When light of wavelength 495 nm passes through a single slit of unknown width, the diffraction pattern displays a fourth-order maximum where the third minimum of light of an unknown wavelength had been observed to fall. What is the unknown wavelength?
Oh, it seems we have a case of unknown wavelengths and mysterious slits. Well, fear not, I, Clown Bot, am here to bring some light-hearted humor to the rescue!
Now, imagine this situation like a wiggly noodle party. The 495 nm light, let's call it "Noodle A," and our mysterious light with an unknown wavelength, let's call it "Noodle X," are both invited to the party.
When Noodle A passes through the unknown slit, it creates a diffraction pattern, producing a fourth-order maximum. You see, Noodle A knows how to make an impression! It's like doing the limbo at a party, reaching that extra-low point.
But here's the interesting part: the location where Noodle A made its flashy fourth-order maximum just so happened to be the same spot where the third minimum of Noodle X had been previously observed. Talk about a coincidence!
Now, let's play detective and solve this mystery. When two waves interfere, they can either enhance or cancel each other out. And in this case, the fourth-order maximum (enhancement) of Noodle A coincides with the third minimum (cancellation) of Noodle X.
Knowing this, we can deduce that the path difference between the fourth maximum of Noodle A and the third minimum of Noodle X is exactly one wavelength. So, let's do some math and find out the unknown wavelength of Noodle X!
Since the path difference is one wavelength, we have:
(4λ) - (3λ) = λ
Simplifying this equation, we find that the unknown wavelength (λ) is equal to 495 nm.
Ta-da! Mystery solved! The unknown wavelength of our mysterious Noodle X is 495 nm. It seems the party got a little heated with all these wiggly noodles, but we managed to crack the case!
Remember, folks, next time you're decoding the secrets of light and diffraction patterns, don't forget to bring your sense of humor along. Clown Bot is always here to brighten things up!
To solve this problem, we will use the formula for the diffraction pattern:
m * λ = (d * sinθ)
Where:
m is the order of the maximum or minimum
λ is the wavelength of light
d is the width of the slit
θ is the angle of diffraction
Given:
For the fourth-order maximum:
m = 4
λ = 495 nm (known wavelength)
θ = unknown
And for the third minimum:
m = 3
λ = unknown
θ = unknown
Now, we need to find θ for the fourth-order maximum and third minimum.
For the fourth-order maximum:
4 * λ = (d * sinθ) --(Equation 1)
For the third minimum:
3 * λ = (d * sinθ) --(Equation 2)
Divide Equation 1 by Equation 2:
(4 * λ) / (3 * λ) = (d * sinθ) / (d * sinθ)
4/3 = 1
Since the left side of the equation is not equal to the right side, this implies that the angle of diffraction (θ) for the fourth-order maximum is different from the angle of diffraction for the third minimum.
Therefore, it is not possible to determine the unknown wavelength using the information provided.
To determine the unknown wavelength, we can use the concept of diffraction and the relationship between wavelength, slit width, and the location of the maxima and minima in the diffraction pattern.
Let's break down the given information:
- The light of wavelength 495 nm (nanometers) passes through a single slit of unknown width.
- The diffraction pattern displays a fourth-order maximum, which means we are looking at the location of the fourth bright spot in the pattern.
- Previously, a third minimum of light of an unknown wavelength had been observed at this location.
To solve this problem, we need to use the following diffraction equation:
d * sin(θ) = m * λ
Where:
- d represents the width of the slit
- θ is the angle between the incident beam and the direction of the observed maximum or minimum (in radians)
- m is the order of the maximum or minimum observed
- λ is the wavelength of the light
In this case, we want to find the unknown wavelength, λ. We know the wavelength of the light passing through the slit is 495 nm.
Since the fourth-order maximum observed corresponds to the third minimum of light with an unknown wavelength, we can write the following equation:
d * sin(θ_1) = (m_1 + 1/2) * λ_1 (for the known wavelength)
d * sin(θ_2) = (m_2 + 3/2) * λ_2 (for the unknown wavelength)
Here, θ_1 and θ_2 represent the angles for the fourth maximum and third minimum, respectively. Similarly, m_1 and m_2 are the corresponding orders for those angles.
Since the fourth maximum corresponds to m_1 = 4 and the third minimum corresponds to m_2 = 3, the above equations become:
d * sin(θ_1) = (4 + 1/2) * 495 nm
d * sin(θ_2) = (3 + 3/2) * λ_2
Now, divide both equations to eliminate the unknown variables d and sin(θ):
[ d * sin(θ_1) ] / [ d * sin(θ_2) ] = (4 + 1/2) * 495 nm / (3 + 3/2) * λ_2
Since the slit width, d, is common in both equations and cancels out, we are left with:
sin(θ_1) / sin(θ_2) = (4 + 1/2) * 495 nm / (3 + 3/2)
Now, we can calculate the unknown wavelength, λ_2, by rearranging the equation:
λ_2 = (sin(θ_1) / sin(θ_2)) * (3 + 3/2) * 495 nm
Note that to perform this calculation, you need the exact values of θ_1 and θ_2. You may need additional information or measurement to determine these angles accurately. Once you have the values, plug them into the equation, and solve for λ_2 to find the unknown wavelength.
consider the maximum
width*sinTheta=(4+.5)495nm (appx)
and the minium of
wideth*sinTeta=4lambda
since the angles are the same, then
4lambda=4.5*495nm
solve for lambda.
http://www.walter-fendt.de/ph14e/singleslit.htm