Unit 2: Refraction and Lenses
Problem Set D
Overview:
Problem Set D targets your ability to use the lens equation to analyze physical situations involving images formed by lenses. There is a blend of problems in the set, beginning with several relatively simple calculations and ending with some very complex analyses.
Lens Equation and Magnification Ratio
Throughout the set, you will make use of two equations. The two equations  the lens equation and the magnification ratio  are shown below:
1 / f = 1/ d_{o} + 1 / d_{i} 
M = h_{i} / h_{o} =  d_{i} / d_{o} 
In the above equations, the variable d_{o} represents the object distance or the distance between the lens surface and the object. The variable d_{i} represents the image distance or the distance between the lens surface and the image. The variable h_{o} represents the object height and the variable h_{i} represents the image height. The variable f stands for the focal length of the lens. The variable M stands for the magnification of the image; it represents how many times bigger the image is than the object.
In Set B, the above equations are used to assist in the solution of physics word problems. Values of two or more quantities are typically expressed in the problem; and the goal of the word problem is to solve for an unknown quantity. Substitutions of known quantities will have to be made into the above equations and proper algebraic manipulations must be performed in order to solve for the unknown quantity.
Developing Effective ProblemSolving Habits
Effective problemsolving habits should be practiced in the solution of these problems. Establishing good habits early in the course will serve a student as more difficult problems appear in later sets and later units. Such habits include the following:
Perhaps one of the most problematic areas of this problem set is dealing with the sign conventions associated with the image distance, focal length and image height. The table below summarizes the sign conventions associated with these quantities.

Sign Convention 

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

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

A positive image distance (d_{i}) corresponds to an image location on the opposite side of the lens as the object. A negative image distance (d_{i}) corresponds to an image located on the same side as the lens as the object. Thus, all real images have positive image distances and all virtual images have negative image distances. 

A positive image height (h_{i}) corresponds to an upright image. A negative image height (h_{i}) corresponds to an inverted image. All upright images (positive h_{i} values) are virtual images located on the object's side of the lens; upright images will thus be virtual images with negative image distances. Likewise, inverted images with their negative h_{i} values are real images that have positive image distance values. 

A converging lens will have a positive focal length (f) and a diverging lens will have a negative focal length (f). 

Magnification values are positive whenever image heights (h_{i}) are positive. Thus, positive M values correspond to upright, virtual images located on the object's side of the lens. And negative M values correspond to inverted, real images located on the side of the lens opposite of the object. 
When reading a problem, give attention to cues within the problem in order to determine the sign on the given quantity. Problem #10 illustrates the importance of this. The problem describes an object located to the left of a diverging lens and a virtual image located to the left of the diverging lens. Applying the above conventions, one can assign a positive value to the object distance and a negative value to the image distance. Failure to recognize such small nuances of problems will result in wrong answers despite the accuracy of the mathematical manipulations. A careful reading of the problem statement along with an understanding of the sign conventions (as stated in the table above) for the variables within the two equations allows one to make proper conceptual decisions. These types of decisions are critical to your success on this problem set. Making the correct decisions have 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.
Additional Readings/Study Aids:
The following pages from The Physics Classroom tutorial may serve to be useful in assisting you in the proper use of the lens equation and magnification ratio and in your ability to make the connection between the image characteristics and the proper sign (+ vs. ) for stated values.
View Sample Problem Set.

Description 


Routine calculation of d_{i}. 


Referring to the previous problem; a routine calculation of M. 


Routine calculation of d_{i}. 


Routine calculation of d_{i}. 


Referring to the previous problem; a routine calculation of M. 


Routine calculation of d_{i}. 


Routine calculation of d_{i}. 


Routine calculation of d_{i}. 


Routine calculation of d_{i}. 


Routine calculation of f; requires conceptual decisionmaking with regards to sign conventions. 


A multistep problem requiring some conceptual decisionmaking with regards to sign conventions. 


Routine calculation of f; requires conceptual decisionmaking with regards to sign conventions. 


Mathematically difficult problem involving two equations and two unknowns and requiring some conceptual decisionmaking with regards to sign conventions. 


A multistep problem requiring some conceptual decisionmaking with regards to sign conventions. 


Mathematically difficult problem involving several steps and the use of the quadratic formula; requires some conceptual decisionmaking with regards to sign conventions. 


Mathematically difficult problem involving several steps and the use of the quadratic formula; requires some conceptual decisionmaking with regards to sign conventions. 


A complex analysis involving many steps and some considerable thinking. 


A complex problem with many steps and some difficult mathematics. 


Referring to the previous problem; a routine calculation of M. 


A complex problem involving routine calculations and several steps. 

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