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Wave Diffraction Calculator

Analyze single slit diffraction patterns, interference minima, and angular spread

Parameters

mⓘ
Hzⓘ
mⓘ
mⓘ
mⓘ
Show Trail

Controls

xⓘ

Calculated Values

Angular Width:
4.00;rad4.00;rad
First Min Angle:
NaN;radNaN;rad
First Min Position:
NaN;mNaN;m
Central Max Width:
NaN;mNaN;m
Diffraction Strength:
2.00;2.00;

Examples

Example 1: Light Diffraction

Visible light (λ=500 nm) through a 0.1 mm slit at 1 m distance.

  • Angular Width: 0.010.01
  • First Min Position: 0.010.01
  • Central Max Width: 0.010.01

Example 2: Sound Diffraction

Sound wave (λ=0.34 m) through a 0.5 m opening at 2 m distance.

  • Angular Width: 1.361.36
  • First Min Position: 1.361.36
  • Central Max Width: 2.722.72

Example 3: Radio Wave Diffraction

Radio wave (λ=1 m) through a 2 m aperture at 5 m distance.

  • Angular Width: 1.001.00
  • First Min Position: 2.502.50
  • Central Max Width: 5.005.00

Visualization

Wave Diffraction

Diffraction is the bending of waves around obstacles or through openings. When a wave passes through a single slit, it creates a characteristic diffraction pattern with a central maximum and alternating minima and maxima.

The angular positions of the minima in single slit diffraction are given by a sin θ = mλ, where a is the slit width, θ is the angle, m is the order number (1, 2, 3...), and λ is the wavelength. The central maximum occurs at θ = 0.

The intensity pattern follows I(θ) = I₀(sin β/β)², where β = (πa sin θ)/λ. The central maximum has the highest intensity, and the intensity decreases as the angle increases.

The angular width of the central maximum is approximately 2λ/a radians. This shows that diffraction is more pronounced when the wavelength is comparable to or larger than the slit width.

Diffraction is fundamental to understanding wave behavior and has applications in optics, acoustics, and quantum mechanics. It demonstrates the wave nature of light and other phenomena.

Key Concepts

  • Single Slit Diffraction: Wave bending through a narrow opening
  • Diffraction Pattern: Characteristic intensity distribution
  • Minima Positions: a sin θ = mλ for destructive interference
  • Central Maximum: Brightest part of the pattern at θ = 0
  • Angular Width: 2λ/a for the central maximum
  • Intensity Function: I(θ) = I₀(sin β/β)²
  • Diffraction Limit: Minimum resolvable detail
  • Wave Nature: Evidence of wave behavior

Real-World Applications

  • Optics: Diffraction gratings and optical instruments
  • Acoustics: Sound diffraction around obstacles
  • Radio Waves: Signal propagation and reception
  • X-ray Crystallography: Structure determination
  • Quantum Physics: Wave-particle duality

Explore Further

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  • Standing Waves

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  • Wave Reflection

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  • Wave Transmission

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Physics Equations

Minima Positions:
asin⁡θ=mλa\sin\theta = m\lambda
Angular Width:
Δθ=2λa\Delta\theta = \frac{2\lambda}{a}
Intensity Function:
I(θ)=I0(sin⁡ββ)2I(\theta) = I_0\left(\frac{\sin\beta}{\beta}\right)^2
Beta Parameter:
β=πasin⁡θλ\beta = \frac{\pi a\sin\theta}{\lambda}
Linear Position:
y=Ltan⁡θy = L\tan\theta

Step-by-Step Solution

See how the main results are calculated.

1

Step 1: Calculate Angular Width

First, we calculate the angular width of the central maximum:

Equation:

Δθ=2λa\Delta\theta = \frac{2\lambda}{a}

Calculation:

Δθ=2×1.0000.500=4.000 rad\Delta\theta = \frac{2 \times 1.000}{0.500} = 4.000 \text{ rad}

Explanation:

The angular width determines how much the wave spreads after passing through the slit.

2

Step 2: Calculate First Minimum Angle

The angle of the first minimum is:

Equation:

sin⁡θ1=λa\sin\theta_1 = \frac{\lambda}{a}

Calculation:

sin⁡θ1=1.0000.500=2.000\sin\theta_1 = \frac{1.000}{0.500} = 2.000

Explanation:

This gives the angle where the first destructive interference occurs.

3

Step 3: Calculate Linear Position

The linear position of the first minimum on the screen is:

Equation:

y=Ltan⁡θy = L\tan\theta

Calculation:

y=2.0×tan⁡(NaN)=NaN my = 2.0 \times \tan(NaN) = NaN \text{ m}

Explanation:

This converts the angular position to a linear distance on the screen.

4

Step 4: Calculate Central Maximum Width

The width of the central maximum is twice the first minimum position:

Equation:

Width=2y1\text{Width} = 2y_1

Calculation:

Width=2×NaN=NaN m\text{Width} = 2 \times NaN = NaN \text{ m}

Explanation:

The central maximum extends from -y₁ to +y₁, giving a total width of 2y₁.

Frequently Asked Questions (FAQ)

What is diffraction?

Diffraction is the bending of waves around obstacles or through openings. It occurs when waves encounter edges or apertures that are comparable in size to the wavelength.

How does slit width affect diffraction?

Smaller slit widths produce wider diffraction patterns. The angular width of the central maximum is approximately 2λ/a, so narrower slits create more pronounced diffraction effects.

What are the positions of the minima in single slit diffraction?

The minima occur at angles given by a sin θ = mλ, where a is the slit width, θ is the angle, m is the order number (1, 2, 3...), and λ is the wavelength.

Why does the central maximum have the highest intensity?

The central maximum occurs where all wavelets from the slit arrive in phase, resulting in constructive interference and maximum intensity. As the angle increases, phase differences reduce the intensity.

What is the diffraction limit?

The diffraction limit is the minimum resolvable detail in an optical system, approximately λ/2 for a circular aperture. It represents the fundamental limit imposed by the wave nature of light.

Practice MCQs

  1. The angular width of the central maximum in single slit diffraction is:
  2. For destructive interference in single slit diffraction:
  3. Diffraction is most pronounced when:
  4. The intensity at the central maximum is:
  5. As the slit width increases, the diffraction pattern:

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