Chapter 3, Problem 14Q

Use Figure 3.4 to specify the region of the electromagnetic spectrum where radiation of each of the following wavelengths is found.

Hint: Change each wavelength to meters before making the comparison.

1. a. 2.0 cm
2. b. 50 um
3. c. 400 nm
4. d. 150 mm

Figure 3.4

The electromagnetic spectrum, highlighting the visible region. Note that 10^x is equivalent to 1 × 10^x.

Interpretation Introduction

Interpretation:

The region of the electromagnetic spectrum for the given wavelength has to be specified.

Concept-Introduction:

Wavelength: The distance between successive peaks in a wave and measured in units of length.

Frequency: The number of waves passed a point in a certain amount of time.

Relation between wavelength and frequency:

$\text{Frequency}\left(\text{ν}\right)\text{=}\frac{\text{speed}\text{of}\text{light}\left(\text{c}\right)}{\text{wavelength}\left(\text{λ}\right)}$

Where, c is constant, the value of c is $\left(\text{3}{\text{.00×10}}^{\text{8}}\text{m/s}\right)$ and represents the maximum velocity that light is able to travel through air.

Conversion of centimeter into meter (cm to m): $\text{1 cm}\text{=}{\text{1×10}}^{\text{-2}}\text{m}$

Explanation of Solution

The wavelength is $\text{2}\text{.0 cm}$

Conversion of centimeter to meter is,

$\begin{array}{l}\text{1 cm}\text{=}{\text{1×10}}^{\text{-2}}\text{m}\\ \\ \text{2}\text{.0 cm}\text{=}?\\ \\ =\frac{\left({\text{1×10}}^{\text{-2}}\text{m}\right)\left(2.0\text{cm}\right)}{\text{1 cm}}=2\times 10^{-2}m\end{array}$

Therefore, the given value of wavelength in meter is $2\times 10^{-2}m$ which fall in the region of Microwave region of the spectrum.

Interpretation Introduction

Interpretation:

The region of the electromagnetic spectrum for the given wavelength has to be specified.

Concept-Introduction:

Wavelength: The distance between successive peaks in a wave and measured in units of length.

Frequency: The number of waves passed a point in a certain amount of time.

Relation between wavelength and frequency:

$\text{Frequency}\left(\text{ν}\right)\text{=}\frac{\text{speed}\text{of}\text{light}\left(\text{c}\right)}{\text{wavelength}\left(\text{λ}\right)}$

Where, c is constant, the value of c is $\left(\text{3}{\text{.00×10}}^{\text{8}}\text{m/s}\right)$ and represents the maximum velocity that light is able to travel through air.

Conversion of micrometer into meter ($\mu m$ to m): $\text{1 μm}\text{=}{\text{1×10}}^{\text{-6}}\text{m}$

Explanation of Solution

The wavelength is $\text{50}\mu \text{m}$

Conversion of micrometer to meter is,

$\begin{array}{l}\text{1 μm}\text{=}{\text{1×10}}^{\text{-6}}\text{m}\\ \\ \text{50 }\mu \text{m}\text{=}?\\ \\ =\frac{\left({\text{1×10}}^{\text{-6}}\text{m}\right)\left(50\mu m\right)}{\text{1 }\mu \text{m}}=50\times 10^{-6}m\end{array}$

Therefore, the given value of wavelength in meter is $50\times 10^{-6}m$ which fall in the region of visible region of the spectrum.

Interpretation Introduction

Interpretation:

The region of the electromagnetic spectrum for the given wavelength has to be specified.

Concept-Introduction:

Wavelength: The distance between successive peaks in a wave and measured in units of length.

Frequency: The number of waves passed a point in a certain amount of time.

Relation between wavelength and frequency:

$\text{Frequency}\left(\text{ν}\right)\text{=}\frac{\text{speed}\text{of}\text{light}\left(\text{c}\right)}{\text{wavelength}\left(\text{λ}\right)}$

Where, c is constant, the value of c is $\left(\text{3}{\text{.00×10}}^{\text{8}}\text{m/s}\right)$ and represents the maximum velocity that light is able to travel through air.

Conversion of nanometer into meter (nm to m): $\text{1 nm}\text{=}{\text{1×10}}^{\text{-9}}\text{m}$

Explanation of Solution

The given wavelength is $\text{400}\text{nm}$

Conversion of nanometer to meter is,

$\begin{array}{l}\text{1 nm}\text{=}{\text{1×10}}^{\text{-9}}\text{m}\\ \\ \text{400}\text{nm}\text{=}?\\ \\ =\frac{\left({\text{1×10}}^{\text{-9}}\text{m}\right)\left(400nm\right)}{\text{1 nm}}=40\times 10^{-8}m\end{array}$

Therefore, the given value of wavelength in meter is $40\times 10^{-8}m$ which fall in the region of violet region of visiblelight of the spectrum. Because visible region begins from the range $10^{-6}-10^{-8}m$, so the obtained range falls at the violet color.

Interpretation Introduction

Interpretation:

The region of the electromagnetic spectrum for the given wavelength has to be specified.

Concept-Introduction:

Wavelength: The distance between successive peaks in a wave and measured in units of length.

Frequency: The number of waves passed a point in a certain amount of time.

Relation between wavelength and frequency:

$\text{Frequency}\left(\text{ν}\right)\text{=}\frac{\text{speed}\text{of}\text{light}\left(\text{c}\right)}{\text{wavelength}\left(\text{λ}\right)}$

Where, c is constant, the value of c is $\left(\text{3}{\text{.00×10}}^{\text{8}}\text{m/s}\right)$ and represents the maximum velocity that light is able to travel through air.

Conversion of millimeter into nanometer (mm to m): $\text{1 mm}\text{=}{\text{1×10}}^{\text{-3}}\text{m}$

Explanation of Solution

The given wavelength is $\text{150}\text{mm}$

Conversion of millimeter to meter is,

$\begin{array}{l}\text{1 mm}\text{=}{\text{1×10}}^{\text{-3}}\text{m}\\ \\ \text{150}\text{mm}\text{=}\text{?}\\ \\ \text{=}\frac{\left({\text{1×10}}^{\text{-3}}\text{m}\right)\left(\text{150 mm}\right)}{\text{1 mm}}\text{=}15\times 10^{-2}m\end{array}$

Therefore, the given value of wavelength in meter is $15\times 10^{-2}m$ which fall in the region of microwave region of the spectrum.