Single Slit Diffraction — Revision Notes

⚡ 30-Second Revision

Diffraction: — Bending of waves around obstacles/apertures.
Single Slit Minima: — $a \sin \theta = nlambda$, where $n = \pm 1, \pm 2, \dots$
Angular Width of Central Max: — $\Delta \theta = \frac{2lambda}{a}$ (in radians)
Linear Width of Central Max: — $W = \frac{2\lambda D}{a}$
Central Max: — Brightest, widest (twice width of secondary maxima).
Secondary Maxima: — Weaker, narrower, intensity decreases with order.
Effect of Medium: — $lambda' = lambda/\mu$, so $W' = W/\mu$.
Proportionalities: — $W \propto \lambda$, $W \propto D$, $W \propto 1/a$.

2-Minute Revision

Single-slit diffraction is the spreading of light as it passes through a narrow slit, producing a pattern of bright and dark fringes. The central feature is a very wide and bright central maximum, flanked by progressively weaker and narrower secondary maxima, separated by dark minima.

The condition for these dark minima is given by $a \sin \theta = nlambda$, where '$a$' is the slit width, '$\theta$' is the angle, '$\lambda$' is the wavelength, and '$n$' is an integer ($\pm 1, \pm 2, \dots$).

The linear width of this central maximum on a screen at distance $D$ is $W = \frac{2\lambda D}{a}$. This formula is crucial: it shows that a wider slit produces a narrower central maximum, and a longer wavelength produces a wider central maximum.

If the entire setup is immersed in a medium of refractive index $\mu$, the wavelength changes to $lambda' = lambda/\mu$, causing the pattern to shrink proportionally, i.e., $W' = W/\mu$. Remember to distinguish this pattern from double-slit interference, where fringes are typically of equal width and intensity.

5-Minute Revision

Single-slit diffraction is a key phenomenon demonstrating the wave nature of light. When monochromatic light passes through a narrow slit of width '$a$' comparable to its wavelength '$\lambda$', it spreads out (diffracts) and forms an interference pattern on a screen. This pattern is characterized by a very bright and wide central maximum at $\theta = 0$, flanked by alternating dark (minima) and bright (secondary maxima) fringes.

Key Equations and Concepts:

1
Condition for Minima (Dark Fringes): — These occur when destructive interference is complete. The angular positions are given by $a \sin \theta = nlambda$, where $n = \pm 1, \pm 2, \dots$. Note that $n=0$ corresponds to the central maximum.
2
Condition for Secondary Maxima (Bright Fringes): — These occur approximately at $a \sin \theta = (n + \frac{1}{2})\lambda$, where $n = \pm 1, \pm 2, \dots$. The central maximum is at $\theta=0$.
3
Intensity Distribution: — The central maximum is the most intense. The intensity of secondary maxima decreases rapidly as $n$ increases (e.g., first secondary maxima are about $4.5\%$ of central max intensity).
4
Width of Central Maximum:

* Angular Width: $\Delta \theta = \frac{2lambda}{a}$ (in radians). This is the angle between the first minimum on either side of the center. * Linear Width: $W = \frac{2\lambda D}{a}$, where $D$ is the distance from the slit to the screen. This is the physical width of the central bright band on the screen.

Important Relationships:

$W \propto \lambda$: Longer wavelength $\implies$ wider central maximum.
$W \propto 1/a$: Narrower slit $\implies$ wider central maximum.
$W \propto D$: Greater screen distance $\implies$ wider central maximum.

Effect of Medium: If the entire setup is immersed in a medium of refractive index $\mu$, the wavelength changes to $lambda' = lambda/\mu$. Consequently, the linear width of the central maximum becomes $W' = W/\mu$. The pattern shrinks.

Example: If a slit of width $0.1\,\text{mm}$ is illuminated by $600\,\text{nm}$ light, and the screen is $2\,\text{m}$ away, the linear width of the central maximum is $W = \frac{2 \times (600 \times 10^{-9},\text{m}) \times (2\,\text{m})}{0.1 \times 10^{-3},\text{m}} = 2.4\,\text{cm}$. If this setup is put in water ($\mu=4/3$), the new width would be $2.4 / (4/3) = 1.8\,\text{cm}$.

Distinction from Double-Slit Interference: Remember that in double-slit interference, fringes are generally of equal width and intensity (within a diffraction envelope), unlike the varying intensity and width in single-slit diffraction.

Prelims Revision Notes

Single-slit diffraction is the spreading of light waves through a narrow aperture, leading to an interference pattern. This pattern is distinct: a very bright and wide central maximum, flanked by much weaker and narrower secondary maxima, separated by dark minima.

Key Conditions:

Minima (Dark Fringes): — Occur at $a \sin \theta = nlambda$, where $a$ is slit width, $\theta$ is angular position, $\lambda$ is wavelength, and $n = \pm 1, \pm 2, \dots$. (Note: $n=0$ is the central maximum).
Secondary Maxima (Bright Fringes): — Occur approximately at $a \sin \theta = (n + \frac{1}{2})\lambda$, where $n = \pm 1, \pm 2, \dots$.

Central Maximum Properties:

Angular Width: — $\Delta \theta = \frac{2lambda}{a}$ (in radians). This is the angular separation between the first minima on either side.
Linear Width: — $W = \frac{2\lambda D}{a}$, where $D$ is the distance from the slit to the screen. This is the physical width of the central bright band.
Intensity: — Highest at the center ($\theta=0$). Secondary maxima have significantly lower intensity (e.g., first secondary maxima are about $4.5\%$ of central maximum intensity).

Dependence on Parameters:

Wavelength ($\lambda$): — $W \propto \lambda$. Longer wavelength (e.g., red light) leads to a wider pattern.
Slit Width ($a$): — $W \propto 1/a$. Narrower slit leads to a wider pattern (more spreading).
Screen Distance ($D$): — $W \propto D$. Greater distance leads to a wider pattern.

Effect of Medium: If the entire setup is immersed in a medium of refractive index $\mu$, the wavelength changes to $lambda' = lambda/\mu$. Consequently, the linear width of the central maximum becomes $W' = W/\mu$. The pattern shrinks.

Distinction from Double-Slit Interference:

Single Slit: — Central max is widest/brightest; secondary maxima are weaker/narrower. Fringes are not equally spaced.
Double Slit: — Fringes are generally equally spaced and of uniform intensity (within a diffraction envelope). Central fringe is a bright fringe of same width as others.

NEET Focus: Be prepared for numerical problems involving the width of the central maximum and conceptual questions comparing single-slit diffraction with double-slit interference or analyzing the effect of changing parameters.

Vyyuha Quick Recall

For Single Slit Minima: 'A Sinful Noodle Lambda' (a sin $\theta$ = n$\lambda$). For Central Max Width: '2 Large Donuts, please, A-side' (2$\lambda$D/a).