Ch. 4 - Applications of Derivatives

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 4 - Applications of Derivatives

Problem 4.5.54

Chapter 4, Problem 4.5.54

54. Fermat’s principle in optics Light from a source A is reflected by a plane mirror to a receiver at point B, as shown in the accompanying figure. Show that for the light to obey Fermat’s principle, the angle of incidence must equal the angle of reflection, both measured from the line normal to the reflecting surface. (This result can also be derived without calculus. There is a purely geometric argument, which you may prefer.)

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Fermat's principle states that light follows the path that takes the least time. In the context of reflection, this means the path of light from point A to point B via the mirror is such that the time taken is minimized.

Consider the path of light from point A to the mirror at point P, and then from point P to point B. The total path length is the sum of the distances AP and PB.

To find the point P on the mirror that minimizes the total path length, we can use calculus. Let the mirror be on the x-axis, and let A and B have coordinates (x1, y1) and (x2, y2) respectively. The point P on the mirror can be represented as (x, 0).

The total path length L is given by the sum of the distances: L = sqrt((x - x1)^2 + y1^2) + sqrt((x - x2)^2 + y2^2). To minimize L, take the derivative of L with respect to x and set it to zero: dL/dx = 0.

Solving the derivative equation will show that the angles of incidence and reflection are equal. This is because the derivative condition leads to the equality of the angles formed by the light path with the normal to the mirror at point P, confirming Fermat's principle.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Fermat's Principle

Fermat's Principle states that light travels between two points along the path that requires the least time. In optics, this principle is used to derive the laws of reflection and refraction. For reflection, it implies that the path taken by light will be such that the angle of incidence equals the angle of reflection, ensuring the shortest travel time.

Angle of Incidence and Reflection

The angle of incidence is the angle between the incoming light ray and the normal (perpendicular) to the surface at the point of contact. The angle of reflection is the angle between the reflected light ray and the same normal. According to the law of reflection, these angles are equal, which is a direct consequence of Fermat's Principle in the context of reflection.

Normal Line to a Surface

The normal line to a surface is an imaginary line perpendicular to the surface at a given point. It is crucial in defining angles of incidence and reflection, as these angles are measured relative to the normal. Understanding the normal line helps in visualizing and calculating the behavior of light as it interacts with reflective surfaces.

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