Monochromatic light of wavelength 605 nm falls on a slit of width 0.095 mm. The slit is located 40 cm from a screen. How far is the center of the central bright band to the first dark band?
We can find the distance from the center of the central bright band to the first dark band using the formula:
x = (λL) / a
where x is the distance we want to find, λ is the wavelength of the light, L is the distance from the slit to the screen, and a is the width of the slit.
First, we need to convert the given values to meters:
λ = 605 nm = 605 × 10^(-9) m
a = 0.095 mm = 0.095 × 10^(-3) m
L = 40 cm = 40 × 10^(-2) m
Now, we can plug the values into the formula:
x = (605 × 10^(-9) m) * (40 × 10^(-2) m) / (0.095 × 10^(-3) m)
x = (24200 × 10^(-11) m^2) / (0.095 × 10^(-3) m)
x ≈ 254736.842 × 10^(-8) m
x ≈ 0.02547 m = 2.547 cm
So, the distance from the center of the central bright band to the first dark band is approximately 2.547 cm.
To determine the distance between the center of the central bright band and the first dark band, we can use the formula for the angular position of the dark bands in a single-slit diffraction pattern. The formula is:
sinθ ≈ m * λ / w
Where:
- θ is the angle of the dark band (measured from the center of the pattern)
- m is the order of the dark band (0, 1, 2, ...)
- λ is the wavelength of the light
- w is the width of the slit
In this case, we want to find the distance between the center of the central bright band (m=0) and the first dark band (m=1). We can set up two equations with m=0 and m=1 to solve for the angle θ0 and θ1 respectively. Subtracting θ1 from θ0 will give us the desired distance on the screen.
For the central bright band (m=0):
sinθ0 = 0 * λ / w = 0
For the first dark band (m=1):
sinθ1 = 1 * λ / w
Since sinθ is small for small angles, we can use the approximation sinθ ≈ θ for small angles. Therefore, we can rewrite the above equations as:
θ0 ≈ 0
θ1 ≈ λ / w
To find the distance between the two bands on the screen, we can use the small angle approximation again:
d ≈ L * (θ1 - θ0)
≈ L * θ1
≈ L * (λ / w)
Where:
- d is the distance between the central bright band and the first dark band on the screen
- L is the distance between the slit and the screen
Plugging in the given values:
λ = 605 nm = 605 * 10^(-9) m
w = 0.095 mm = 0.095 * 10^(-3) m
L = 40 cm = 40 * 10^(-2) m
d ≈ (40 * 10^(-2)) * (605 * 10^(-9) / (0.095 * 10^(-3)))
≈ (40 * 10^(-2)) * (0.605 / 0.095)
≈ (40 * 10^(-2)) * 6.3684
Using a scientific calculator or performing the calculation, we get:
d ≈ 15.684 cm
Therefore, the center of the central bright band is approximately 15.684 cm away from the first dark band on the screen.
To find the distance between the center of the central bright band and the first dark band, we need to understand the concept of interference patterns in a single-slit diffraction experiment.
In a single-slit diffraction experiment, light waves passing through a narrow slit will create an interference pattern on a screen. The pattern consists of alternating bright and dark bands resulting from the constructive and destructive interference of the light waves.
The formula for the location of the bright and dark bands in a single-slit diffraction pattern is given by the equation:
dsinθ = mλ
Where:
- d is the width of the slit
- θ is the angle between the line from the slit to the central bright band or dark band and the line perpendicular to the screen
- m is an integer representing the order of the bright or dark band (m = 0 for the central bright band)
In this case, we are asked to find the distance between the center of the central bright band and the first dark band. Since we are dealing with the central bright band and first dark band, m = 0 for the central bright band and m = 1 for the first dark band.
We are given the following information:
- Wavelength of light, λ = 605 nm = 605 × 10^-9 m
- Width of the slit, d = 0.095 mm = 0.095 × 10^-3 m
- Distance from the slit to the screen, L = 40 cm = 40 × 10^-2 m
To find the distance between the center of the central bright band and the first dark band, we need to find the angles θ for both the central bright band and the first dark band, and then calculate their difference.
Step 1: Calculate the angle for the central bright band (m = 0):
Using the equation dsinθ = mλ, substitute the known values:
0.095 × 10^-3 m * sinθ = 0 * 605 × 10^-9 m
sinθ = 0
Since sinθ = 0, we know that θ = 0 degrees. This means the central bright band is located straight ahead, along the line perpendicular to the screen.
Step 2: Calculate the angle for the first dark band (m = 1):
Using the equation dsinθ = mλ, substitute the known values:
0.095 × 10^-3 m * sinθ = 1 * 605 × 10^-9 m
Rearranging the equation to solve for sinθ:
sinθ = (1 * 605 × 10^-9 m) / (0.095 × 10^-3 m)
sinθ = 6.3684 × 10^-6
Taking the inverse sine (arcsin) of sinθ to find the angle θ:
θ = arcsin(6.3684 × 10^-6)
θ ≈ 0.364 degrees
Step 3: Calculate the distance between the central bright band and the first dark band:
Since the central bright band is located straight ahead (θ = 0 degrees) and the first dark band is located at an angle θ ≈ 0.364 degrees, the distance between them can be calculated using trigonometry.
Using the formula: distance = L * tanθ
distance = (40 × 10^-2 m) * tan(0.364 degrees)
distance ≈ 0.091 m or 9.1 cm
So, the center of the central bright band is approximately 9.1 cm away from the first dark band.