Two parallel narrow slits are at a distance of 0.840 mm apart. Each of them has a width of 0.140 mm. The double slit is illuminated with parallel light of wavelength 546.1 nm.
(a) Find the ratio of the intensity of the central maximum to that of its adjacent maximum. (1 : 0.912)
(b) Certain interference maxima are missing. Find the position of the missing maximum that is nearest to the central maximum. (13.41' from central maximum)
To find the ratio of the intensity of the central maximum to that of its adjacent maximum, we can use the formula for the intensity of the interference pattern:
I(theta) = I_0 * cos^2[(pi * d * sin(theta)) / (lambda)]
Where:
I(theta) is the intensity at an angle theta
I_0 is the intensity of the unperturbed wave
d is the distance between the slits
theta is the angle at which the maxima occur
lambda is the wavelength of the light
(a) In the case of the central maximum (theta = 0), the equation simplifies to:
I(0) = I_0.
For the adjacent maximum (theta = theta_1), the equation becomes:
I(theta_1) = I_0 * cos^2[(pi * d * sin(theta_1)) / (lambda)].
To find the ratio, we divide the two intensities:
Ratio = I(0) / I(theta_1)
= I_0 / (I_0 * cos^2[(pi * d * sin(theta_1)) / (lambda)])
= 1 / cos^2[(pi * d * sin(theta_1)) / (lambda)]
= 1 / cos^2[(pi * 0.840 mm * sin(theta_1)) / (546.1 nm)]
= 1 / cos^2[1.533 * sin(theta_1)]
To find the value of the ratio, we need to find the value of theta_1. For the first adjacent maximum, the condition for constructive interference is given by:
d * sin(theta_1) = lambda.
Plugging in the values, we have:
0.840 mm * sin(theta_1) = 546.1 nm.
Converting the units:
0.840 mm * sin(theta_1) = 0.5461 mm.
Simplifying:
sin(theta_1) = 0.5461 mm / 0.840 mm,
sin(theta_1) = 0.6501.
To find the value of the ratio, we substitute this value of sin(theta_1) into the previous equation:
Ratio = 1 / cos^2[1.533 * 0.6501]
= 1 / cos^2[0.99663]
= 1 / 0.912,
≈ 1 : 0.912.
Therefore, the ratio of the intensity of the central maximum to that of its adjacent maximum is approximately 1 : 0.912.
(b) To find the position of the missing maximum that is nearest to the central maximum, we need to find the angle at which it occurs. For the missing maximum nearest to the central maximum, the condition for destructive interference is given by:
d * sin(theta) = (m + 1/2) * lambda,
where m is the order of the missing maximum.
Rearranging the equation for theta, we have:
sin(theta) = (m + 1/2) * lambda / d.
Since we are interested in the position nearest to the central maximum, we can assume m = 0. Plugging in the values, we get:
sin(theta) = 0.5 * 546.1 nm / 0.840 mm
= 0.5 * 0.5461 / 0.840
= 0.2731 / 0.84
≈ 0.3250.
To find the angle theta, we take the inverse sine:
theta ≈ arcsin(0.3250)
≈ 18.58°.
To convert this angle to minutes, we multiply by 60:
theta ≈ 18.58° * 60
≈ 1114.8'.
Therefore, the position of the missing maximum nearest to the central maximum is approximately 1114.8' from the central maximum.
To find the ratio of the intensity of the central maximum to that of its adjacent maximum, we need to consider the interference pattern formed by the double slits.
The formula to calculate the position of the maxima in an interference pattern is given by the equation:
y = (m * λ * L) / d
where:
y is the distance from the central maximum to the mth maximum,
λ is the wavelength of light,
L is the distance between the double slits and the screen,
d is the distance between the two slits, and
m is the order of the maximum.
In this case, we are considering the central maximum (m = 0) and its adjacent maximum (m = ±1).
First, let's calculate the distance between the double slits and the screen (L). This information is not provided in the question. However, we can assume that the screen is far away compared to the distance between the slits (d). Therefore, we can consider L to be large enough (approximately infinity).
For the central maximum (m = 0), the equation becomes:
y0 = (0 * λ * L) / d = 0
This means that the central maximum is at the center of the interference pattern. The intensity at the central maximum is given by:
I0 = I1 + I2 + 2√(I1 * I2) * cos(δ)
where I1 and I2 are the individual intensities of light passing through each slit, and δ is the phase difference between the two waves coming from the slits.
Since the slits are illuminated with parallel light, we can assume that both slits receive equal intensities of light. Therefore, I1 = I2 = I.
Also, since the slits are narrow, the phase difference between the two waves can be approximated as zero (δ ≈ 0).
Plugging these values into the formula for I0, we get:
I0 = I + I + 2√(I * I) * cos(0) = 2I + 2√(I * I) * 1 = 4I
Moving on to the adjacent maximum (m = ±1), the equation becomes:
y1 = (±1 * λ * L) / d
Since L is large compared to d, we can assume that y1 is small compared to L.
Now, to calculate the intensity at the adjacent maximum (I1 and I2), we can use the formula:
I = I1 + I2 + 2√(I1 * I2) * cos(δ)
Since the slits are illuminated with parallel light and the phase difference between the two waves from the slits is almost zero (δ ≈ 0), we can assume that the intensity at each slit is the same (I1 = I2 = I/2).
Plugging these values into the formula for I, we get:
I = (I/2) + (I/2) + 2√((I/2) * (I/2)) * cos(0)
I = I + √(I^2) * 1
I = I + I = 2I
Therefore, the ratio of the intensity of the central maximum to that of its adjacent maximum is:
(I0 / I1) = (4I / 2I) = 2
Therefore, the ratio of the intensity of the central maximum to that of its adjacent maximum is 1:2.
Now, let's move on to finding the position of the missing maximum nearest to the central maximum. In an interference pattern, the missing maxima occur when the path difference (Δy) between the waves from the two slits is equal to an integer multiple of the wavelength (λ).
The formula to calculate the path difference is:
Δy = (m * λ * L) / d
where m is the order of the maximum.
For the missing maximum nearest to the central maximum, we can assume m = ±2.
Setting Δy equal to λ, we can find the value of L:
(2 * λ * L) / d = λ
Simplifying, we get:
2L / d = 1
Therefore, L = d / 2.
Now, using the value of L, we can calculate the position of the missing maximum nearest to the central maximum:
Δy = (2 * λ * L) / d = (2 * λ * (d / 2)) / d = λ
Therefore, the position of the missing maximum nearest to the central maximum is at a distance of λ from the central maximum.
Given the wavelength of light (λ = 546.1 nm), we can substitute this value into the formula to calculate the position:
Position = λ = 546.1 nm = 13.41' (approximately)
Therefore, the missing maximum nearest to the central maximum is located at a distance of 13.41' from the central maximum.