What are the wavelengths of electromagnetic wave in free space that have the following frequencies?
(a) 3.00 10^19 Hz
pm
(b) 7.50 10^9 Hz
cm
To a wave, wavelength * frequency = velocity.
The speed of light in vacuum = 2.99792458e8 m/s ¡Ö 3*10^8 m/s.
(a) wavelength = velocity/frequency = 3*10^8/(3*10^19) = 10^-11 m.
(b) wavelength = 3*10^8/(7.5*10^9) = 0.04 m.
To find the wavelength of an electromagnetic wave in free space, you can use the relationship between wavelength and frequency. The equation that relates wavelength (λ), frequency (f), and the speed of light (c) is:
λ = c / f
Where:
λ = wavelength in meters (m)
f = frequency in hertz (Hz)
c = speed of light in a vacuum, approximately 3.00 × 10^8 m/s
(a) For a frequency of 3.00 × 10^19 Hz:
λ = c / f = (3.00 × 10^8 m/s) / (3.00 × 10^19 1/s)
To simplify the calculation, we can write 3.00 × 10^19 as (3.00 × 10^19) / (10^19) = 3.00 × 10^0 = 3.00
λ = (3.00 × 10^8 m/s) / (3.00 × 10^0 1/s) = 1.00 × 10^8 m
Therefore, the wavelength is 1.00 × 10^8 meters.
(b) For a frequency of 7.50 × 10^9 Hz:
λ = c / f = (3.00 × 10^8 m/s) / (7.50 × 10^9 1/s)
To simplify the calculation, we can write 7.50 × 10^9 as (7.50 × 10^9) / (10^9) = 7.50 × 10^0 = 7.50
λ = (3.00 × 10^8 m/s) / (7.50 × 10^0 1/s) = 4.00 × 10^-2 m
However, the question asks for the wavelength in centimeters (cm). To convert meters to centimeters, remember that there are 100 centimeters in 1 meter. Therefore:
λ = (4.00 × 10^-2 m) * (100 cm/m) = 4.00 cm
Therefore, the wavelength is 4.00 centimeters.
To find the wavelengths of the electromagnetic waves, we can use the formula:
λ = c / f
where λ represents the wavelength, c is the speed of light in a vacuum (approximately 3.00x10^8 m/s), and f is the frequency of the electromagnetic wave.
(a) For a frequency of 3.00x10^19 Hz:
λ = (3.00x10^8 m/s) / (3.00x10^19 Hz)
Calculating this expression, we get:
λ = 10^-11 m
To convert this to picometers (pm), we multiply by 10^12:
λ = 10^-11 m x 10^12 pm/m = 10^1 pm = 10 pm
Therefore, the wavelength of the electromagnetic wave with a frequency of 3.00x10^19 Hz is 10 pm.
(b) For a frequency of 7.50x10^9 Hz:
λ = (3.00x10^8 m/s) / (7.50x10^9 Hz)
Calculating this expression, we get:
λ = 0.04 m
To convert this to centimeters (cm), we multiply by 100:
λ = 0.04 m x 100 cm/m = 4 cm
Therefore, the wavelength of the electromagnetic wave with a frequency of 7.50x10^9 Hz is 4 cm.