- Identify the wave equation \(\lambda = \frac{c}{f}\) connecting wavelength, speed of light, and frequency.
- Substitute the given values \(\lambda = \frac{3.0 \times 10^8}{8.8 \times 10^{17}}\).
- Divide the coefficients and the powers of 10 to obtain \(\lambda \approx 0.3409 \times 10^{-9} \text{ m}\).
- Convert to standard scientific notation to arrive at \(\lambda \approx \boxed{3.4 \times 10^{-10} \text{ m}}\).
Explanation
- Problem Analysis and Variables
We are given the frequency of ultraviolet (UV) light as \(f = 8.8 \times 10^{17} \text{ Hz}\) and the speed of light as \(c = 3.0 \times 10^8 \text{ m/s}\). The relationship between the speed of light (\(c\)), its wavelength (\(\lambda\)), and its frequency (\(f\)) is given by the wave equation:
\[c = \lambda \cdot f\]
Our objective is to determine the wavelength \(\lambda\) and express it in standard scientific notation.
- Setting up the Calculation
To find the wavelength \(\lambda\), we rearrange the wave equation to solve for \(\lambda\):
\[\lambda = \frac{c}{f}\]
Substituting the given values into the equation:
\[\lambda = \frac{3.0 \times 10^8 \text{ m/s}}{8.8 \times 10^{17} \text{ Hz}}\]
- Performing the Calculation and Formatting
First, we divide the coefficients:
\[\frac{3.0}{8.8} \approx 0.340909\]
Next, we divide the powers of 10 using the laws of exponents:
\[\frac{10^8}{10^{17}} = 10^{8-17} = 10^{-9}\]
Combining these parts, we get:
\[\lambda \approx 0.340909 \times 10^{-9} \text{ m}\]
To express this in standard scientific notation (where the coefficient is between 1 and 10), we shift the decimal point one place to the right, which adjusts the exponent by decreasing it by 1:
\[0.340909 \times 10^{-9} = 3.40909 \times 10^{-1} \times 10^{-9} = 3.40909 \times 10^{-10} \text{ m}\]
Rounding to two significant figures, as provided in the initial data, we have \(\lambda \approx 3.4 \times 10^{-10} \text{ m}\).
Examples
Understanding the wavelength of light is fundamental to fields like medical imaging and telecommunications. For instance, UV light is used in sterilization processes because its short wavelength allows it to interact effectively with the DNA of microorganisms, neutralizing them. Knowing the relationship between frequency and wavelength helps scientists and engineers design tools that operate at specific frequencies for safer and more efficient applications in medicine and technology.