Two lasers are shining on a double slit, with slit separation d. Laser 1 has a wavelength of d/20, whereas laser 2 has a wavelength of d/15. The lasers produce separate interference patterns on a screen a distance 4.90m away from the slits
a)What is the distance Δymax−max between the first maxima (on the same side of the central maximum) of the two patterns?
b)What is the distance Δymax−min between the second maximum of laser 1 and the third minimum of laser 2, on the same side of the central maximum?
a) Δymax−max = d/10
b) Δymax−min = d/5
To solve these problems, we need to use the concept of interference in double-slit setups. The interference pattern occurs due to the superposition of the waves emanating from each slit. The resulting pattern on the screen depends on the wavelength of the light and the separation between the slits.
Let's start with part (a) and find the distance Δymax-max between the first maxima of the two patterns.
1. Calculate the angles θ1 and θ2 for the first maxima using the formula: sin(θ) = mλ/d, where m is the order of the maximum, λ is the wavelength, and d is the separation between the slits.
- For Laser 1: λ1 = d/20
sin(θ1) = λ1/d
- For Laser 2: λ2 = d/15
sin(θ2) = λ2/d
2. Now, we can use the small-angle approximation sin(θ) ≈ tan(θ) to find the angles in radians.
3. Since the angles are small, the distance Δymax-max is approximately given by:
Δymax-max = L * (tan(θ1) - tan(θ2)), where L is the distance between the screen and the slits.
Substitute the given values and compute Δymax-max.
Moving on to part (b), we need to find the distance Δymax-min between the second maximum of laser 1 and the third minimum of laser 2.
1. Calculate the angles θ3 and θ4 for the second maximum of Laser 1 and the third minimum of Laser 2 using the same formula as before.
- For Laser 1: sin(θ3) = λ1/d
- For Laser 2: sin(θ4) = λ2/d
2. Convert the angles to radians using the small-angle approximation.
3. The distance Δymax-min is given by:
Δymax-min = L * (tan(θ3) - tan(θ4))
Substitute the given values and compute Δymax-min.
Remember to double-check all the calculations and units to ensure accuracy.
To solve this problem, we can use the formula for the position of the maxima in a double-slit interference pattern:
y = λL / d
where:
y is the distance from the central maximum to the nth maximum,
λ is the wavelength of the light,
L is the distance from the slits to the screen,
and d is the separation between the slits.
a) For laser 1, the wavelength is (d/20) and for laser 2, the wavelength is (d/15). The distance between the slits and the screen is given as 4.90 m.
To find the distance Δymax−max between the first maxima of the two interference patterns, we subtract the two distances:
Δymax−max = [(λ2 * L) / d] - [(λ1 * L) / d]
Substituting the values, we have:
Δymax−max = [(d/15 * 4.90) / d] - [(d/20 * 4.90) / d]
Simplifying the expression, we get:
Δymax−max = (4.90/15) - (4.90/20)
Calculating this expression gives us:
Δymax−max ≈ 0.3267 m
Therefore, the distance Δymax−max between the first maxima of the two interference patterns is approximately 0.3267 m.
b) To find the distance Δymax−min between the second maximum of laser 1 and the third minimum of laser 2, we need to determine the distances y2 for laser 1 and y3 for laser 2.
For laser 1, the distance y2 is given by:
y2 = λ1 * L / d
For laser 2, the distance y3 is given by:
y3 = λ2 * L / d
Now, we can find the distance Δymax−min by subtracting y3 from y2:
Δymax−min = y2 - y3
Substituting the values, we have:
Δymax−min = [(d/20 * 4.90) / d] - [(d/15 * 4.90) / d]
Simplifying the expression, we get:
Δymax−min = (4.90/20) - (4.90/15)
Calculating this expression gives us:
Δymax−min ≈ 0.1567 m
Therefore, the distance Δymax−min between the second maximum of laser 1 and the third minimum of laser 2 is approximately 0.1567 m.