Single Slit Diffraction: Videos & Practice Problems

Single slit diffraction results in a unique pattern of alternating bright and dark fringes due to the interference of light waves. The central bright spot is notably larger and brighter than the others, which are uniform in width. The dark fringes are calculated using the equation $\sin (\theta$_m)=mλ/d, where m indexes the dark fringes. Understanding this concept is crucial for grasping wave behavior and interference patterns in physics.

Single Slit Diffraciton

Single slit diffraction is a phenomenon that occurs when light passes through a narrow opening, resulting in a diffraction pattern characterized by alternating bright and dark spots on a screen. Unlike the double slit experiment, where two sources of light interfere, the single slit experiment involves light emanating from a single source, with different parts of the slit emitting light at varying angles. This leads to a central bright spot that is notably larger and brighter than the other bright fringes, which are uniform in width.

The central bright fringe in a single slit diffraction pattern is twice the width of the other bright fringes. The dark fringes, which are the points of destructive interference, can be mathematically described using the equation:

sin(θ_m) = mλ/d

In this equation, θ_m represents the angle of the m-th dark fringe, m is the index of the dark fringe (starting from 1), λ is the wavelength of the light, and d is the width of the slit. Notably, there is no m = 0 index for dark fringes in a single slit setup.

To determine the width of the central bright spot, one can analyze the geometry of the situation. For example, if a laser with a wavelength of 450 nanometers passes through a slit of width 0.1 millimeters and the screen is positioned 1.4 meters away, the first dark fringe can be calculated using the aforementioned equation. After finding the angle θ₁, the height of the triangle formed can be determined using trigonometric relationships, specifically the tangent function:

tan(θ) = opposite/adjacent

In this case, the opposite side corresponds to the height y₁, and the adjacent side is the distance to the screen. The width of the central bright spot is then calculated as:

W = 2y₁

Thus, if y₁ is found to be 6.4 millimeters, the total width of the central bright fringe would be 12.8 millimeters. This illustrates the unique characteristics of single slit diffraction and the mathematical relationships that govern the behavior of light in such scenarios.

Light from a 600 nm laser is shown through a single slit of unknown width. If a screen is placed 4.5 m behind the slit captures a diffraction patter with a central bright fringe of width 20 mm, what is the width of the single slit?

• A. 0.13 mm
• B. 0.11 mm
• C. 0.44 mm
• D. 0.26 mm

Answer: (D)

Number of Dark Fringes on a Screen

In this example, we explore the diffraction pattern created by a laser light passing through a single slit. The laser emits light at a wavelength of 500 nanometers, and the slit has a width of 0.5 millimeters. The screen, positioned 3.5 meters from the slit, is 2 centimeters wide. Our goal is to determine how many dark fringes can fit on the screen.

To visualize the problem, we consider the central axis and the maximum angle, θ_max, at which light can exit the slit and still reach the screen. If light exceeds this angle, it will not be captured by the screen, resulting in no visible diffraction pattern. The dark fringes, which alternate with bright fringes, are evenly spaced angularly. The angle φ between the central axis and the first dark fringe is crucial for our calculations.

The relationship for the angles of dark fringes in a single slit is given by the equation:

sin(θ_m) = mλ/d

For the first dark fringe (where m = 1), this simplifies to:

sin(θ₁) = λ/d

Substituting the known values, where λ = 500 × 10^-9 m and d = 0.5 × 10^-3 m, we find:

sin(θ₁) = (500 × 10^-9) / (0.5 × 10^-3) = 0.001

This results in θ₁ ≈ 0.057 degrees, which is also the angle φ between successive dark fringes.

Next, we need to calculate θ_max. This angle can be determined using the dimensions of a right triangle formed by the screen's width and the distance to the slit. The height of the triangle is half the screen width (1 cm), and the base is 3.5 meters. Using the tangent function:

tan(θ_max) = opposite/adjacent = (1 × 10^-2) / (3.5)

Calculating this gives us θ_max ≈ 0.16 degrees.

To find the total number of dark fringes that can fit on the screen, we use the formula:

Number of dark fringes = 2 × θ_max / φ

Substituting the values:

Number of dark fringes = 2 × 0.16 / 0.057 ≈ 5.6

This indicates that 5 dark fringes can fit on the screen, with some bright fringes partially visible at the edges. The calculation illustrates the interplay between light diffraction, slit width, and screen dimensions, providing insight into wave behavior in optics.

Here’s what students ask on this topic:

What is single slit diffraction and how does it differ from double slit diffraction?

Single slit diffraction occurs when light passes through a narrow slit and produces a pattern of alternating bright and dark fringes due to interference. The central bright spot in single slit diffraction is notably larger and brighter than the other fringes, which are uniform in width. In contrast, double slit diffraction involves two slits, resulting in multiple sources of light that interfere. The central bright spot in double slit diffraction is the same width as the other bright spots. The key difference lies in the central bright spot's size and brightness, and the number of light sources causing interference.

How is the location of dark fringes in single slit diffraction calculated?

The location of dark fringes in single slit diffraction is calculated using the equation:

$\theta =\frac{\mathrm{m\lambda }}{d}$

where $m$ is the fringe order (1, 2, 3, ...), $\lambda$ is the wavelength of the light, and $d$ is the width of the slit. This equation gives the angle $\theta$ at which the dark fringes occur. Note that there is no $\mathrm{m=0}$ dark fringe in single slit diffraction.

Why is the central bright spot in single slit diffraction larger and brighter than the other fringes?

The central bright spot in single slit diffraction is larger and brighter because it results from the constructive interference of light waves that travel nearly parallel paths through the slit. This central maximum is twice as wide as the other bright fringes and has a higher intensity due to the greater number of light waves contributing to the constructive interference at this central point. The other bright fringes are formed by light waves that interfere constructively at specific angles, resulting in uniform but narrower and less intense bright spots.

What is the equation for the width of the central bright spot in single slit diffraction?

The width of the central bright spot in single slit diffraction can be determined by first finding the angle $\theta$ for the first dark fringe using the equation:

$\theta =\frac{\lambda }{d}$

Then, the width $w$ of the central bright spot on a screen at distance $L$ from the slit is given by:

$w=2L\cdot \theta$

This accounts for the distance from the central maximum to the first dark fringe on either side.

How does interference cause the bright and dark fringes in single slit diffraction?

In single slit diffraction, interference occurs because light waves leaving different parts of the slit travel at different angles. When these waves meet, they can interfere constructively or destructively. Constructive interference happens when waves align with their peaks and troughs matching, resulting in bright fringes. Destructive interference occurs when a peak of one wave meets the trough of another, canceling each other out and creating dark fringes. This alternating pattern of bright and dark fringes is the diffraction pattern observed on the screen.