Physics 301 Fall 2001

Imaging with a Lens

Introduction

We will explore the image-forming properties of lenses. Thin lenses can be grouped into two categories, positive (converging) lenses, and negative (diverging) lenses. A positive lens is one that causes incident parallel rays to converge at a focal point on the opposite side of the lens. A negative lens is one that causes incident parallel rays to emerge from the lens as though they emanated from a focal point on the incident side of the lens. The two types of lenses are illustrated below.

Use the mouse to determine the position of the lens and of the focal point. The
difference in these positions should be the focal length. Select the lens by
clicking on it: the focal length *f* will be shown. Is it about what you
expected?

The parallel rays in the figures above represent a *point source* infinitely
far away to the left. (This is a tricky concept! Think about it.)
More generally, light rays emanating from any point (the "object") on the central axis
will be brought to a focus (the "image", real or virtual) at some other point on the
central axis. If the object is not at infinity, but at the finite position p, then a
lens of focal length f forms an image at the position q determined by the
*thin lens equation*

Here it is understood that *p* is measured from the center of the lens, and is considered
positive if the object is to the left of the lens, and negative to the right. Similarly
*q* is measured from the center of the lens and is positive to the right of the lens,
negative to the left (opposite sign convention from *p*). What is *q* in the special
case that *p* goes to infinity? (Compare the figures at the top of this page.)

Experiment with the
setup pictured below
by dragging the object (source of the
light rays) with the mouse. Note that when the object is
off-axis the image is off-axis too. If the object is not too far off-axis, then the
distances *p, q,* and *f* are still given by the thin-lens equation. This
determines the horizontal position of the image. The vertical
distance of the image from the axis is most easily
determined by the special ray through the center of the
lens, which is undeviated: as you move the object vertically, the image also moves
vertically as if at the opposite end of a lever which pivots at the center of
the lens. Try to make sense of these statements in the figure below,
and in particular check the thin lens equation.

Drag the object around and notice how the image behaves. As the object
approaches the focal point of the lens, the image retreats to infinity (note
the parallel rays). What
happens if you bring the object inside the focal length of the lens? Something
very interesting! The image reappears as a virtual image on the same side
of the lens as the object (i.e., *q* is now negative: check how the
thin lens equation describes this). An eye looking in from the right would
receive rays *as if* from this image. It is called "virtual" because
you have to extrapolate the rays backward to find it: the light rays themselves
don't actually intersect to form a "real" image. By the way, in this configuration
the lens is acting as a magnifying glass: think why this is.

It makes sense to compare the size of the object and the image. We define the magnification M to be the ratio of the size I of the image to the size O of the object, but by similar triangles this is also the ratio of the image distance q to the object distance p:

**Procedure:**

**Determination of focal length:**

In this experiment you will measure the focal length of two converging lenses using two different methods. For each measurement, record the focal length along with estimated errors in your lab notebook. The experimental setup consists of an optical rail along which a light source, screen, and mounted lens can travel in a straight line. The rail is grated so that the object and image distances can be measured directly off the rail. The light source contains a white square with two crossed arrows and will serve as the object for the majority of the experiment.

Method 1:

Select two dissimilar converging lenses.
From the thin lens equation we know that the image distance, *q*, is approximately equal to the
focal length, *f*, when the object distance, *p*, is very large. We can use this observation
to approximate the focal length of a lens. With each lens, attempt to image a distant
object (the sun, overhead lights, distant buildings) and classify the lens as either
converging or diverging. Determine the focal lengths of the converging lenses by
measuring the distance between the center of the lens and the location of the image.
Which lens has the longer focal length? Which lens forms the larger image of a
distant object? Which lens forms the *brighter* image? What factors do
you think determine the brightness?

Method 2:

Place the light source at one end of the optical rail. Place the screen on the other
end of the rail and a converging lens between the light source and the screen. By moving
either the lens or the screen or both, focus an image of the crossed arrows on the screen.
Measure the object distance, image distance, object size and image size (and orientation).
Repeat for a total of 5 distinct object distances spread over a range of 1.5*f* to 4*f*.
Determine the focal length of the lens by making a plot of *1/q* vs. *1/p*. The
focal length of the lens can be determined from the equation of the best fit line. How
does this compare to your value from method 1? Which method gives more accurate results?
Calculate the magnification at each object distance using two different methods: first
using the image and object size, then using the image and object distances. How do these
two values compare? Repeat the above procedure for the second positive lens.
With an image focused on the screen, slide the screen back and forth along the optical
bench. Observe carefully how the colors of the border change as the image passes through
the point of focus. The index of refraction of the lens unfortunately
depends on the wavelength of the light. As a result the focal length of the lens,
ultimately determined by ray tracing and Snell's Law, is
slightly different for each wavelength. Since our white light source is made up of all
wavelengths, different colors are focused at slightly different image distances. This
is referred to as chromatic aberration of a lens.