In a Fraunhofer diffraction experiment involving a single slit, the pattern of light and dark fringes is determined by the interference of light waves passing through the slit. The position of the minima and maxima can be calculated using specific formulas related to the slit width and the wavelength of the light used. Let's break down the problem step by step to find the angle of the first secondary maximum.
Understanding the Single Slit Diffraction Pattern
In a single slit diffraction setup, the first minimum occurs at an angle θ given by the formula:
Here, a is the width of the slit, m is the order of the minimum (with m = 1 for the first minimum), and λ is the wavelength of the light used. For the first minimum, we have:
- m = 1
- λ = 400 nm = 400 x 10-9 m
Finding the Slit Width
Given that the first minimum occurs at an angle of 30 degrees, we can rearrange the formula to find the slit width a:
Substituting the known values:
- a = (400 x 10-9 m) / sin(30°)
- sin(30°) = 1/2
Thus, we have:
- a = (400 x 10-9 m) / (1/2) = 800 x 10-9 m = 800 nm
Calculating the First Secondary Maximum
The first secondary maximum occurs between the first and second minima. The positions of the secondary maxima can be approximated using the formula:
For the first secondary maximum, we set m = 1:
- sin(θ) = (1 + 0.5)(400 x 10-9 m) / (800 x 10-9 m)
This simplifies to:
- sin(θ) = (1.5)(400 x 10-9) / (800 x 10-9)
- sin(θ) = (600 x 10-9) / (800 x 10-9) = 0.75
Determining the Angle
Now, we can find the angle θ:
This corresponds to the option:
Final Answer
Therefore, the direction of the first secondary maximum is given by option b. sin-1(3/4).
