Two mirrors of a laser separated by 8.0 cm form an optical cavity in which standing waves of laser light can be set up. Each standing wave has an integral number n of half wavelengths in the 8.0 cm length, where n is large and the waves differ slightly in wavelength. Near λ = 533 nm, determine the difference in wavelength between adjacent standing waves i.e., the least difference in wavelength between any two standing waves about that wavelength.
someone pls explain how to do this ;) no clue
To determine the difference in wavelength between adjacent standing waves in the optical cavity, we can use the formula:
Δλ = λ/n
Where:
Δλ is the difference in wavelength between adjacent standing waves.
λ is the central wavelength (533 nm).
n is the number of half wavelengths in the 8.0 cm length.
First, we need to determine the number of half wavelengths, n, in the 8.0 cm length. Since there are an integral number of half wavelengths, we can set up the equation:
8.0 cm = (λ/2) * n
We know that λ is given as 533 nm, so we need to convert it to centimeters:
λ = 533 nm = (533 nm) * (1 cm / 10^7 nm)
λ = 5.33 * 10^(-5) cm
Now, we plug in the values into the equation:
8.0 cm = (5.33 * 10^(-5) cm / 2) * n
Simplifying the equation:
8.0 cm = 2.665 * 10^(-5) cm * n
Dividing both sides by 2.665 * 10^(-5) cm:
n = 8.0 cm / (2.665 * 10^(-5) cm)
n ≈ 300580
Now, we can calculate the difference in wavelength using the formula:
Δλ = λ/n
Plugging in the values:
Δλ ≈ (5.33 * 10^(-5) cm) / 300580
Δλ ≈ 1.78 * 10^(-10) cm
Therefore, the difference in wavelength between adjacent standing waves is approximately 1.78 * 10^(-10) cm.
To determine the difference in wavelength between adjacent standing waves, you need to first understand the concept of standing waves in an optical cavity.
In an optical cavity, two mirrors are placed facing each other, with a distance of 8.0 cm between them. When a laser is emitted into the cavity, the light travels back and forth between the mirrors, creating standing waves. These standing waves have specific wavelengths that fulfill the condition of having an integral number of half wavelengths within the 8.0 cm length.
The formula for the wavelength of these standing waves is given by:
λ = 2L/n
Here, λ is the wavelength of the standing wave, L is the distance between the mirrors (8.0 cm in this case), and n is the number of half wavelengths. In this scenario, n is a large number.
Now, you need to find the difference in wavelength between adjacent standing waves. This can be calculated by finding the difference between the wavelengths of two consecutive standing waves.
Let's assume the wavelength of a standing wave is λ1, and the wavelength of the adjacent standing wave is λ2. The difference in wavelength (Δλ) can be calculated as:
Δλ = λ2 - λ1
To determine the difference in wavelength, you need to find the values of λ1 and λ2. Given that the waves differ slightly in wavelength and the wavelength you're interested in is near 533 nm, you can start by substituting the known values into the equation for the wavelength of the standing waves:
λ1 = 2L/n
λ1 = 2(8.0 cm)/n
Next, find the value of n that corresponds to the wavelength nearest to 533 nm. You can do this by rearranging the equation to solve for n:
n = 2L/λ1
Substitute the values of L = 8.0 cm and λ = 533 nm (which is equal to 5.33 x 10^-7 m) into the equation:
n = 2(8.0 cm)/(5.33 x 10^-7 m)
Now, find the value of n and round it to the nearest whole number since it was mentioned that n is a large integral number.
After obtaining the value of n, you can now use it to calculate the wavelength λ2 using the same formula:
λ2 = 2L/n
λ2 = 2(8.0 cm)/n
Finally, substitute the calculated values of λ1 and λ2 into the formula for the difference in wavelength:
Δλ = λ2 - λ1
Evaluating this expression will give you the difference in wavelength between adjacent standing waves near 533 nm.
L is about 533 nm
n (L1 / 2) = .08
(n+1) L2 / 2 = .08
n L1 = .16
(n+1) L2 = nL2 + L2 = .16
n(L1-L2) = L2
but L2 is about 533*10^-9 meters
n (L1-L2) = 533*10^-9
nL1 is about .16
n is about .08 / (533*10^-9 * 2) = 7.5*10^4
so
L1 - L2 = 533*10^-9 / 7.5*10^4