavelength: λ = f c .
Substitute the given values and calculate the wavelength: λ = 4.0 × 1 0 14 3.0 × 1 0 8 = 7.5 × 1 0 − 7 m . The final answer is 7.5 × 1 0 − 7 m .
Explanation
Understanding the Problem We are given the frequency of an infrared wave and asked to find its wavelength. We know that the speed of light in a vacuum is a constant, approximately 3.0 × 1 0 8 meters per second. The relationship between the speed of light ( c ), wavelength ( λ ), and frequency ( f ) is given by the formula c = λ f .
Rearranging the Formula To find the wavelength, we need to rearrange the formula to solve for λ . Dividing both sides of the equation by f , we get: λ = f c
Substituting the Values Now, we substitute the given values into the formula. The speed of light c = 3.0 × 1 0 8 m/s, and the frequency f = 4.0 × 1 0 14 Hz. Therefore, λ = 4.0 × 1 0 14 Hz 3.0 × 1 0 8 m/s Dividing 3.0 by 4.0 gives 0.75 , and dividing 1 0 8 by 1 0 14 gives 1 0 − 6 . So, we have λ = 0.75 × 1 0 − 6 m We can also write this as λ = 7.5 × 1 0 − 7 m
Final Answer Therefore, the wavelength of the infrared wave is 7.5 × 1 0 − 7 meters.
Examples
Understanding the wavelength of electromagnetic waves, like infrared waves, is crucial in various applications. For example, in telecommunications, different wavelengths are used for transmitting data wirelessly. In medicine, infrared radiation is used in thermal imaging to detect temperature variations, which can help diagnose certain conditions. Knowing the relationship between frequency and wavelength allows engineers and scientists to design and optimize these technologies.
The wavelength of the infrared wave with a frequency of 4.0 × 1 0 14 Hz is calculated to be 7.5 × 1 0 − 7 m . Therefore, the correct answer is B. The calculation uses the formula λ = f c with the known speed of light.
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