Question

an infrared wave traveling through a vacuum has a frequency of 4.0 × 10¹⁴ hz. what is the waves wavelength?
7.5 × 10⁷ m
7.5 × 10⁻⁷ m
1.2 × 10¹⁴ m
1.2 × 10⁻¹⁴ m

Explanation:

Step1: Recall the wave - speed formula

The speed of light in a vacuum \(c = 3\times10^{8}\text{ m/s}\), and the formula relating wave speed \(c\), frequency \(f\), and wavelength \(\lambda\) is \(c=\lambda f\). We need to solve for \(\lambda\), so \(\lambda=\frac{c}{f}\).

Step2: Substitute the given values

Given \(c = 3\times10^{8}\text{ m/s}\) and \(f=4.0\times 10^{14}\text{ Hz}\). Then \(\lambda=\frac{3\times 10^{8}}{4.0\times 10^{14}}\).
Using the rule of exponents \(\frac{a^{m}}{a^{n}}=a^{m - n}\), we have \(\lambda=\frac{3}{4}\times10^{8-14}\).
\(\frac{3}{4}=0.75 = 7.5\times10^{-1}\), so \(\lambda=7.5\times10^{-1}\times10^{-6}\).

Step3: Simplify using exponent rules

Using \(a^{m}\times a^{n}=a^{m + n}\), we get \(\lambda=7.5\times10^{-7}\text{ m}\).

Answer:

B. \(7.5\times10^{-7}\text{ m}\)