Coherent light of frequency 6.38×1014 Hz passes through two thin slits and falls on a screen 80.0 cm away. You observe that the third bright fringe occurs at ± 3.08 cm on either side of the central bright fringe.
How far apart are the two slits?
At what distance from the central bright fringe will the third dark fringe occur?
We can use the equation for the location of the bright fringes in a double-slit experiment:
y = (mλL) / d
where y is the distance from the central bright fringe to the m-th bright fringe on the screen, λ is the wavelength of the light, L is the distance from the slits to the screen, d is the distance between the two slits, and m is an integer representing the order of the bright fringe.
To find the distance between the slits, we can rearrange the equation to solve for d:
d = (mλL) / y
Using the given values, we can solve for d:
λ = 3.00 × 10^8 m/s / 6.38 × 10^14 Hz = 4.69 × 10^-7 m
L = 80.0 cm = 0.8 m
m = 3
y = 3.08 cm = 0.0308 m
d = (mλL) / y = (3)(4.69 × 10^-7 m)(0.8 m) / 0.0308 m = 3.86 × 10^-6 m
Therefore, the distance between the two slits is approximately 3.86 micrometers.
To find the location of the third dark fringe, we can use the equation:
y = [(2m + 1)λL] / (2d)
where m is an integer representing the order of the dark fringe.
We want to find the distance from the central bright fringe to the third dark fringe, so we can set m = 1:
y = [(2 × 1 + 1)λL] / (2d) = (3λL) / (2d)
Using the values we calculated earlier:
y = (3)(4.69 × 10^-7 m)(0.8 m) / (2)(3.86 × 10^-6 m) = 0.244 m
Therefore, the third dark fringe is located 0.244 meters away from the central bright fringe.
Note: It's important to keep track of units when using these equations. In our calculations, we converted the given values to SI units (meters) and made sure the final answers were also in meters.
To find the distance between the two slits, we can use the equation for the fringe spacing in a double-slit interference pattern:
y = (λL) / d
Where:
y is the fringe spacing
λ is the wavelength of the light
L is the distance between the screen and the double slits
d is the distance between the two slits
We are given:
λ = 6.38×10^14 Hz
L = 80.0 cm = 0.8 m
We need to find d.
First, we need to convert the frequency to wavelength using the speed of light equation:
c = λ f
Where:
c is the speed of light = 3.00 × 10^8 m/s
f is the frequency
λ = c / f
= (3.00 × 10^8 m/s) / (6.38×10^14 Hz)
≈ 4.70 × 10^(-7) m
Now, we can plug the values into the fringe spacing equation:
3.08 cm = (4.70 × 10^(-7) m) (0.8 m) / d
Simplifying the equation:
0.0308 m = (4.70 × 10^(-7) m) (0.8 m) / d
Multiplying both sides by d:
0.0308 m * d = (4.70 × 10^(-7) m) (0.8 m)
Dividing both sides by 0.0308 m:
d = [(4.70 × 10^(-7) m) (0.8 m)] / 0.0308 m
Simplifying the equation:
d ≈ 1.22 × 10^(-5) m
Therefore, the distance between the two slits is approximately 1.22 × 10^(-5) m.
To find the distance from the central bright fringe to the third dark fringe, we can use the equation for the dark fringe position:
y = (λL) / (2d)
Where y is the distance from the central bright fringe to the dark fringe.
We are given:
λ = 4.70 × 10^(-7) m
L = 0.8 m
d = 1.22 × 10^(-5) m
We need to find y.
Plugging the values into the equation:
y = [(4.70 × 10^(-7) m) (0.8 m)] / (2(1.22 × 10^(-5) m))
Simplifying the equation:
y ≈ 1.22 × 10^(-3) m
Therefore, the distance from the central bright fringe to the third dark fringe is approximately 1.22 × 10^(-3) m.
To solve this problem, we can use the principles of interference of light waves. The pattern of bright and dark fringes observed on a screen when light passes through two thin slits is called the interference pattern.
Let's start by finding the distance between the two slits, which we'll call "d".
The formula for the fringe spacing, also known as the fringe width (symbolized by "β"), is given by:
β = (λ * D) / d
Where:
- λ is the wavelength of the light
- D is the distance from the slits to the screen
- d is the distance between the two slits
We are given the frequency of the light, not its wavelength. However, we can use the relationship:
λ = c / ν
Where:
- λ is the wavelength
- c is the speed of light (approximately 3.00 x 10^8 m/s)
- ν is the frequency of the light
Plugging in the given frequency:
λ = c / ν
λ = (3.00 x 10^8 m/s) / (6.38 x 10^14 Hz)
Now, we can calculate the wavelength of the light.
Next, we can calculate the fringe spacing β using the given information that the third bright fringe occurs at ± 3.08 cm on either side of the central bright fringe. We can assume that the distance from the central fringe to the first bright fringe (n = 1) is equal to the distance from the first bright fringe to the second bright fringe (n = 2), and so on.
Therefore, the distance from the central bright fringe to the third bright fringe (n = 3) is twice the fringe spacing β.
Knowing the distance from the central bright fringe to the third bright fringe, we can calculate β.
Finally, we can use the formula to find the distance between the two slits, d:
d = (λ * D) / β
Plugging in the known values, we can find the distance between the two slits.
To find the distance from the central bright fringe to the third dark fringe, you can use a similar approach. Just keep in mind that for dark fringes, the path length difference between the two waves is equal to an odd multiple of half the wavelength (λ/2).
With this information, you can follow the same steps outlined above, but instead of finding the β for the bright fringes, you will calculate the β for the dark fringes using the given observations.
I hope this explanation helps you understand how to solve the problem.