A certain type of laser emits light that has a frequency of 4.7 1014 Hz. The light, however, occurs as a series of short pulses, each lasting for a time of 2.8 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 2.3 108 m/s. How many wavelengths are there now in one pulse?
Well, let's dive into this question with a splash of humor!
(a) To find the number of wavelengths in one pulse, we can use the formula:
Number of wavelengths = (speed of light in a vacuum) / (frequency)
The speed of light in a vacuum is approximately 3.0 x 10^8 m/s. So, for one pulse, the number of wavelengths would be:
Number of wavelengths = (3.0 x 10^8 m/s) / (4.7 x 10^14 Hz)
Just crunching some numbers, you'll get approximately 6.38 wavelengths in one pulse. Not quite as refreshing as taking a dip in the pool, but still interesting!
(b) Now, when the light enters the pool of water, the speed slows down to 2.3 x 10^8 m/s. However, the frequency remains the same. So, to calculate the number of wavelengths in one pulse in water, we can use the same formula as before:
Number of wavelengths = (speed of light in water) / (frequency)
Plugging in the values, you'll get:
Number of wavelengths = (2.3 x 10^8 m/s) / (4.7 x 10^14 Hz)
Let's crunch the numbers again and find out the answer. You'll get approximately 0.49 wavelengths in one pulse. That's like swimming in a tiny puddle instead of a pool!
So, to summarize:
(a) In air, there are approximately 6.38 wavelengths in one pulse.
(b) In water, there are approximately 0.49 wavelengths in one pulse.
Keep in mind, though, that water can really make a splash in how it affects light!
To answer these questions, we need to use the formula:
Wavelength (λ) = Speed of Light (c) / Frequency (f)
(a) How many wavelengths are there in one pulse?
Given:
Frequency (f) = 4.7 * 10^14 Hz
Time (t) = 2.8 * 10^-11 s
To find the number of wavelengths in one pulse, we need to calculate the distance the light travels during one pulse. This distance is equal to the speed of light multiplied by the time of one pulse.
Distance = Speed of Light * Time = c * t
To find the number of wavelengths, we divide the distance by the wavelength:
Number of Wavelengths = Distance / Wavelength
Substituting the formula for distance and rearranging the formula for wavelength, we have:
Number of Wavelengths = (c * t) / (c / f) = c * t * f / c = f * t
Number of Wavelengths = (4.7 * 10^14 Hz) * (2.8 * 10^-11 s)
Now, we can calculate the number of wavelengths:
Number of Wavelengths = 1.316 * 10^4 wavelengths
Therefore, there are approximately 1.316 * 10^4 wavelengths in one pulse.
(b) The light enters a pool of water. The frequency remains the same, but the speed of light slows down to 2.3 * 10^8 m/s.
To find the number of wavelengths in one pulse in water, we use the same formula:
Number of Wavelengths = Frequency * Time = f * t
Number of Wavelengths = (4.7 * 10^14 Hz) * (2.8 * 10^-11 s)
Now, we need to find the speed of light in water. We can use the formula:
Speed of Light in Water = Speed of Light in Vacuum / Index of Refraction
Given the speed of light in water is 2.3 * 10^8 m/s, we can calculate the index of refraction (n):
n = Speed of Light in Vacuum / Speed of Light in Water = c / v
n = c / (2.3 * 10^8 m/s)
Next, we substitute the formula for wavelength using the speed of light in water and rearrange the formula:
Number of Wavelengths = (Speed of Light in Water * Time) / (Speed of Light in Water / Frequency)
Number of Wavelengths = (v * t) / (v / f)
Number of Wavelengths = v * t * f / v
Number of Wavelengths = f * t
Number of Wavelengths = (4.7 * 10^14 Hz) * (2.8 * 10^-11 s)
Therefore, the number of wavelengths in one pulse in water is approximately 1.316 * 10^4 wavelengths, the same as in vacuum.
To answer these questions, we need to understand the relationship between frequency, wavelength, and speed of light.
The formula relating these quantities is:
c = λ * ν
Where:
c is the speed of light,
λ is the wavelength,
and ν is the frequency.
Let's start with the first question:
(a) How many wavelengths are there in one pulse?
Given: Frequency (ν) = 4.7 * 10^14 Hz
Pulse duration = 2.8 * 10^-11 s
To find the number of wavelengths in one pulse, we need to calculate the distance covered by light during the pulse duration.
Speed of light (c) = 3 * 10^8 m/s (approximate value)
Distance covered (d) = Speed of light * Pulse duration
d = (3 * 10^8 m/s) * (2.8 * 10^-11 s)
d = 8.4 * 10^-3 m
Now, we can find the number of wavelengths by dividing the distance covered by the wavelength.
Number of wavelengths = Distance covered / Wavelength
Number of wavelengths = d / λ
Rearranging the equation, we have:
λ = d / Number of wavelengths
Substituting the values, we get:
λ = (8.4 * 10^-3 m) / Number of wavelengths
We're given the frequency (ν), which can be converted to the wavelength (λ) using the equation c = λ * ν.
Rearranging the equation, we can solve for λ:
λ = c / ν
Substituting the given values, we can calculate the wavelength.
λ = (3 * 10^8 m/s) / (4.7 * 10^14 Hz)
λ ≈ 6.38 * 10^-7 m
Now, we can find the number of wavelengths in one pulse:
λ = (8.4 * 10^-3 m) / Number of wavelengths
Rearranging the equation, we get:
Number of wavelengths = (8.4 * 10^-3 m) / λ
Number of wavelengths = (8.4 * 10^-3 m) / (6.38 * 10^-7 m)
By dividing these values, we can find the number of wavelengths in one pulse.
(b) The light enters a pool of water. The frequency remains the same, but the speed of light slows down.
Given: Speed of light in water (c') = 2.3 * 10^8 m/s
To find the number of wavelengths now, we need to calculate the distance covered by light in the pulse duration.
Distance covered (d') = Speed of light in water * Pulse duration
d' = (2.3 * 10^8 m/s) * (2.8 * 10^-11 s)
d' = 6.44 * 10^-3 m
Using the equation c = λ * ν, we can calculate the new wavelength in water.
λ' = c' / ν
λ' = (2.3 * 10^8 m/s) / (4.7 * 10^14 Hz)
λ' ≈ 4.89 * 10^-7 m
Finally, we can find the number of wavelengths in one pulse in water:
Number of wavelengths in water = (6.44 * 10^-3 m) / λ'
Dividing these values will give us the number of wavelengths in one pulse in water.
Remember to calculate all the values accurately and consider significant figures in the final answer.