**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
**s**ine function relates the angle
measure of one of the acute angles to the ratio of the lengths of the
side **o**pposite the angle and the
length of the **h **hypotenuse. The
**c**osine function relates the angle
measure of one of the acute angles to the ratio of the lengths of the
side **a**djacent the angle and the
length of the **h**ypotenuse. 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
**o**pposite the angle and the length
of the side **a**djacent the angle.
The meaning of the functions can be easily remembered by the
mnemonic

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:

where n_{1} = index of refraction value of material 1 (on
one side of the boundary)

n_{2} = 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.

**Additional Readings/Study
Aids:**

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

View Sample Problem Set.

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