A monochromatic light illuminates a double slits system with a slit separation, d =0.3 mm. The maximum of the first order occurs at y = 4.0 mm on a screen placed at 1.0 m from the slits. Find the:
a)Wavelength
b)Distance between the central maximum and the minimum of the third order on the screen.
c)Separation between third maxima and first minima
d)Angle between third maxima and first minima.
To find the answers to the given questions, we can use the equations related to the double slit interference pattern. Let's solve each part step-by-step:
a) To find the wavelength of the monochromatic light, we can use the equation: λ = (d * sinθ) / m
Where:
λ is the wavelength of light
d is the slit separation
θ is the angle of the maximum relative to the central maximum on the screen
m is the order of the maximum
In this case, we have the position of the first order maximum (y = 4.0 mm) on the screen at a distance of 1.0 m from the slits. The central maximum will be at y = 0 mm.
The distance y can be related to the angle θ using the small angle approximation, which states that for small angles, sinθ ≈ tanθ ≈ y / L, where L is the distance between the slits and the screen (1.0 m in this case).
Using this approximation, we can calculate the value of sinθ:
sinθ = y / L
sinθ = (4.0 mm) / (1.0 m) = 0.004
Now, we can rearrange the formula to solve for the wavelength (λ):
λ = (d * sinθ) / m
λ = (0.3 mm) * (0.004) / 1
λ = 0.0012 mm = 1.2 μm
Therefore, the wavelength of the monochromatic light is 1.2 μm.
b) To find the distance between the central maximum and the minimum of the third order on the screen, we can use the equation:
y = (λ * L) / (d * m)
Where:
y is the distance on the screen
λ is the wavelength of light
L is the distance between the slits and the screen
d is the slit separation
m is the order of the minimum
In this case, we want to find the distance for the third order minimum (m = 3). Using the previously calculated values:
y = (λ * L) / (d * m)
y = (1.2 μm) * (1.0 m) / (0.3 mm * 3)
y = (1.2 μm) * (1.0 m) / (0.9 mm)
y = 1.33 mm
Therefore, the distance between the central maximum and the minimum of the third order on the screen is 1.33 mm.
c) To find the separation between third maxima and first minima, we can use the equation:
Δy = λ * L / d
Where:
Δy is the separation between maxima and minima
λ is the wavelength of light
L is the distance between the slits and the screen
d is the slit separation
Using the previously calculated values:
Δy = λ * L / d
Δy = (1.2 μm) * (1.0 m) / (0.3 mm)
Δy = 4.0 mm
Therefore, the separation between the third maxima and the first minima is 4.0 mm.
d) The angle between the third maxima and first minima can be calculated using the equation:
tanθ = Δy / L
Where:
tanθ is the tangent of the angle between the maxima and minima
Δy is the separation between maxima and minima
L is the distance between the slits and the screen
Using the previously calculated values:
tanθ = Δy / L
tanθ = 4.0 mm / 1.0 m
tanθ = 0.004
The angle θ can be found by taking the inverse tangent of the tangent value:
θ = arctan(0.004)
Therefore, the angle between the third maxima and the first minima is approximately equal to 0.23 degrees.
To find the answers to these questions, we need to use the concept of interference of light waves and the principles of Young's double-slit experiment. Let's break down the steps to solve each part of the problem:
a) To find the wavelength of the monochromatic light, we can use the formula:
λ = dy / D
Where λ represents the wavelength, d is the slit separation, y is the distance from the central maximum (in this case, the first order) to the screen, and D is the distance from the slits to the screen.
Given that d = 0.3 mm, y = 4.0 mm, and D = 1.0 m (1000 mm), we can plug in these values into the formula:
λ = (0.3 mm * 4.0 mm) / (1000 mm)
= 1.2 mm^2 / 1000 mm
= 0.0012 mm
= 1.2 x 10^-6 m
Therefore, the wavelength of the monochromatic light is 1.2 x 10^-6 m.
b) The distance between the central maximum and the minimum of the third order depends on the interference pattern formed. In the case of Young's double-slit experiment, the distance between consecutive maxima or minima is given by:
Δy = λ * D / d
Where Δy represents the distance between two consecutive maxima or minima.
Given that λ = 1.2 x 10^-6 m, D = 1.0 m (1000 mm), and d = 0.3 mm, we can calculate:
Δy = (1.2 x 10^-6 m * 1000 mm) / 0.3 mm
= 4 x 10^-3 m
= 4 mm
Therefore, the distance between the central maximum and the minimum of the third order on the screen is 4 mm.
c) The separation between the third maxima and the first minima can be calculated by:
Δy = (3λ * D) / d
Using the same values we previously calculated, we find:
Δy = (3 * 1.2 x 10^-6 m * 1000 mm) / 0.3 mm
= 12 x 10^-3 m
= 12 mm
Therefore, the separation between the third maxima and the first minima is 12 mm.
d) The angle between the third maxima and the first minima can be calculated using the formula:
θ = Δy / L
Where θ represents the angle, Δy is the distance between two consecutive maxima or minima, and L is the distance from the slits to the screen.
Given that Δy = 4 mm and L = 1.0 m (1000 mm), we can calculate:
θ = 4 mm / 1000 mm
= 0.004
Therefore, the angle between the third maxima and the first minima is 0.004 radians.
These are the answers to the given questions.