Objective:
The objective is to verify the formula of spherical mirrors by forming the image of an object in a converging (concave) mirror.
Equipment:
A computer with internet connection, a calculator (The builtin calculator of the computer may be used.), a ruler, a few sheets of paper, and a pencil
Theory:
where d_{o} is the object distance from the mirror, d_{i } is the image distance from the mirror, and f is the focal length of the mirror. f = ½ R with R being the radius of the mirror. Assigning a () sign to d_{i} indicates advance knowledge of a virtual image and obtaining a () value for the image through solving this equation for d_{i} indicates a virtual image as well.
For a diverging mirror, ( f ), the focal length should be given a (  ) sign when using the above equation, because for a diverging mirror, even the focal point is virtual. In general, anything virtual is negative, and anything real is positive.
Magnification: Magnification is the ratio of the image size to the object size. The absolute value of magnification is given by the following formula:
Ray Diagrams for a Converging Mirror:
Case ( I ): Object at infinity ( d_{o} >> 2f )
Rays coming from very far away are practically parallel. If such rays are also parallel to the main axis of the mirror, the image forms at ( F ), the focal point of the mirror, as shown:
Image: 1)Real 2) Inverted 3) A'B'<<AB 4) d_{i} = f


Case ( II ): Object beyond 2f ( d_{o} > 2f )
Image: 1)Real 2) Inverted 3) A'B' < AB 4) f < d_{i} < 2f


Infinite rays emerge from object AB, even from point A of it.. Two rays are selected. Important Ray 1 that travels parallel to the main axis passes through F, after reflection. Important Ray 2 that goes through F leaves the mirror parallel to the mail axis, after reflection. The intersection of these two rays, form A’, the image of A. A third ray could have been drawn whose reflection would pass through A’ as well. It is shown as a dotted line in the diagram. This third important ray is the one that passes through the center ( C ) and it reflects back on itself. For a ray passing through C, the angle of incidence is zero and so is the angle of reflection; therefore, it reflects back on itself.
Case ( III ): Object at 2f ( d_{o} = 2f )
Image: 1)Real 2) Inverted 3) A'B' = AB 4) d_{i} = 2f


Case ( IV ): Object between f and 2f ( f < d_{o} < 2f )
Image: 1)Real 2) Inverted 3) A'B' > AB 4) d_{i} > 2f


Case ( V ): Object at F( d_{o} = f )
Image: 1)Real 2) Inverted 3) A'B' >> AB 4) d_{i} → ∞


Case ( V' ): Object slightly beyond F( d_{o} ~ f )
Image: 1)Real 2) Inverted 3) A'B' >> AB 4) d_{i} → ∞


Case ( VI ): Object within f ( d_{o} < f )
Image: 1)Virtual 2) Upright 3) A'B' >AB 4) d_{i} behind the mirror


Practice Page:
Use two out of three important rays emerging from A to form its image (A’) and complete each of the following ray diagrams:
Note that A’ is found by the intersection of rays reflected from the mirror. Also state the image conditions.






A diverging (convex) mirror forms an image of an object that is always virtual, upright, smaller than the object, and behind the mirror.
A ray
diagram is shown below:
Image: 1)Virtual 2) Upright 3) A'B' <AB 4) d_{i} behind the mirror


Practice on Image in a Diverging Mirror
Complete the ray diagram shown below:
Procedure:
Click on the following link: http://surendranath.tripod.com/Applets.html . Click on the "Applet Menu", then on "Optics", and then click on "Spherical Mirrors and Lenses." The applet asks you to click on it to start. Click on the applet. The screen for the experiment (or the Optical Bench) appears. On the top left dropdown menu, click on "Concave Mirror" if it is not already on the bench. Two yellow spots on the main axis of the mirror show its center and focal point Each square is a 20 units by 20 units. If you hold the mouse on the mirror itself where it crosses the main axis, you can move it left or right and place it where you like. The mirror allows you select a focal length for it. If you place the mouse on F and move it left or right, the focal length changes and so does the radius (or the location of C accordingly because R = 2f). As you move F, the concavity of the mirror changes. The more concave a mirror gets, the shorter its focal length becomes. We are going to think of (mm) as the unit of length in this experiment; therefore, if the applet says f = 100.0, it means f = 100.0mm. On this basis, each square on the applet is 20mmX20mm.
Important: In the Mirror's Formula, all distances are measured from the mirror itself for the formula to predict correctly.
Experiment:
Part 1, Concave Mirror (Converging Mirror):
 Make sure that the converging mirror is exactly in the middle (20 squares on the left and 20 squares on the right of it). Set F (the Focal Point) at120mm, or let f = 120mm. Read the top of the applet to make sure it reads f = 120mm.
 Place the mouse at the tip of the object (the red arrow) and make its height equal to 4 squares or 80.0mm. Also move it to the left far enough at d_{o} = 400.0mm. Check the values of y ( the object height), and d_{o} on the top to assure their correctness.
 Measure (Estimate) the position of d_{i }on the applet and record your estimate in Table 1. Estimate the height of the image, y', on the applet and record your estimate in Table 1. These will be your measured values. In estimation, each square has a length of 20.0mm.
 Use the given values of d_{o} and f to calculate the expected d_{i}. This will be the accepted value for d_{i}. Use this d_{i} and d_{o} to find the magnification, M. Then use the magnification, M to calculate y' (the image height). This will be the accepted value for y'. Record all values in Table 1.
 Keeping the same object height of y = 80.0mm, and f = 120mm, repeat the above steps for the following d_{o} values: d_{o} = 240.0mm, 170.0mm, 130.0mm, 121.0mm, 119.0mm, and 70.0mm. Note: In cases that the image goes out of screen, just calculate the accepted value for d_{i} and y', and leave the space for the measured values blank and do not calculate their respective %errors.
Part 2, Convex Mirror (Diverging Mirror):
 Change the mirror to a convex one by clicking on the "Convex Mirror" in the top left dropdown window. A convex Mirror appears. With the mouse, place the mirror exactly at the16th square from the right side of the screen. The screen is 60squares wide. Now, there must be exactly 16 squares to the right of the mirror and 44 to its left.
 Move the virtual focal point F to 8 squares from the mirror such that f = 160.0mm. This places the virtual center of the mirror at the rightmost edge of the matrix (R = 320mm). Make sure the applet reads the same. Of course, you know that the focal point and focal length of a convex mirror are both virtual. The f to be used in calculations is actually f = 160.0mm.
 Keeping the object height still at y = 80.0mm, move the object to 3 squares from the mirror at d_{o} = 60.0mm. Doublecheck your readings for f, d_{o} , and y.
 Again, measure (estimate) d_{i} and y'. These will be your measured (estimated) values.
 Using d_{o} , f, and y, calculate the accepted values for d_{i} , M, and y'.
 Finally calculate the necessary %errors.
 Keeping the same object height of y = 80.0mm, and f = 160.0mm, repeat the above steps for the following d_{o} values: d_{o} = 120.0mm, 230.0mm, 320.0mm, 500.0mm, and 720.0mm. Do all calculations for each case and record the values in Table 1.
Data:
Given and Measured:
Case 
Object Distance d_{o} (mm) 
Focal Length f (mm) 
Measured (Your Estimate) Image Dist. d_{i} (mm) 
Accepted (By Calculation) Image Dist. d_{i} (mm) 
Absolute
Value of Magnifi cation M 
Object Height y (mm) 
Measured Image Size y' (mm) 
Accepted Image Size y' (mm) 
% Error on d_{i}

% Error on y'

Concave
Mirror: 



80.0 





1 
400.0 
+120. 



80.0 




2 
240.0 
+120. 



80.0 




3 
170.0 
+120. 



80.0 




4 
130.0 
+120. 



80.0 




5 
121.0 
+120. 



80.0 




6 
119.0 
+120. 



80.0 




7 
70.0 
+120. 



80.0 




Convex
Mirror: 
Virtual
Image Position ( d_{i }) is
Negative


8 
120.0 
160.0 





80.0 




9 
230.0 
160.0 





80.0 




10 
320.0 
160.0 





80.0 




11 
500.0 
160.0 





80.0 




12 
720.0 
160.0 





80.0 















Table 1
Calculations:
Use the Mirror’s Formula and the Magnification Formula to perform calculations.
Comparison of the Results:
Calculate a %error on d_{i} and y' for each case using the usual %error formula.
Conclusion: To be explained by students
Discussion: To be explained by students