Light of wavelength 589 nm incident on a pair of slits...

Light of wavelength 589 nm incident on a pair of slits produces an interference pattern on a distant screen in which the separation between adjacent bright fringes at the center of the pattern is $0.530 \mathrm{cm} .$ A second light source, when incident on the same pair of slits, produces an interference pattern on the same screen with a separation of $0.640 \mathrm{cm}$ between adjacent bright fringes at the center of the pattern. What is the wavelength of the second source? [Hint: Is the small-angle approximation justified?]

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Key Concepts

Double-Slit Interference

This concept involves the phenomenon where coherent light waves pass through two closely spaced slits and interfere with each other, producing a pattern of bright and dark fringes on a screen. The interference arises from the path difference between the two waves, leading to constructive interference (bright fringes) when the path difference is an integer multiple of the wavelength, and destructive interference (dark fringes) when it is a half-integer multiple.

Fringe Spacing

Fringe spacing refers to the distance between adjacent bright or dark fringes in an interference pattern. In the context of the double-slit experiment, this spacing is directly related to the wavelength of the light, the distance between the slits, and the distance from the slits to the screen, typically expressed by a linear equation under the small-angle approximation.

Small-Angle Approximation

The small-angle approximation is a mathematical simplification used when the angles involved in an interference or diffraction problem are very small. Under this assumption, trigonometric functions such as sine or tangent of the angle can be approximated by the angle itself (when measured in radians), which simplifies the relationships and equations used to analyze the interference pattern.

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