Ray Optics: Reflection and Mirrors
Reflection and Mirrors: Problem Set Overview
This set of 26 problems targets your ability to use the law of reflection, to understanding the relationship between image distance and object distance for plane mirrors, and to use the mirror equation and magnification ratio to solve problems which relate object and image characteristics to the focal length of concave and convex mirrors. Problems range in difficulty from the very easy and straightforward to the very difficult and complex. The more difficult problems are colorcoded as blue problems.
The Law of Reflection
Light rays follow a rather predictable pattern when it comes to reflection off a plane mirror surface. The angle at which the light ray approaches the mirror surface is equal to the angle at which it departs from the mirror. This is known as the law of reflection. In physics, the angles of approach are measured with respect to the normal line to the surface. The normal line is the imaginary line that is perpendicular to the mirror at the point that the light ray strikes the mirror. The angle between the normal line and the approaching or incident ray is known as the angle of incidence. Similarly, the angle between the reflected ray and the same normal line is known as the angle of reflection. According to the law of reflection, the angle of incidence is equal to the angle of reflection. A more detailed and exhaustive discussion of the law of reflection and associated terms can be found at The Physics Classroom Tutorial.
Characteristics of Plane Mirror Images
Objects placed in front of plane mirrors will have a corresponding image located behind the mirror. The distance from the image to the mirror is always identical to the distance from the object to the mirror. So if a person stands 2.0 meters in front of the mirror, then the image will be located an identical 2.0 meters behind the mirror. Such an image is a virtual image; when viewing such a virtual image in the mirror, it would seem as though light is coming from a location 2.0 meters behind the mirror. If you were to walk behind the mirror and look at this socalled virtual image location, there would be nothing physical present there. It only seems to the observer as thought light is coming from this location to the eye when viewing the image of the person in the mirror. A more detailed and exhaustive discussion of plane mirror image characteristics can be found at The Physics Classroom Tutorial.
Curved Mirror Mathematics
Most of the problems in this problem set pertain to curved mirrors  both the concave and the convex varieties.The two equations of relevance for these problems are the mirror equation and the magnification equation. The mirror equation relates the image distance to the object distance and the focal length. The mirror equation is
1/f = 1/d_{o} + 1/d_{i}
The variable d_{o} represents the object distance or the distance between the mirror surface and the object. The variable d_{i} represents the image distance or the distance between the mirror surface and the image. The variable f stands for the focal length of the mirror. In some problems, the focal length is not stated; rather, the radius of curvature of the spherical mirror is stated. The radius of curvature (R) is simply twice the focal length value (R = 2•f). Like any equation in physics, the mirror equation can be used to solve for an unknown variable through algebraic substitution and rearrangement. Given that there are three quantities present in the mirror equation, two of them must be known in order to solve for the third unknown quantity.
A curved mirror usually causes an image to be either magnified or reduced in size relative to the size of the object. The magnification ratio is a number which expresses the amount of magnification or reduction. The magnification ratio is simply the ratio of the image size to the object size. It is often calculated using the equation
M = h_{i} / h_{o}
In this equation, the variable M represents the magnification, the variable h_{i} represents the image height, and the variable h_{o} represents the object height. It ends up that the ratio of the image to object heights is equivalent to the ratio of the image to the object distance. And so the magnification equation is often written as
M = h_{i} / h_{o} =  d_{i} / d_{o}
The negative sign in the above equation is related to what could be the most problematic aspect of this set of problems. The variables in these two equations can be either positive or negative. The positive and negative nature is determined by the actual characteristics of the images which are formed and the mirrors which are used in the specific problems. The table below summarizes the socalled sign conventions for the six variables of these two equations.
Variable

Sign Convention

d_{o}

For our purposes,the object distance (d_{o}) will always be positive.

h_{o}

For our purposes, the object height (h_{o}) will always be positive.

d_{i}

A positive image distance (d_{i}) corresponds to an image location on the same side of the mirror as the object. A negative image distance (d_{i}) corresponds to an image located behind the mirror. As such, real images will always be characterized by positive d_{i} values; virtual images will have negative d_{i} values

h_{i}

A positive image height (h_{i}) corresponds to an upright image. A negative image height (h_{i}) corresponds to an inverted image. Since all upright images (positive h_{i} values) are virtual images located behind the mirror; upright images will thus be virtual images with negative d_{i} values. Similarly, inverted images with their negative hi values are real images that have positive d_{i} values.

f

A concave mirror will have a positive focal length (f) and a convex mirror will have a negative focal length (f).

M

Magnification values are positive whenever image heights (h_{i}) are positive. Thus, positive M values correspond to upright, virtual images located behind the mirror surface. And negative M values correspond to inverted, real images located on the object's side of the mirror.

When reading a problem, it is important to give attention to cues within the problem in order to determine the sign on the given quantity. For example, the distance from the focal point to a mirror is often stated. This is simply a distance value corresponding to the absolute value of the focal length. Whether the focal length is positive or negative is dependent upon whether the mirror is concave or convex. A careful reading of the problem and an understanding of the sign convention on focal length (as stated in the table above) allows one to make the decision about the sign on f. As a second example, some problems describe an image being located a stated distance from a curved mirror. The stated value is simply the absolute value of the image distance. Whether the d_{i} value is positive or negative depends upon whether the image is in front of or behind the mirror. A careful reading of the problem statement along with an understanding of the sign convention for image distance (as stated in the table above) allows one to make the decision about the sign on d_{i}. These types of decisions are critical to your success on this problem set. Making the correct decisions has nothing to do with your mathematical skills; rather, they are tests of your conceptual understandings and your willingness to read a problem carefully and to give attention to details which may be important.
There are a few instances in this problem set in which the mirror equation must be used to solve for an unknown variable but only one of the other two variable values are known. Such problems usually have a statement of the effect: "the image is real and three times the size of the object." Such a statement reveals information about the magnification of the image. Since the ratio of the image to object height is equal to the (negative of the) ratio of the image distance to object distance, we can say that size and height can be treated synonymously. Stating that the image is three times the size of the object is stating that the ratio h_{i}/h_{o} is either +3 or 3. Determining whether h_{i}/h_{o} is +3 or 3 demands an understanding of the sign conventions (as discussed in the above table). The h_{o} value is always positive (for our purposes). The h_{i} value is positive for upright images and negative for inverted images. Since this statement asserts that the image is real (and thus inverted), a 3 value must be assigned to the h_{i}/h_{o} ratio. Since hi/ho is equal to d_{i}/d_{o}, the 3 value can be equated with d_{i}/d_{o}. This stream of logic allows one to write an expression for d_{i} in terms of d_{o}. This expression for d_{i} in terms of d_{o} can be substituted into the mirror equation in order to transform it into a single equation with a single unknown. Customary algebraic manipulations can then be performed in order to solve for d_{i} or for d_{o}.
The table below summarizes the process of transforming a verbal statement into a mathematical equation which ultimately is used to substitute into the mirror equation.
Verbal Statement

Mathematical Equivalent

Expression of d_{i} in terms of d_{o}

"... the image is real and three times the size of the object."

h_{i} / h_{o} = 3 = d_{i} / d_{o}

d_{i} = +3d_{o}
or d_{o} = (+1/3)•d_{i}

Habits of an Effective ProblemSolver
An effective problem solver by habit approaches a physics problem in a manner that reflects a collection of disciplined habits. While not every effective problem solver employs the same approach, they all have habits which they share in common. These habits are described briefly here. An effective problemsolver...

...reads the problem carefully and develops a mental picture of the physical situation. If needed, they sketch a simple diagram of the physical situation to help visualize it.

...identifies and records the known and unknown quantities in an organized manner. Equates given values to the symbols used to represent the corresponding quantity  e.g., d_{o} = 24.2 cm; d_{i}= 16.8 cm; f = ???.

...plots a strategy for solving for the unknown quantity; the strategy will typically center around the use of physics equations be heavily dependent upon an understanding of physics principles.

...identifies the appropriate formula(s) to use, often times writing them down. Where needed, they perform the needed conversion of quantities into the proper unit.

...performs substitutions and algebraic manipulations in order to solve for the unknown quantity.
Read more...
Additional Readings/Study Aids:
The following pages from The Physics Classroom Tutorial may serve to be useful in assisting you in the understanding of the concepts and mathematics associated with these problems.
Problem Sets and Audio Guided Solutions
Reflection and Mirrors Problem Set
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Reflection and Mirrors Audio Guided Solutions
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