In a double-slit experiment, the distance between slits is 5.0 mm and the slits are 1.0 m from t

step 1 In this problem, we have a double slit experiment, so we have two slits separated by a distance

In a double-slit experiment, the distance between slits is...

In a double-slit experiment, the distance between slits is 5.0 $\mathrm{mm}$ and the slits are 1.0 $\mathrm{m}$ from the screen. Two interference patterns can be seen on the screen: one due to light of wavelength $480 \mathrm{nm},$ and the other due to light of wavelength 600 $\mathrm{nm} .$ What is the separation on the screen between the third-order $(m=3)$ bright fringes of the two interference patterns?

Key Concepts

Fringe Order

The fringe order, denoted by the integer m, classifies the different interference maxima (bright fringes) based on their position relative to the central maximum. Higher order fringes (larger m values) are located further from the center and their positions can be calculated using the formula derived from the interference condition. The order number also indicates the number of wavelengths by which the paths differ between the two slits.

Wavelength Dependence

The position of the interference fringes is directly proportional to the wavelength of the light used. Different wavelengths yield different fringe spacings even under identical geometric conditions. This concept is crucial for understanding how variations in the wavelength lead to shifts in the interference pattern observed on the screen.

Small-Angle Approximation

In many double-slit experiments, when the screen is much farther away than the slit separation, the angles involved are small. This allows the approximation sin? ? tan? ? ? (with ? measured in radians). This simplification enables the fringe position to be calculated directly with a linear relationship y = (m?L)/d, where L is the distance to the screen, making the analysis more straightforward.

Constructive Interference Condition

For two waves to interfere constructively, their path difference must equal an integer multiple of the wavelength, often expressed as m? = d sin?. This ensures that the peaks of one wave align with the peaks of the other, resulting in a bright fringe on the screen. The integer m represents the order of the fringe and plays a key role in determining its position.

Double-Slit Interference

This concept refers to the phenomenon where waves emanating from two coherent sources, such as two slits, overlap and superimpose on a screen. The constructive and destructive interference of these waves gives rise to a pattern of alternating bright and dark fringes. It is an important demonstration of the wave nature of light and other waves.

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