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 answers, we need to use the principles of interference, specifically the formula for the intensity of the interference pattern created by a double slit:

I = I_0 * cos^2(πd sinθ / λ)

where:
- I is the intensity at a given point in the interference pattern
- I_0 is the intensity of the light source
- d is the separation between the two slits
- θ is the angle between the line from the center of the double slit to the point at which the intensity is being measured and the normal to the screen (this angle is very small for small angles of θ, so we can approximate sinθ ≈ θ in radians)
- λ is the wavelength of the light

Now, let's apply this formula to the given problem:

(a) Ratio of the intensity of the central maximum to that of its adjacent maximum:

In a double slit interference pattern, the intensity of the central maximum is given by I_c = I_0. The intensity of each adjacent maximum is given by I_a = I_0 * cos^2(πd / λ). Therefore, the ratio of the intensity of the central maximum to that of its adjacent maximum is:

I_c / I_a = I_0 / (I_0 * cos^2(πd / λ))
= 1 / cos^2(π * 0.840 mm / 546.1 nm)

To find the value, we need to convert the distances to the same unit. Let's convert mm to nm:

0.840 mm = 840 μm = 840,000 nm

Now, substitute the values into the equation and calculate:

I_c / I_a = 1 / cos^2(π * 840000 nm / 546.1 nm)
≈ 1 / cos^2(4386.906)

Using a calculator, the ratio is approximately 1 : 0.912.

(b) Position of the missing maximum nearest to the central maximum:

To find the position of the missing maximum nearest to the central maximum, we need to determine the angle θ_m that corresponds to the missing maximum. We know that the intensity is zero at this angle:

I = I_0 * cos^2(πd sinθ / λ) = 0

cos^2(πd sinθ / λ) = 0

This implies that πd sinθ_m / λ = (2n + 1)π/2

Simplifying, we get:

sinθ_m = (2n + 1)λ / 2d

where n is an integer representing the position of the missing maximum.

We are looking for the missing maximum nearest to the central maximum, which corresponds to n=1.

Substituting the given values, we have:

sinθ_m = (2 * 1 + 1) * 546.1 nm / (2 * 0.840 mm)
= 3 * 546.1 nm / 0.840 mm

Convert mm to nm:

0.840 mm = 840 μm = 840,000 nm

Now, calculate:

sinθ_m = 3 * 546.1 nm / 840,000 nm
= 0.001963

To find the angle θ_m, we can use the approximation that sinθ ≈ θ for small angles:

θ_m ≈ 0.001963 radians

To convert this angle to minutes of arc, multiply it by 60:

θ_m ≈ 0.001963 * 60
= 0.11778

Therefore, the position of the missing maximum nearest to the central maximum is approximately 0.11778 minutes of arc from the central maximum.