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Ultraviolet light is typically divided into three categories. UV-A, with wavelengths between 400 nm and 320 nm, has been linked with malignant melanomas. UV-B radiation, which is the primary cause of sunburn and other skin cancers, has wavelengths between 320 nm and 280 nm. Finally, the region known as UV-C extends to wavelengths of 100 nm. (a) Find the range of frequencies for UV-B radiation. (b) In which of these three categories does radiation with a frequency of 7.9 * 1014 Hz belong

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Answer:

a) The UV-B has frequencies between [tex]9.375x10^{14}Hz[/tex] and [tex]1.071x10^{15}Hz[/tex]

b) The radiation with a frequency of [tex]7.9x10^{14}Hz[/tex] belong to the UV-A category.

Explanation:

(a) Find the range of frequencies for UV-B radiation.

Ultraviolet light belongs to the electromagnetic spectrum, which distributes radiation along it in order of different frequencies or wavelengths.

Higher frequencies:  

  • Gamma ray
  • X ray
  • Ultraviolet rays
  • Visible region

Lower frequencies:

  • Infrared
  • Microwave
  • Radio waves

That radiation is formed by electromagnetic waves, which are transverse waves formed by an electric field and a magnetic field perpendicular to it. Any of those radiations will have a speed of [tex]3x10^{8]m/s[/tex] in vacuum.

The velocity of a wave can be determined by means of the following equation:

[tex]c = \nu \cdot \lambda[/tex] (1)  

Where c is the speed of light, [tex]\nu[/tex] is the frequency and [tex]\lambda[/tex] is the wavelength.

Then, from equation 1 the frequency can be isolated.

[tex]\nu = \frac{c}{\lambda}[/tex] (2)

Before using equation 2 to determine the range of UV-B it is necessary to express [tex]\lambda[/tex] in units of meters in order to match with the units from c.

   

[tex]\lambda = 320nm . \frac{1m}{1x10^{9}nm}[/tex] ⇒ [tex]3.2x10^{-7}m[/tex]

[tex]\lambda = 280nm . \frac{1m}{1x10^{9}nm}[/tex] ⇒ [tex]2.8x10^{-7}m[/tex]

[tex]\nu = \frac{3x10^{8}m/s}{3.2x10^{-7}m}[/tex]

[tex]\nu = 9.375x10^{14}s^{-1}[/tex]

[tex]\nu = 9.375x10^{14}Hz[/tex]

[tex]\nu = \frac{3x10^{8}m/s}{2.8x10^{-7}m}[/tex]

[tex]\nu = 1.071x10^{15}Hz[/tex]

Hence, the UV-B has frequencies between [tex]9.375x10^{14}Hz[/tex] and [tex]1.071x10^{15}Hz[/tex]

(b) In which of these three categories does radiation with a frequency of [tex]7.9x10^{14}Hz[/tex] belong.

The same approach followed in part A will be used to answer part B.

Case for UV-A:

[tex]\lambda = 400nm . \frac{1m}{1x10^{9}nm}[/tex] ⇒ [tex]4x10^{-7}m[/tex]

[tex]\nu = \frac{3x10^{8}m/s}{4x10^{-7}m}[/tex]

[tex]\nu = 7.5x10^{14}s^{-1}[/tex]

[tex]\nu = 7.5x10^{14}Hz[/tex]

Hence, the UV-A has frequencies between [tex]7.5x10^{14}Hz[/tex] and [tex]9.375x10^{14}Hz[/tex].

Therefore, the radiation with a frequency of [tex]7.9x10^{14}Hz[/tex] belongs to UV-A category.

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