ChemPhys 173/273

Unit 2: Refraction and Lenses

Problem Set B

Overview:

Problem Set B targets your ability to use Snell's law and trigonometric functions of sine, cosine and tangent in order to analyze physical situations involving the refraction of light at boundaries. The first 10 questions pertain exclusively to the use of trigonometric functions. It provides practice for those who are first learning how to use such functions and review for those who have previously learned it. The last 10 questions involve the use of Snell's law and other skills (including a trigonometric analysis) in order to analyze refraction situations at boundaries.

Trigonometric Analysis:

There is a unique relationship between the angles within a right triangle and the ratio of the length of the sides. Trigonometric functions are mathematical functions which relate the length of the sides of a right triangle to the angles within the triangle.

Any triangle has three angles. A right triangle has two acute angles and one right angle. Trigonometric functions are typically used to express the relationship between the measure of one of the acute angles and the ratio of the length of the sides. Each of the acute angle is formed by the intersection of one of the short sides and the hypotenuse side of the triangle. The short side which forms the angle (along with the hypotenuse) is referred to as the adjacent side. The third side is then referred to as the side opposite the angle. The sine function relates the angle measure of one of the acute angles to the ratio of the lengths of the side opposite the angle and the length of the h hypotenuse. The cosine function relates the angle measure of one of the acute angles to the ratio of the lengths of the side adjacent the angle and the length of the hypotenuse. And finally the tangent function relates the the angle measure of one of the acute angles to the ratio of the lengths of the side opposite the angle and the length of the side adjacent the angle. The meaning of the functions can be easily remembered by the mnemonic

SOH CAH TOA

The equations defining these functions are listed below.

In Physics, there are numerous physical situations in which a student would want to relate the angle in a right triangle to the length of one or more of its sides. In such instances, a trigonometric function is chosen and used to analyze the physical situation. For instance, the refraction of light at a boundary causes the angle that the light path makes with the normal (and also with the boundary) to change. It is different on one side of the boundary than on the other side of the boundary. If this light path is projected through space on either side of the boundary it will likely strike a physical object. There is subsequently a triangle (or even a pair of triangles) formed by the boundary, the light ray and physical objects present in the vicinity. This is best illustrated in problem #17 of this set of problems in which a light ray is sent from a submarine at an angle to the water surface. The light ray refracts at the boundary and strikes the top of a tall building on the water's edge. Refraction principles can be used to relate the angle of incidence (i) to the angle of refraction (r). These angles are in turn related to the distance measurements shown on the diagram.

As a discipline concerned about the relationships present in the physical environment, physics is clearly interested in questions concerning dimensions, directions, angle measures and the like. These three trigonometric functions, when combined with physics principles, allows one to make quantitative predictions and conclusions concerning the dimensions and angles associated with the origin and the destination of the path of a light ray. Problems #17 - #20 will provide several illustrations of these ideas in action.

Refraction and Snell's Law:

A light ray will undergo refraction (a change in direction of its path) at a boundary between two materials whenever it approaches the boundary at an angle of incidence other than zero degrees. This refraction occurs in a rather predictable manner as expressed by the Snell's law equation:

n1 • sin ( 1 ) = n2 • sin ( 2 )

where n1 = index of refraction value of material 1 (on one side of the boundary)

n2 = index of refraction value of material 2 (on the opposite side of the boundary)

1 = angle of refraction of material 1 (on one side of the boundary)

2 = angle of refraction of material 2 (on the opposite side of the boundary)

Knowing the index of refraction values of the two materials on opposite sides of the boundary and the angle of incidence allows one to make a calculation of the angle of refraction. In a similar manner, knowing the two angles which a light ray makes with the normal line for both sides of the boundary and knowing one of the index of refraction values of the two materials allows one to determine the second index of refraction value.

The following pages from The Physics Classroom tutorial may serve to be useful in assisting you in accomplishing the above tasks.

Angles of Incidence and Refraction | Light Speed and n Values | Snell's Law | Ray Tracing

View Sample Problem Set.

 Problem Description Audio Link 1 Use of SOH CAH TOA to determine an angle measure in a right triangle from two sides of known length. 2 Use of SOH CAH TOA to determine an angle measure in a right triangle from two sides of known length. 3 Use of SOH CAH TOA to determine an angle measure in a right triangle from two sides of known length. 4 Use of SOH CAH TOA to determine an angle measure in a right triangle from two sides of known length. 5 Use of SOH CAH TOA to determine the length of a side of a right triangle from knowledge of an angle and another length. 6 Use of SOH CAH TOA to determine the length of a side of a right triangle from knowledge of an angle and another length. 7 Use of SOH CAH TOA to determine the length of a side of a right triangle from knowledge of an angle and another length. 8 Use of SOH CAH TOA to determine the length of a side of a right triangle from knowledge of an angle and another length. 9 Application of SOH CAH TOA to a physical situation in order to determine a distance. 10 Application of SOH CAH TOA to a physical situation in order to determine a distance. 11 A multi-step problem involving Snell's law, the law of reflection and simple geometric principles. 12 A double-layer problem involving the use of Snell's law to determine the angle of incidence. 13 Routine calculation of an angle of refraction using Snell's law. 14 Referring to the previous problem; application of geometric principles and Snell's law to trace the path of light through a prism and out its side. 15 Application of basic geometry and Snell's law to analyze the light path through a layer. 16 Referring to the previous problem; routine calculation of an angle of refraction using Snell's law. 17 A complex analysis of the path of light using SOH CAH TOA and Snell's law. 18 A complex analysis of the path of light using SOH CAH TOA and Snell's law. 19 A complex analysis of the path of light using SOH CAH TOA, Snell's law and the law of reflection. 20 A complex analysis of the path of light using SOH CAH TOA and Snell's law.

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