Double-Slit Fringe Spacing Equation, Derivation & Rearrangements

Core idea

Overview

Under the small-angle approximation, the distance between interference fringes is proportional to the screen distance and the wavelength of light used. As the slit separation increases, the spacing between these fringes decreases inversely.

When to use: Use this formula to find the distance between neighboring bright fringes on a screen when light passes through two slits.

Why it matters: This calculation allows for the experimental determination of light wavelength using simple laboratory geometry.

Symbols

Variables

D = Fringe Spacing, L = Screen Distance, $λ$ = Wavelength, d = Slit Spacing

D

Fringe Spacing

m

L

Screen Distance

m

λ

Wavelength

m

d

Slit Spacing

m

Walkthrough

Derivation

Derivation of Distance between adjacent maxima

This derivation explains how the distance between adjacent bright fringes in a double-slit interference pattern is determined by the wavelength of light, the distance to the screen, and the separation between the slits.

The slits are very narrow, acting as point sources of light.
The distance to the screen (L) is much larger than the distance between the slits (d).
The angle of observation (theta) is small, allowing for approximations like sin(theta) ≈ tan(theta) ≈ theta.

1

Path Difference for Constructive Interference

For constructive interference to occur, meaning a bright fringe (maximum), the path difference between the light waves arriving from the two slits must be an integer multiple of the wavelength (lambda). Here, 'm' represents the order of the maximum.

Δ r = mλ (m = 0, 1, 2, ...)

Note: Remember that m=0 corresponds to the central maximum.

2

Geometric Relationship for Path Difference

Consider the geometry of the double-slit experiment. 'd' is the distance between the slits, and 'theta' is the angle from the center of the slits to a point on the screen. The path difference between the waves from the two slits can be related to 'd' and 'theta' using trigonometry.

Δ r = d sin θ

Note: Visualize a right-angled triangle formed by the slits and the point on the screen.

3

Combining Conditions for Maxima

By equating the two expressions for the path difference (from step 1 and step 2), we get the condition for the angular positions of the bright fringes.

d sin θ = mλ

Note: This equation tells us the angle at which we'll see each bright fringe.

4

Approximation for Small Angles

Since we assume the distance to the screen (L) is much larger than the slit separation (d), the angle 'theta' for the observed fringes will be small. For small angles, sin(theta), tan(theta), and theta (in radians) are approximately equal. The tangent of the angle is also equal to the ratio of the distance 'y' from the central maximum to the fringe on the screen, divided by the distance 'L' to the screen.

sin θ \approx tan θ \approx \frac{y}{L}

Note: This approximation is crucial for simplifying the equation and is valid for typical double-slit experiments.

5

Expressing Angular Position in Terms of Screen Position

Substituting the small-angle approximation (from step 4) into the equation for maxima (from step 3), we relate the position ' $y_{m}$ ' of the m-th maximum on the screen to the experimental parameters.

d (\frac{y _{m}}{L}) = mλ

Note: Here, $y_{m}$ signifies the position of the m-th fringe from the center.

6

Position of Adjacent Maxima

Rearranging the equation from step 5 to solve for $y_{m}$ gives us the position of the m-th bright fringe from the center of the interference pattern.

y_{m} = \frac{mλ L}{d}

Note: This equation allows us to calculate the exact location of each bright band.

7

Distance Between Adjacent Maxima

The distance between two adjacent maxima (e.g., the (m+1)-th fringe and the m-th fringe) is found by subtracting the position of the m-th fringe from the position of the (m+1)-th fringe. This calculation shows that the distance between adjacent fringes is constant and independent of the order 'm'.

D = y_{m + 1} - y_{m} = (\frac{( m + 1 ) λ L}{d}) - (\frac{mλ L}{d}) = \frac{λ L}{d}

Note: This is the final equation you'll use to calculate fringe spacing.

Result

D = y_{m + 1} - y_{m} = (\frac{( m + 1 ) λ L}{d}) - (\frac{mλ L}{d}) = \frac{λ L}{d}

Source: University Physics textbooks, e.g., Young & Freedman, Serway & Jewett

Free formulas

Rearrangements

Solve for $L = \frac{D d}{λ}$

Solve for Screen Distance (L)

L = \frac{D d}{λ}

Isolate the screen distance by multiplying the fringe spacing by the slit spacing and dividing by the wavelength.

Difficulty: 3/5

Solve for $λ = \frac{D d}{L}$

Solve for Wavelength ( $λ$ )

λ = \frac{D d}{L}

Isolate the wavelength by multiplying the fringe spacing by the slit spacing and dividing by the screen distance.

Difficulty: 3/5

Solve for $d = \frac{L λ}{D}$

Solve for Slit Spacing (d)

d = \frac{L λ}{D}

Isolate the slit spacing by dividing the product of screen distance and wavelength by the fringe spacing.

Difficulty: 3/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

The graph shows fringe spacing (D) increasing linearly with screen distance (L), forming a straight line through the origin. For a student, this means that if you move the screen further away from the double slits, the bright and dark fringes will spread out more. The most important feature is this direct proportionality: doubling the screen distance doubles the fringe spacing. This relationship is directly represented by the formula D = Lλ/d, where λ and d are held constant.

Graph type: linear

Why it behaves this way

Intuition

Imagine shining a light through two very narrow, parallel slits. This creates a pattern of bright and dark bands (fringes) on a screen behind the slits. This equation tells you how far apart two of those bright bands are.

D

Distance between adjacent maxima

This is the spacing between the centers of two consecutive bright bands (fringes) on your screen. Think of it as how much space there is between one bright spot and the next bright spot.

L

Distance from slits to screen

This is how far away your screen is from the double slits. A larger 'L' means the fringe pattern will be spread out more, making the distances between the bright bands larger.

λ

Wavelength of light

This is the color of the light you are using. Different colors of light have different wavelengths. For example, red light has a longer wavelength than blue light. Longer wavelengths will spread out more, leading to wider fringe spacing.

d

Distance between the slits

This is how far apart the two narrow slits are from each other. If the slits are very close together, the light waves emerging from them will interfere more dramatically, resulting in wider fringe spacing. If they are further apart, the fringes will be closer together.

Free study cues

Insight

Canonical usage

This equation is used to calculate the fringe spacing (D) in a double-slit interference pattern, given the wavelength of light (lambda), the distance to the screen (L), and the spacing between the slits (d).

Common confusion

Students may confuse the slit spacing (d) with the fringe spacing (D), or use inconsistent units for the lengths (e.g., meters for L and millimeters for d).

Dimension note

This equation does not inherently produce a dimensionless quantity. All variables have physical units of length.

Unit systems

$D$ m or mm · The fringe spacing is a measure of distance.

$L$ m · The distance to the screen is a measure of length.

lambdam or nm · The wavelength of light is a measure of length.

$d$ m or mm · The slit spacing is a measure of distance.

Ballpark figures

Quantity:
Quantity:
Quantity:

One free problem

Practice Problem

A screen is 2.0 m away, the slit spacing is 1.0e-3 m, and the wavelength is 5.0e-7 m. What is the fringe spacing?

Screen Distance2 m

Wavelength5e-7 m

Slit Spacing0.001 m

Solve for: $D$

Hint: Multiply the screen distance by the wavelength, then divide by the slit spacing.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

When estimating fringe spacing in a standard classroom laser diffraction experiment, Double-Slit Fringe Spacing is used to calculate Fringe Spacing from Screen Distance, Wavelength, and Slit Spacing. The result matters because it helps predict motion, energy transfer, waves, fields, or circuit behaviour and check whether the answer is plausible.

Study smarter

Tips

Ensure that all input units are converted to meters before calculating.
Remember that the small-angle approximation is required for this specific linear relationship.

Avoid these traps

Common Mistakes

Placing the slit spacing (d) in the numerator instead of the denominator.
Failing to convert units like millimeters or nanometers into base meters.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

This derivation explains how the distance between adjacent bright fringes in a double-slit interference pattern is determined by the wavelength of light, the distance to the screen, and the separation between the slits.

Use this formula to find the distance between neighboring bright fringes on a screen when light passes through two slits.

This calculation allows for the experimental determination of light wavelength using simple laboratory geometry.

Placing the slit spacing (d) in the numerator instead of the denominator. Failing to convert units like millimeters or nanometers into base meters.

When estimating fringe spacing in a standard classroom laser diffraction experiment, Double-Slit Fringe Spacing is used to calculate Fringe Spacing from Screen Distance, Wavelength, and Slit Spacing. The result matters because it helps predict motion, energy transfer, waves, fields, or circuit behaviour and check whether the answer is plausible.

Ensure that all input units are converted to meters before calculating. Remember that the small-angle approximation is required for this specific linear relationship.

References

Sources

University Physics Volume 2 (OpenStax) - Chapter 17.3 Double-Slit Interference
Halliday, Resnick, and Walker, Fundamentals of Physics
Hecht, Optics
NIST CODATA
Hecht, Eugene. Optics.
Jenkins, Francis A., and Harvey E. White. Fundamentals of Optics.
Wikipedia, "Double-slit experiment"
University Physics textbooks, e.g., Young & Freedman, Serway & Jewett

Double-Slit Fringe Spacing

Overview

Variables

Derivation

Path Difference for Constructive Interference

Geometric Relationship for Path Difference

Combining Conditions for Maxima

Approximation for Small Angles

Expressing Angular Position in Terms of Screen Position

Position of Adjacent Maxima

Distance Between Adjacent Maxima

Rearrangements

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Double Slit Interference Fringe Orders

Frequently Asked Questions

Sources