Question

Elucidate the diffraction of monochromatic light by narrow slit. Determine the expression for angular width of central maximum.

Hint:

For single-slit diffraction, the intensity is zero (minimum) when path difference is \(n\lambda\) (from slit edges). Don't confuse these conditions! The central maximum in diffraction is twice as wide as the secondary maxima.

Answer 1

Elucidation of Diffraction:
Diffraction is the phenomenon of the bending or spreading of waves as they pass through a narrow opening (aperture) or around an obstacle. When a monochromatic plane wavefront is incident on a narrow slit, each point on the portion of the wavefront that passes through the slit acts as a source of secondary spherical wavelets, according to Huygens' principle. These wavelets spread out in all directions and interfere with one another. This interference results in a characteristic diffraction pattern on a screen placed far from the slit. The pattern consists of a bright central maximum, which is flanked on both sides by a series of alternating dark bands (minima) and weaker bright bands (secondary maxima).
Expression for Angular Width of Central Maximum:

Step 1: Condition for the First Minimum
Consider a slit of width 'a'. The central maximum is spread between the first minima on either side of the center. To find the location of the first minimum, we consider the wavelets from the top and bottom edges of the slit. The first minimum occurs at an angle \(\theta\) where the path difference between these two extreme rays is exactly one wavelength (\(\lambda\)). \[ \text{Path Difference} = a \sin\theta \] For the first minimum, this path difference is equal to \(\lambda\): \[ a \sin\theta = \lambda \]

Step 2: Derivation
From the condition for the first minimum, we have \(\sin\theta = \frac{\lambda}{a}\).
For typical diffraction setups, the angle \(\theta\) is very small, so we can use the small-angle approximation: \(\sin\theta \approx \theta\) (where \(\theta\) is in radians). \[ \theta = \frac{\lambda}{a} \] This angle \(\theta\) represents the angular position of the first minimum from the center. It is also the angular half-width of the central maximum.
The total angular width of the central maximum is the angle between the first minimum on one side of the center and the first minimum on the other side. \[ \text{Total Angular Width} = \theta + \theta = 2\theta \] Substituting the expression for \(\theta\): \[ \text{Total Angular Width} = \frac{2\lambda}{a} \]

Step 3: Final Answer:
The expression for the angular width of the central maximum in single-slit diffraction is \(\frac{2\lambda}{a}\), where \(\lambda\) is the wavelength of the light and 'a' is the width of the slit.