An infrared sensor detects electromagnetic radiation with a frequency of 4.0 \times 10^{14… — JAMB UTME

An infrared sensor detects electromagnetic radiation with a frequency of $4.0 \times 10^{14}\text{ Hz}$ . A second sensor detects ultraviolet radiation with a wavelength of $300\text{ nm}$ . Given that the speed of light in a vacuum is $c = 3.0 \times 10^8\text{ m/s}$ , what is the ratio of the wavelength of the infrared radiation to the wavelength of the ultraviolet radiation?

$2.5$ Answer
B
$0.4$
C
$25.0$
D
$1.33$

Answer

The ratio of the wavelength of the infrared radiation to the wavelength of the ultraviolet radiation is 2.5.

Using

c = f \lambda

, the infrared wavelength is

\lambda = \frac{3.0 \times 10^8}{4.0 \times 10^{14}} = 7.5 \times 10^{-7}\text{ m}

. Comparing this to the ultraviolet wavelength of

300\text{ nm} = 3.0 \times 10^{-7}\text{ m}

gives a ratio of

\frac{7.5 \times 10^{-7}}{3.0 \times 10^{-7}} = 2.5

Step-by-Step Solution

Calculate the wavelength of the infrared radiation using the wave equation

c = f \lambda

\lambda_{\text{IR}} = \frac{3.0 \times 10^8\text{ m/s}}{4.0 \times 10^{14}\text{ Hz}} = 7.5 \times 10^{-7}\text{ m}

The speed of all electromagnetic waves in a vacuum is constant (

c = 3.0 \times 10^8\text{ m/s}

Convert the ultraviolet wavelength to meters for consistent units.

\lambda_{\text{UV}} = 300\text{ nm} = 300 \times 10^{-9}\text{ m} = 3.0 \times 10^{-7}\text{ m}

Units must be in standard meters before calculating the dimensionless ratio.

Compute the ratio of the infrared wavelength to the ultraviolet wavelength.

\text{Ratio} = \frac{7.5 \times 10^{-7}\text{ m}}{3.0 \times 10^{-7}\text{ m}} = 2.5

Dividing the two wavelengths in the same units yields the desired ratio.

Key Concept

Wave Equation and Electromagnetic Spectrum Properties

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