Geometric Optics

   Geometric Optics

The study of optics is ancient. The Greeks were the first to enunciate the Law of Reflection, which applies to a wide variety of reflective processes. Refraction, another optical process, was also studied by a variety of civilizations. It  involves the bending of light’s direction and the changing of its speed and wavelength as it goes from one transparent medium to another. Over the centuries, Greeks as well as Arabs studied the underlying physics of it.  Building from this repository of science, seventeenth century Holland became a world leader in the field. Ultimately, Willebrord Snellius in 1626 was able to  enunciate a mathematical foundation for  the Law of Refraction;a law confirmed by more modern scientific and mathematical investigations. This budding science was accompanied by the creation of lenses, one of which Galileo purchased to study the motion of planets. Eventually the optics of lenses was mathematically formulated in what is known as the lensmaker’s equation. In this lab we will explore how refraction works in a  double convex lens and see ho = we can apply a simplified version of this equation.

Rather than restate what is explained in great detail elsewhere (including your textbook), you are directed to the website  “The Physics Classroom”. Go to the section labelled

Refraction and the Ray Model of Light – Lesson 5 – Image Formation by Lenses

Review the topics on

• The Anatomy of a Lens ( if you wish, try their interactive exercise)
• Refraction by Lenses Note that this interactive exercise covers both Converging AND Diverging lenses.  The PhET Geometric Optics app only covers the Converging Lens.
• Image Formation Revisited ( omit the section on Plane and Curved Mirrors)
• Converging Lens-Ray Diagrams. Pay attention to the rules of refraction when an object is beyond 2f ( f is the focal distance), between f and 2f and in front of f
• Converging Lens-Object-Image Relation
• The Mathematics of Lenses

The exercise here will involve investigating double convex lenses, with  set-ups at home and the PhET app GEOMETRIC OPTICS and an app from The Physics Classroom.

Note that your responses to the numbered questions that you encounter in this lab should be completed on a separate word processed sheet. Please scan and copy all diagrams. If you cannot scan and copy, write what you think the diagram should look like.

Now perform  two exercises at home  that will illustrate the principle of refraction in double sided convex lenses. YOU ARE NOT REQUIRED TO DO THIS HOME SECTION IF YOU DO NOT HAVE A MAGNIFYING GLASS OF SOME KIND.   Should you not be able to perform these exercises, then describe in words and diagrams what should be observed.

Exercise 1. Obtain a magnifying glass or double convex  lens.  ( They are cheap)  Also, obtain some kind of screen: Preferably a dark surface,  but even a white sheet of paper will do.  Now open the curtains and stand with your back to a window that looks out on the street. Hold the magnifying glass in one hand in front of the window and place the screen on the other side of the lens. Let the light outside, with its arrangement of various objects outside, come through the lens. You should observe some kind of blob on the screen. Now move either the screen or the lens, or both,  until the blob comes sharply into view. Try various positions of the lens and screen before and after your blob is focussed just so you can get a feeling of how this works.

Number your response to the questions and write full sentences., not just one word answers.

Question1  When the blob comes sharply into focus, what image do you observe?

Question 2 :  What is the orientation of the image you observe?

Question 3.   Why is this image  oriented the way it is? Use a diagram if you want.

By bringing the image into focus you can now measure, in centimeters, the focal length of the lens, f. This is the  distance between the screen and lens. If need be,  get another person to measure while you hold them.

This type of double convex lens is called a converging lens.

Question 4: Why is that?  Explain or diagram why light is converging through this lens.

Exercise 2.  Clear a table so that you are able to measure distances  from one end of the table. Also make space available in front of the table so you can move back and forth.

Place a lit candle, firmly attached to a stable candle holder near the far corner of the table. The candle should be on your LEFT as you face the table. (Of course practice fire safety, making sure there  is nothing flammable near the lit candle). Darken the room so images will be more visible.

Note that you need to vertically align the flame and the center of the lens so that the image appears somewhere in the center of the screen.

Now, as you did in front of the window, with your back to the flame,  arrange the lens and screen  to line up with the flame at the top of the candle. Use common sense to  easily arrange this so you do not have to do any contortions to make this work. Remember you are working with a flame.

If the distances are longer than your arms, try to find a way to support the screen so that you can obtain an image on it some distance from the flame.

With tape,  mark out and label three equal spaces to the right of the candle. Each space should be equal to  the length of the focal length that you measured at the window. Label them 1, 2, 3. The lens will be moved in steps from 3f to 2f, 2f, halfway between 2f and f,   to f and to  one point  in front of f.

If  possible,  create a  thin rigid object that is the length of f.  Place this to the right of the lens. This helps get perspective on how the image distance between the lens and the screen which is to the right of the lens, is changing relative to the length of f.

Place the upright lens just a little to the left  of position 3 (three). Now  slowly move the lens to the left towards position 2 (two). In essence you are creating the situation where the object distance is between 3f and 2f. At the same time adjust the screen to get a sharp image of the flame.  Note and record what is happening to the size and position of the image.

Stop at position 2 and note size and position ( relative to the lens)  of the sharp image on the screen. This is the image distance when the object distance is at 2f.

Now move the lens slowly to the left from  position 2 to position 1. Again adjust the screen position (image distance). Here the object distance is being varied from 2f to f.     Vary the position of the screen at the same time so that you can obtain a sharp image of the candle. Be prepared that small changes in object distance, p, can generate large changes in image distance, q.

Question 5:  How does the  flame’s image on the screen  change in terms of size and position and orientation as you vary the object distance ( flame to lens) from

1. 3f to 2f (position 3 to 2)
2. b) 2f to f (position 2 to1)

Question 6. How does the size and position of the image appear when the flame is a distance of 2f from the lens?

Now slowly  move the lens closer than f  (position 1)  to the flame.  The object distance is now less than f.

Question 7. Can you obtain an image on the screen ? Why or why not.

Now turn around and look at the flame through the lens when it is closer than f and observe the image.

Question 8. How is the image you observe now  different from the images you obtained before?

Question 9. Why is this lens now called a magnifying glass?

IF YOU ARE UNABLE TO DO THE PREVIOUS EXERCISE 2 AT HOME, THEN,  BASED ON WHAT YOU HAVE LEARNED,  RESPOND TO THE QUESTIONS

Now that you have had real life experience with a lens, go to the simulation where you can better control the process and verify the mathematics underlying this process.

Go to this  PhET site    Geometric Optics – Refraction | Lens | Optics

(Light and Radiation. Next, select Geometric Optics).

If you are in Chrome, It is likely that when you click the app you will go to a screen that blocks Adobe flash. If that happens, go to the URL window on top. To the left of the phet address, see a small icon of a lock. Click it.  It opens to a menu. The top line  says Flash. Click the window to see the choice Allow. Click that.  You  may or may not have to  click the “Try Anyway “ line to see a “Reload “ box in the upper left. Click it. This should get you to the Geometric Optics simulation.

If  this does not work you may have to switch to Firefox browser and try again. You may have to try a bunch of trial and error strategies to finally open up the simulation. Be patient with it.

Once you are in, the screen should look like this:

Note the controls on top:

First play with the controls to better understand what you can manipulate. Note that you can move the lens itself and the upright pencil box on the left. A little trial and error will get you there.  Be patient.  Also explore the controls on top to vary the values of curvature, refractive index and diameter.  Click the various choices on the left and right sides of the box.

The object used in this app is  an upright pencil.There is a horizontal line that goes through the center of curvature of the lens. By eye, position the pencil so that it is half above and half below this line. Clicking the box “Screen” produces a lamp and screen.  Leave that box unchecked.

Click “Marginal Rays” on the left side of the Control Box and only Ruler on the lower right side.

Once you are familiar with the controls, investigate the geometry of this situation.

Note that images created may be real or virtual. Real images can be thrown upon a screen and are inverted. Virtual images cannot be thrown on a screen and are upright. as you might have discovered with the home exercises you did.

You will now verify the lens equation by  setting distance values of p ( lens to object) and measuring the distance values of q (lens to image).

The simplified lensmaker’s equation used here is;

1/f  = 1/p + 1/q

Rearranging terms we get the following:

   q=  f x p /( p-f)

You will be collecting data for four conditions.  Each condition requires you to measure the focal distance. To find the focal length f of each condition, place the zero end of the ruler on the vertical line in the center of the lens and record the measured distance to the X mark to the right of the lens. The ruler is scaled to centimeters, but these are obviously not actual centimeters. Also note that all distances are measured relative to that vertical line in the center of the lens.

There are four tables to record data. Each table is arranged to record 5 sets of distances of the object (p) for that particular f. They are two distances beyond 2f , one position at 2f and two positions between 2f and f.  Set the two  distances of p beyond 2f and two distances between 2f and f in such a way the image is still visible on the right of your screen and whose image distance from the lens, q, can be measured.  For each of these positions record the distance for both p, the object distance and q, the image distance .  The next column should record your mathematically derived value of q based on the lens equation.

Two columns are also  provided to allow for a description of the image size relative to the object ( Much Bigger,  Bigger,  Same,  Smaller,  Much Smaller,  None)  and orientation (Inverted, Upright)  when the object is between f and the lens. These need not be measured.

The four conditions are as follows.

Curvatures will be set at  0.5 and 1.0 meters

The Index of Refraction will be  set at 1.5 and 1.79

A sample table below is presented for the following condition.

Curvature = 0.5 meters and Index of Refraction=  1.5

Use this for your first run.

Enter the actual  measured value of q  and in the next column  record the calculated value of q.

Record  the Relative Size  of the image compared to the object and note its Orientation.  You need not show your calculations.

In the next three conditions, fill in the tables below. First measure f.  Then arrange the position of the pencil at beyond 2f, at  2f  halfway between f and 2f , and finally at f

Question 10:  How does q vary as p is varied?

Question 11:  How would you describe the  image size ( compared to the object size) as p changes both beyond 2f , at 2f and between 2f and f.

Question 12:  In general, how close were your calculated values of q to the measured values you obtained?

Question 13;  What happens to the image when p is at f. Why is that?

Again, set the Curvature = 0.5 m and Index =1.5. Place the object at 2f. Vary the size of the lens.

Question 14: How does changing the  diameter of the lens affect the image?  Why?

To create  virtual images in the Geometric Optics app,  click the box labelled “Virtual Image” under Change Object. Set the Curvature = 0.5 meters and Index = 1.5.

Place the object  just in front of f between f and the lens. Slowly move the object in small increments towards the lens. Record your observations.

Question 15; This is a bit more complex.  Why do you perceive an image at all if in fact there is no image to be found on the screen? Here, you have to throw in some Psychology and Neuroscience (as well as Art Perspective ) to really get the correct interpretation.

Now create a virtual image with a double concave lens.

Go back to the Physics Classroom website  for Refraction and Lenses.

https://www.physicsclassroom.com/Physics-Interactives/Refraction-and-Lenses/Optics-Bench/Optics-Bench-Refraction-Interactive

Once you are in the app, select Diverging at the top of the screen.

You can enlarge the box by dragging a small arrow at the right bottom corner of the box.

Select the arrow as the object ( not the candle).

Keep f constant at 20 centimeters. Move the arrow to three positions.

Question 16:  How does the virtual image created here compare to the virtual image created by a double convex converging lens?

Note the distance/height data available on the right side of the box. Place the object at a point beyond f. Calculate and record the two ratios. (ignore the sign)

One ratio is:                  Object Distance/Image Distance .

The other ratio is:        Object Height /Image Height.

Question 17 : How do these two ratios compare to each other?

Do this for a second object position as well and record your values and again compare their values.

Now confirm the mathematical  foundation for this. Here, the lens equation is a bit different.  Since the focal length and image are on the left side of the converging lens, both are given negative values.

1/-f  = 1/p + 1/-q

Select one position for p. Use the f value of 20cm. Calculate the value of q (image distance) from the given values of  f and p.

Question 18;  How close is the  calculated value of q to the listed value on the table?

Ray Diagram

Now draw one Ray Diagram of a convex lens ( converging) with a ruler. Let f be 20 centimeters as the default position of the app.  Place your 10 centimeter arrow upright at 43.8 centimeters from the midline of the lens.  Draw your ray diagram to scale.  One centimeter on paper equals two centimeters of the app. The bottom of the arrow should rest on the reference line. Follow the three rules of drawing ray diagrams (taken from the Physics Classroom)

• Any incident ray traveling parallel to the principal axis of a converging lens will refract through the lens and travel through the focal point on the opposite side of the lens.
• Any incident ray traveling through the focal point on the way to the lens will refract through the lens and travel parallel to the principal axis.
• An incident ray that passes through the center of the lens will in effect continue in the same direction that it had when it entered the lens.

Assume that the ray coming from the bottom of the upright arrow will continue along the reference line to the right.

Carefully draw and extend three rays from the tip of the arrow. The first, parallel to the reference line is drawn to the lens (the height of the arrow tip just passes through the top of the lens) and bent accordingly. The second is drawn from the arrow tip to f and continues to the lower edge of the lens and bent. The third is drawn from the arrow tip to the center of the vertical line in the lens ( considered the center of curvature) and extended in a straight line beyond.

Hopefully, if drawn carefully enough the three rays converge at a common point, which would represent the tip of the arrow image.

Question 19: Does the ray diagram work?

Last question 20:  Does the calculated value of q in your ray diagram match the actual measured value of q in your ray diagram?  % error?