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Difficulty: Very hardElectromagnetic Waves and Electromagnetic Spectrum

An electromagnetic wave propagating in a vacuum has a frequency of 6.0×1014 Hz6.0 \times 10^{14}\text{ Hz}. It passes from the vacuum into a dense glass block with a refractive index of 1.501.50. Given that the speed of light in vacuum is c=3.0×108 m/sc = 3.0 \times 10^8\text{ m/s}, what is the wavelength of the wave inside the glass block, and to which region of the electromagnetic spectrum does the wave belong based on its vacuum properties?

  1. 3.33×10−7 m3.33 \times 10^{-7}\text{ m}, Visible lightAnswer
  2. B
    5.00×10−7 m5.00 \times 10^{-7}\text{ m}, Visible light
  3. C
    7.50×10−7 m7.50 \times 10^{-7}\text{ m}, Infrared
  4. D
    2.25×10−7 m2.25 \times 10^{-7}\text{ m}, Ultraviolet

Answer

The wavelength of the wave inside the glass block is 3.33×10−7 m3.33 \times 10^{-7}\text{ m}, and the wave belongs to the Visible light region.
The vacuum wavelength of the wave is λ0=c/f=5.0×10−7 m\lambda_0 = c/f = 5.0 \times 10^{-7}\text{ m}, placing it in the visible light spectrum. Upon entering the glass medium (n=1.50n = 1.50), the wave frequency remains unchanged while its wavelength is compressed by the factor nn, giving λ=(5.0×10−7)/1.50=3.33×10−7 m\lambda = (5.0 \times 10^{-7})/1.50 = 3.33 \times 10^{-7}\text{ m}.

Step-by-Step Solution

1
Calculate the vacuum wavelength of the electromagnetic wave
λ0=cf=3.0×108 m/s6.0×1014 Hz=5.0×10−7 m\lambda_0 = \frac{c}{f} = \frac{3.0 \times 10^8\text{ m/s}}{6.0 \times 10^{14}\text{ Hz}} = 5.0 \times 10^{-7}\text{ m}
The wave equation in vacuum relates wave speed, frequency, and wavelength by c=fλ0c = f \lambda_0.
2
Classify the electromagnetic spectral region
Visible light region
A vacuum wavelength of 5.0×10−7 m5.0 \times 10^{-7}\text{ m} (500 nm500\text{ nm}) falls within the visible light band (400 nm−700 nm400\text{ nm} - 700\text{ nm}).
3
Calculate the wavelength inside the glass block
λ=λ0n=5.0×10−7 m1.50=3.33×10−7 m\lambda = \frac{\lambda_0}{n} = \frac{5.0 \times 10^{-7}\text{ m}}{1.50} = 3.33 \times 10^{-7}\text{ m}
When passing into a medium with refractive index nn, frequency remains constant while the wave speed and wavelength are reduced by a factor of nn.

Key Concept

Electromagnetic Wave Refraction and Spectrum Classification
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Topics

Electromagnetic Waves and Electromagnetic Spectrum

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