When using a double slit with light of wavelength 504 nm, it is observed that the 4-th order maximum of the interference pattern occurs at the positions of the first minimum in the diffraction pattern. If the width of the slits is 0.24 mm, what is in mm the separation of the slits?
Oh, you've stumbled into a tricky question! Well, let's see if I can make it a little less serious for you.
Alright, so we have a double slit, a wavelength of 504 nm, and some interference patterns going on. Now, we're told that the 4th order maximum coincides with the first minimum. It's like a game of hide and seek, but they found each other!
Now, to find the separation of the slits, we can use some handy-dandy formulas. The formula for the position of the nth order maximum is given by:
x = (n * λ * L) / d
Here, x is the position of the maximum, λ is the wavelength, L is the distance from the double slit to the screen, and d is the separation of the slits. We know the wavelength (504 nm), and we're given that the 4th order maximum coincides with the first minimum, so n = 4 and x = first minimum.
But wait, there's more! In the diffraction pattern, the first minimum occurs at an angle θ, given by:
sin(θ) = λ / d
And we're given that the width of the slits is 0.24 mm.
So, if we set up our equations right, we can do some math and find the separation of the slits. And I'm happy to do that! But remember, I may be a clown bot, but I do take my math seriously. So, let's crunch those numbers and find out the separation of the slits!
To solve this problem, we can use the formula for the position of the nth-order maximum in an interference pattern produced by a double slit:
y = (n * λ * L) / d
Where:
- y is the position of the nth-order maximum,
- λ is the wavelength of light,
- L is the distance from the slits to the screen, and
- d is the separation between the slits.
In this problem, we are given:
- λ = 504 nm = 504 * 10^-9 m,
- n = 4,
- y = position of the 4th-order maximum = position of the first minimum in the diffraction pattern, and
- d = 0.24 mm = 0.24 * 10^-3 m.
We need to find the separation of the slits (d).
Rearranging the formula, we get:
d = (n * λ * L) / y
Substituting the given values:
d = (4 * (504 * 10^-9) * L) / y
We don't know the distance (L) from the slits to the screen, so we cannot solve for d directly. However, we are given that the 4th-order maximum occurs at the position of the first minimum in the diffraction pattern. This happens when the path difference between the two slits is one wavelength.
Considering this condition, we have:
y = λ * L / d
Substituting L = (y * d) / λ into the previous equation:
d = (4 * (504 * 10^-9) * ((y * d) / λ)) / y
Simplifying further:
d = (4 * (504 * 10^-9) * d) / λ
Now, let's solve for d:
d * λ = 4 * (504 * 10^-9) * d
Simplifying:
d * λ = 2.016 * 10^-6 * d
Dividing both sides by d:
λ = 2.016 * 10^-6
Since d is not canceled out, we can solve for d:
d = λ / (2.016 * 10^-6)
Substituting the value of λ:
d = 504 * 10^-9 m / (2.016 * 10^-6)
Converting from meters to millimeters:
d = (504 / 2.016) * 10^-9 * 10^3
d ≈ 250 mm
Therefore, the separation of the slits is approximately 250 mm.