Lenses and Lens SystemsGoals
• Familiarity with the meanings of the following optics terms: converging, magnification, object, image, focus, object distance, image distance, and focal length.
• To be able to construct ray diagrams for a single lens and a two lens system for converging lenses
• To establish that 1/i + 1/p = 1/f
• To use the above equation to measure the focal length of a lens
• To be able to design and construct a two lens system to achieve a desired magnification

Terminology

Figure 1.Ray diagram showing the image formation by a “thin” converging lens.

With a “thin” lens the ray paths are based on the central plane of the lens, the pecked line in the diagram.
The intersection of the rays on the right is used to predict the location of the top of the image.

Object distance, p, is the distance between the lens and the object.
Image distance, i, is the distance between the lens and the image.
Focal point, F, is shown. This is where light from an infinitely distant object would be focused.
Focal length, f, is the distance from the lens to the focal point. Lens strengths are known by their focal lengths.

Notice that the size of the image is different from the size of the object and that the image is upside down. Magnification, m, is the ratio of the image height to the object height and the value of m is negative if the image is inverted. The in Figure 1, m = –2.5.

Part I. Determining the Relationship Between p, i, and f (Object, Image, and Focal lengths)
• Setup the optical bench with a light source, just one lens, and a screen.
• Find positions of the equipment that give a clear sharp image on the screen.
• Measure the object and image distances, p and i, and record them in two columns in one row of a spreadsheet that is set up like Table 1 below.
• Record data for other object and image lengths that also produce sharp images.
• Use the Excel functions to calculate a values for 1/i and 1/p.
• Graph 1/i vs. 1/p, with a best fit straight line and equation of the line.
• Derive an equation relating i, p, and f.
o To do this you will have to interpret the physical meaning of the y intercept.

Write your empirical equation that links i, p, and f.

Assume that the magnification can be expressed as: m = -i/p

Eliminate i in your equation and solve for p in terms f and m: p =

Eliminate p in your equation and solve for i in terms f and m: i =

Part 2. Determining the Focal Lengths of Two Lenses
Use your graph intercept from part 1 to find the focal length of your lens. f1 =

Repeat the process for another lens and find its focal length. f2 =

Part 3. Theory

Confirm that your equation from part 1 is equivalent to the thin lens equation: 1/f = 1/p + 1/i (1)

Assumption: magnification can be expressed as: m = -i/p (2)

Confirm that your equation for p in terms f and m is equivalent to: p = f (1 – 1/m) (3)

Confirm that your equation fori in terms f and m is equivalent to: i = – f (m – 1) (4)

Part 4. Design and Construction of a Two Lens System
Construct an optical system that uses your two lenses to produce an upright image one-sixth the size of the original. Do this in two stages, the first lens producing a magnification of – 1/2 and the second a magnification of -1/3. The final magnification is the product of the magnifications of all the stages, thus mtotal = (- 1/2 ) x (– 1/3 ) = + 1/6

Use the value of magnification, m = – ½ , of the first lens and equation 3 to calculate the first object distance, p1. Then use equation 4 to find the image distance for the first lens. Temporarily locate the image using the screen.

Calculate p2 using equation 3 again and place the second lens at a distance p2 from the image formed by the first lens. Calculate the image distance for the second lens. Locate this image on the screen and check the sign and size of the total magnification of the system.
Figure 2.Sketch of the two lens system.