Question

what is the frequency of an electromagnetic wave that has a wavelength of 8.5 × 10² m in a vacuum? (the speed of light in a vacuum is 3.00 × 10⁸ m/s.)

a. 2.8 × 10⁻⁶ hz
b. 1.2 × 10⁻³ hz
c. 2.6 × 10¹¹ hz
d. 3.5 × 10⁵ hz

Explanation:

Step1: Recall the formula for wave speed

The formula relating the speed (\(v\)), wavelength (\(\lambda\)), and frequency (\(f\)) of a wave is \(v = \lambda f\). We need to solve for frequency, so we rearrange the formula to \(f=\frac{v}{\lambda}\).

Step2: Substitute the given values

We know that the speed of light in a vacuum \(v = 3.00\times 10^{8}\space m/s\) and the wavelength \(\lambda=8.5\times 10^{2}\space m\). Substituting these values into the formula for frequency:
\(f=\frac{3.00\times 10^{8}}{8.5\times 10^{2}}\)

Step3: Simplify the expression

First, divide the coefficients: \(\frac{3.00}{8.5}\approx0.3529\). Then, use the rule of exponents for division \(a^{m}/a^{n}=a^{m - n}\) for the powers of 10: \(10^{8}/10^{2}=10^{8 - 2}=10^{6}\). Now multiply the two results: \(0.3529\times 10^{6}=3.529\times 10^{5}\approx3.5\times 10^{5}\space Hz\) (Wait, no, wait. Wait, I made a mistake here. Wait, \(3.00\times 10^{8}\div(8.5\times 10^{2})=\frac{3.00}{8.5}\times10^{8 - 2}\). \(\frac{3.00}{8.5}\approx0.3529\), and \(10^{6}\), so \(0.3529\times 10^{6}=3.529\times 10^{5}\)? Wait, no, that's not right. Wait, no, \(3.00\times 10^{8}\) divided by \(8.5\times 10^{2}\) is equal to \((3.00\div8.5)\times10^{8 - 2}\). \(3.00\div8.5\approx0.3529\), and \(10^{6}\), so \(0.3529\times 10^{6}=3.529\times 10^{5}\)? But wait, let's check again. Wait, maybe I messed up the exponent. Wait, \(8.5\times 10^{2}\) is 850, and \(3.00\times 10^{8}\) is 300000000. So 300000000 divided by 850. Let's compute that: 300000000 ÷ 850 ≈ 352941.18, which is approximately \(3.5\times 10^{5}\space Hz\). Wait, but let's check the options. Option D is \(3.5\times 10^{5}\space Hz\). Wait, but wait, maybe I made a mistake in the formula? Wait, no, the formula \(v = \lambda f\) is correct for electromagnetic waves in vacuum, where \(v = c\) (speed of light). So \(f=\frac{c}{\lambda}\). So \(c = 3\times 10^{8}\), \(\lambda=8.5\times 10^{2}\). So \(f=\frac{3\times 10^{8}}{8.5\times 10^{2}}=\frac{3}{8.5}\times 10^{6}\approx0.3529\times 10^{6}=3.529\times 10^{5}\approx3.5\times 10^{5}\space Hz\), which is option D. Wait, but let me check the calculation again. Wait, 3e8 divided by 8.5e2: 3e8 / 8.5e2 = (3 / 8.5) 10^(8 - 2) = 0.3529 10^6 = 352941.18 Hz, which is 3.529×10^5 Hz, which is approximately 3.5×10^5 Hz, so option D.

Answer:

D. \(3.5 \times 10^{5}\space Hz\)