A certain type of laser emits light that has a frequency of 4.0 × 1014 Hz. The light, however, occurs as a series of short pulses, each lasting for a time of 2.3 × 10-11 s. (a) How many wavelengths are there in one pulse? (b) The light enters a pool of water. The frequency of the light remains the same, but the speed of the light slows down to 2.3 x 108 m/s. How many wavelengths are there now in one pulse?
To answer these questions, we'll use the equation:
wavelength (λ) = speed of light (c) / frequency (f)
where c is a constant equal to the speed of light in a vacuum (approximately 3.0 x 10^8 m/s).
(a) How many wavelengths are there in one pulse?
In this case, we know the frequency (f) is 4.0 x 10^14 Hz and the time duration of each pulse (t) is 2.3 x 10^-11 s.
Step 1: Calculate the speed of light during the pulse.
The speed of light in a vacuum is approximately 3.0 x 10^8 m/s, but we need to determine the speed during the pulse. Since speed is defined as distance traveled divided by time taken, we can calculate it by dividing the wavelength by the pulse duration:
speed during the pulse (v) = λ / t
Step 2: Calculate the wavelength.
Using the given frequency (f) and the speed during the pulse (v), we can rearrange the equation to solve for the wavelength:
wavelength during the pulse (λ) = v / f
(b) How many wavelengths are there now in one pulse, after the light enters water?
In this case, the frequency remains the same, but the speed of light changes to 2.3 x 10^8 m/s.
Step 1: Calculate the wavelength after entering water.
Using the new speed of light during the pulse (v) and the frequency (f), we can again rearrange the equation to solve for the wavelength:
wavelength after entering water (λ) = v / f
Now let's go ahead and calculate the values for both cases:
(a) How many wavelengths are there in one pulse?
Step 1: Calculate the speed during the pulse.
v = λ / t = c = 3.0 x 10^8 m/s
Step 2: Calculate the wavelength.
λ = v / f = (3.0 x 10^8 m/s) / (4.0 x 10^14 Hz)
λ ≈ 7.5 x 10^-7 m
(b) How many wavelengths are there now in one pulse, after the light enters water?
Step 1: Calculate the wavelength after entering water.
v = 2.3 x 10^8 m/s (given speed in water)
λ = v / f = (2.3 x 10^8 m/s) / (4.0 x 10^14 Hz)
λ ≈ 5.75 x 10^-7 m
Therefore, the number of wavelengths in one pulse before entering water is approximately 7.5 x 10^-7 m / 2.3 x 10^-11 s ≈ 3.26 x 10^4 wavelengths/pulse. After entering water, the number of wavelengths becomes approximately 5.75 x 10^-7 m / 2.3 x 10^-11 s ≈ 2.50 x 10^4 wavelengths/pulse.