A laser beam is incident at an angle of 30.0^∘ from the...

A laser beam is incident at an angle of $30.0^{\circ}$ from the vertical onto a solution of corn syrup in water. The beam is refracted to $19.24^{\circ}$ from the vertical. (a) What is the index of refraction of the corn syrup solution? Assume that the light is red, with vacuum wavelength $632.8 \mathrm{nm} .$ Find its (b) wavelength, (c) frequency, and (d) speed in the solution.

Key Concepts

Speed of Light in a Medium

The speed at which light travels in a medium is lower than in a vacuum due to the optical density of the medium. It can be calculated using the expression v = c/n, where c is the speed of light in vacuum and n is the index of refraction of the medium. This relationship is vital in optical physics, as it influences how light travels through different materials and impacts the design of devices such as lenses and optical fibers.

Frequency Constancy

The frequency of light is invariant across the boundary between different media, meaning that even though the speed and wavelength may change due to refraction, the frequency remains constant. This constancy is a result of the continuity in the oscillation rate of the electromagnetic wave as it propagates, which is essential for understanding energy conservation in optical systems.

Index of Refraction

The index of refraction is a dimensionless number that describes how much light slows down in a medium compared to its speed in a vacuum. It is defined as n = c/v, where c is the speed of light in a vacuum and v is the speed in the medium. This concept is fundamental to understanding how light propagates through different materials and affects phenomena such as refraction and dispersion.

Snell's Law

Snell's law relates the angles of incidence and refraction for a wave crossing the interface between two media with different indices of refraction. Mathematically, it is expressed as n1*sin(?1) = n2*sin(?2), where n1 and n2 are the indices of refraction of the initial and secondary media, respectively, and ?1 and ?2 are the angles measured from the normal. This principle is crucial for predicting how light bends when entering materials like water or corn syrup solutions.

Wavelength in a Medium

When light enters a medium other than vacuum, its wavelength changes due to the change in speed, even though its frequency remains constant. The wavelength in a medium can be calculated by dividing the vacuum wavelength by the index of refraction of the medium. This concept is important in applications involving optical instruments and fiber optics, where precise wavelength control is needed.

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