My name is Ashley, and I was a student of Professor Wall the summer of 2005.  As part of an extra-credit assignment, I was asked to research a project involving the use of lenses.   First, I will summarize what I learned form Professor Wall and literature (referenced at the end).  Next, I'll show how I was asked to relate the information with actual lenses.

A Summary

  Optics is a science dealing with the behavior of light.  One area of optics is the way light interacts with different instruments, like lenses.  "A lens is used to concentrate or disperse light and to form images."  (Miller, 1999).  There are many different shapes of lenses, but they are grouped into two categories:  converging (convex) and diverging (concave). 

The shape of the lens determines the way the light is shaped to form an image.  In a converging lens, parallel light enters and is brought together at a focal point (f) which is determined by the index of refraction of the lens and the medium it is in.  Diverging lenses, on the other hand, cause parallel rays to spread out.  (Walker, 2004).  All the light rays are brought together again at a focal point.  Tracing these light rays from an object (p) though the lens and its focal point can show where the image (q) is located.  (Miller, 1999).  When the image is on the opposite side of the lens to the object, it is called real.  It is normally upside down which is denoted by a negative sign.  If the image is on the same side of the lens as the object, it is called virtual (with a positive sign).  There are 3 principal rays that are traced for lenses (illustration below):

1)    A parallel ray goes through a converging lens to cross the focal

     point.  An exception happens when the object is between the focal

     point and the lens.  The ray diagram acts like it is going through a

     diverging lens.  The parallel ray goes though a diverging lens to

     diverge on the other side, but then extrapolates to the focal point on

     the side of the object.

2)  A ray passes through the lens at the thin spot of the lens and is 

      not bent in either of the lenses. 

3)  A ray passes through the focal point on the side of the object and

     then through a converging lens which bends it parallel.  In a

     diverging lens, the ray acts as if it is going to the focal point on the

     opposite side of the object but is refracted parallel once it goes 

     through the lens.  The parallel line again extrapolates to the side of   

     the object.

  These relationships are shown through the triangles formed along the path of ray #2.  The height of the object (hp) and the height of the image (hq) are equal.  Also shown below, the dp is the distance of the object from the lens, and the dq is the distance of the image from the lens.  hp/dp = hq/dq.

         Ray # 1 forms a triangular equivalence where hp/f = -hq/dq-f

   The distances of the focal point (f), the object (p), and the image (q) can be related with these equations in the “Thin-Lens Equation.”

                    1/f = 1/p + 1/q

   The magnification (M) of the lens is defined as the –image distance divided by the object distance. 

                    M = -q/p

   Since the negative signs show the location as related to the lens or the orientation of the image, here is a brief summary:

   Lenses used in instruments have the same basic principals.  For example, the eye has a converging lens that when relaxed can see infinitely far.  The image of the object is placed upside down (because it’s a converging lens) on the retina.  The lens can also be manipulated by ciliary muscles to focus on a near object to a certain point.  This is called the near point (N).  The average young person has a near point of around 25 cm.  Sometimes defects occur and a person could have myopia (nearsightedness) or hypermetropia (farsightedness).  These conditions may be treated with lenses!

  Nearsightedness is caused by an over-converged lens, which limits the person to only see clearly to a far-point.  By placing a diverging lens in front of the converging lens, the distant object’s image is placed at that person’s far-point.  The same is true for in reverse for a farsighted person.  They have a near point that is farther away than the normal person’s near point.  They must use a converging lens to place the near object’s image past there near point. 

   A magnifying glass is used in the same way. “The magnifier brings the near point closer to the eye.”  (Walker, 2004).  Therefore, if a person’s near-point is “N” cm, the object can only come “N” cm away from the eye and that is as much detail as can be seen by the unaided eye.  The closer that the object could get to the eye would increases the height of the object to the eye, and also, increases the image height given on the retina.  However, if the person put a magnifying lens in front of there eye with focal length of “f” cm (f < N), then the object is brought “N” – “f” closer to the eye and the object to the eye becomes greater; therefore, it puts a greater sized image on the retina.

                             M  =  N/f

  This is the point that the image is at infinity, and the eye is relaxed at the minimum eyestrain.  But, the magnification can increase if the image where put at the near point.  This is the maximum magnification of the lens.

                              M = 1 + N/f

   The telescope is a two lens system that brings infinitely far objects up close.  An astronomical telescope uses two converging lenses, an eyepiece and an objective lens.  The objective lens (like the magnifying glass) brings a far distant object closer and puts the image up-side down on the focal point of the eyepiece.  The objective lens’s image becomes the eyepiece lens’s object.  The eyepiece lens puts the image at infinity, so that, the eye views the image without strain.  The magnification is defined as the objective lens’s focal length (fo) divided by the eyepiece lens’s focal length (fe).  The image will be inverted.

                              M = fo/fe

   The Galilean telescope uses a converging lens for the objective lens and a diverging lens as the eyepiece.  The difference is the placing of the image by the diverging lens.  It is right-side up and placed 25 cm from the eye.

(Miller, 1999).

The Project

   The project for the assignment consisted of finding the magnification for minimum eyestrain and the maximum magnification of two different lenses using a camera as an eye.  The camera and the eye are very similar in function.  Also, by using two lenses, astronomical and Galilean telescopes can be made. 

   This was Minnie Mouse.  She was the object set at the near point of the camera at 15.7 cm away from the camera lens.

 The focal length was found by placing the lens at a point before it blows up the image (shown below).  This was 6.3 cm from the object. 

 By pulling the object in closer to 9.7 cm away from the camera and the lens 6.3 cm away from the object still, the image was put at infinity for minimum eyestrain (shown below).  Using the magnification for minimum eyestrain equation M = N/f, the magnification for the lens was 1.5X.  When putting the image 3.2 cm closer to the lens or at 6.5 cm from the camera, the maximum magnification of 2.5X was found.  This was the point at which the image is shown at the near point.

 The next magnifying glass was more powerful.  Its focal length was measured the same way as before and was at 2.6 cm from the lens.  

 Minnie was then moved to 7.4 cm away from the camera with the lens still 2.6 cm away from Minnie.  The minimum magnification was 2.8x.

The maximum should be 3.8x by moving the object 1.4 cm closer to the camera (or 6 cm from the camera).  The image was then formed at the near point.

 This was a picture of Bill the bullet almost 2 m away.

 An astronomical telescope was made by using the above lens of 1.5x and 2.8x.  The magnification of a telescope is fo/fe, and in this case was 1.9x. 

 This was an example of a Galilean telescope (using a converging objective lens and a diverging eyepiece).

   Some error was seen in the calculations of proving image distance due to the camera’s ability to refocus after moving the object.  Another limitation in the project was also seen in taking the pictures for the maximum magnification.  These pictures would not focus and were not include.

All information was taken from:

 Miller, Franklin Jr.  College Physics, 5^th edition. Binnacle Publishing Group,

San Francisco:  1999.

 Walker, James S.  Physics, 2^nd edition.  Pearson Education, Inc., New Jersey: 

2004.

Wall, Dave.  2005 2^nd Summer Session Class.