Fermat's principle, also known as the principle of least time, states that light travels between two points along the path that requires the least time, compared to other nearby paths.
In more detail, Fermat's law implies:
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Light minimizes travel time: Light doesn't necessarily take the shortest distance between two points, but rather the path that allows it to traverse the distance in the least amount of time. This is crucial when light travels through different media with varying refractive indices (how much they slow down light).
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Derivation of optical laws: From Fermat's principle, we can derive the fundamental laws of geometrical optics:
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Law of Reflection: The angle of incidence (the angle at which light strikes a surface) is equal to the angle of reflection (the angle at which light bounces off the surface).
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Law of Refraction (Snell's Law): When light passes from one medium to another (e.g., from air to water), it bends (refracts). Snell's Law quantifies this bending based on the refractive indices of the two media. Specifically, n₁sinθ₁ = n₂sinθ₂, where n₁ and n₂ are the refractive indices of the two media, and θ₁ and θ₂ are the angles of incidence and refraction, respectively.
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Mathematical Formulation: Fermat's principle can be expressed mathematically using calculus of variations, where the time taken for light to travel along a path is minimized.
Example:
Imagine light traveling from point A in air to point B in water. A straight line from A to B might be the shortest distance, but light travels slower in water. Therefore, the path of least time will involve bending (refraction) at the air-water interface, so the light spends less time in the water, even if the overall path is slightly longer than a straight line.
In essence, Fermat's principle provides a powerful and elegant way to understand and predict the behavior of light as it propagates through various media.
