To find the frequency of an electromagnetic wave given its wavelength, we can use the fundamental relationship between the speed of light, wavelength, and frequency. The formula that relates these three quantities is:
[tex]\[ f = \frac{c}{\lambda} \][/tex]
Where:
- [tex]\( f \)[/tex] is the frequency of the wave,
- [tex]\( c \)[/tex] is the speed of light in a vacuum, which is approximately [tex]\( 3.0 \times 10^8 \)[/tex] meters per second,
- [tex]\( \lambda \)[/tex] (lambda) is the wavelength of the wave.
Given in the problem:
- The wavelength [tex]\( \lambda \)[/tex] is [tex]\( 2.7 \times 10^{-10} \)[/tex] meters.
We need to find the frequency [tex]\( f \)[/tex].
Step-by-step solution:
1. Write down the known values:
- Speed of light, [tex]\( c = 3.0 \times 10^8 \)[/tex] m/s
- Wavelength, [tex]\( \lambda = 2.7 \times 10^{-10} \)[/tex] m
2. Substitute these values into the formula for frequency:
[tex]\[ f = \frac{c}{\lambda} = \frac{3.0 \times 10^8 \, \text{m/s}}{2.7 \times 10^{-10} \, \text{m}} \][/tex]
3. Perform the division to solve for [tex]\( f \)[/tex]:
[tex]\[ f = \frac{3.0 \times 10^8}{2.7 \times 10^{-10}} \][/tex]
This division can be tackled by dividing the numerators and subtracting the exponents of 10 in the denominator from the exponent in the numerator:
[tex]\[ f = \frac{3.0}{2.7} \times 10^{8 - (-10)} \][/tex]
[tex]\[ f = \frac{3.0}{2.7} \times 10^{8 + 10} \][/tex]
[tex]\[ f = \frac{3.0}{2.7} \times 10^{18} \][/tex]
4. Simplify the fraction:
[tex]\[ \frac{3.0}{2.7} \approx 1.111111111111111 \][/tex]
5. Combine the simplified fraction with the power of 10:
[tex]\[ f \approx 1.111111111111111 \times 10^{18} \, \text{Hz} \][/tex]
So, the frequency of the electromagnetic wave with a wavelength of [tex]\( 2.7 \times 10^{-10} \)[/tex] meters is approximately [tex]\( 1.1111111111111112 \times 10^{18} \)[/tex] Hz.
