OBJECTIVE

Compare the observed and predicted behavior of the image of a positive lens. Determine focal length and f-number of the lens and explore object distance as a function of image distance.

Use two lenses to function as a simple telescope; determine theoretical and observed magnification.

Use a ready-made refracting telescope to observe a distant object with 2 different magnifications. Compare calculated and observed magnification.

THEORY

The interface between air and a laboratory optical lens is modeled as a spherical refracting surface. A converging (positive) lens is thicker in the middle than at the edges. If such a lens is symmetric, then a light ray falling on it meets a convex surface. On entering, the ray meets a higher refractive index, bending toward the line normal to its point of entry.

On leaving, the ray crosses a concave boundary. Meeting a lower refractive index, the light bends away from the normal at that point. This occurs with all non-axial rays traversing the lens. The net effect is convergence toward the optical axis - as in the following diagram, which poses a light source and film aligned on the optical axis of the lens.

Note that the light rays generated from a point source will spread and then be converged by the lens. If the object distance p is greater than the lens focal length f (see discussion below), a real inverted image will form on the film surface.

Focus and focal length

Light from a far-away source (e.g., starlight) strikes the lens with nearly parallel rays. The lens will converge all the rays from that source through a single region on the central axis of the ray system. If that axis corresponds with the lens optical axis (i.e., the starlight is approaching "dead on"), the focus will occupy a region centered on the common axis. As the distance p to such a source approaches infinity, the region shrinks to a point on the axis line. We call this the focal point of the lens. Its distance from the lens is the lens parameter f, focal length.

Under these conditions (infinite p and axial source), a real image formed on the axis at this distance will be most sharply defined, and less well defined elsewhere.

For some purposes (see Telescope discussion below) we consider a converging lens to have two focal points located on the optical axis of the lens, equidistant from the lateral axis of the lens.

The f-number of a lens is found by dividing its focal length by its light aperture diameter. For a simple converging lens having diameter d, this ratio is

Thin lens equation

In this course we will not be sending all light sources to infinity before refracting their rays. A practical lens will intercept light rays at oblique angles to its axis. If we wish to know how far to place the film from the lens in order to obtain a sharp image, we have to deal somehow with the angles of rays that will fall on the lens.

As the above figure suggests, the more distant the source, the smaller will be the angles made by its rays with the optical axis. Some rays from a very close point source will fall on the lens at angles closer to 90º. The range of angles impinging on the lens from a given point source, is a nonlinear function of source distance p.

The lens will bend rays of a given angle predictably, and this behavior is indexed by its focal length f. The lens refracts a system of rays from a point source, in a manner that depends on their range of approach angles. For a given range of ray angles, the converging lens will concentrate those rays in a region whose distance from the lens is a nonlinear function of focal length f. This distance is the best place to place the film in order to obtain a sharp image of the object.

The relation between p, f, and i for a converging thin lens is:

(In reciprocal land, you can find the focal length by adding the object distance to the image distance.)

Telescope

The telescope uses two converging lenses (or lens systems, for practical telescopes). An objective lens or system refracts light from the object to form a real image. The eyepiece lens refracts light from the real image and forms a virtual image seen by the viewer. It is necessary for the focal points of both lenses to coincide. (See Fig. 35-18 in our text.)

The lateral magnification of a telescope is found by dividing the focal length of the eyepiece into the focal length of the objective.

The magnification is negative because the image is inverted.

PROCEDURE

An incandescent light source was fixed at one end of a rail. A 50mm x 150mm converging lens was positioned on the rail, so that the light source (object) was outside the focal length of the lens. A thin acrylic sheet was placed opposite the lens from the object, in order to pick up the refracted image.

We moved the lens fixture along the rail in increments, repositioning the film to achieve a focused image. We recorded each new lens position and its corresponding new image position. These data were used to generate a graph of i (image distance) vs. p (object distance), and a graph of (1 / i) vs. (1 / p).

We used the same 150mm focal length lens, together with a 50mm lens, to set up a quick and easy two-handed telescope. We practiced focusing on visible lab features, comparing sizes of the images viewed with and without the "scope".

We also used a telescope to view features inside and outside the physics lab, comparing image sizes to determine magnification factor.

DATA and OBSERVATIONS

We recorded p and i for 22 positions of the lens and film plate relative to the stationary light source. (See data table and Results section).

In preliminary testing we defined focal image position as being within the smallest discernible interval within which moving the plate fixture slightly along the rail would neither cause a certain portion of the image pattern to become "fuzzier", nor cause the "aura" around that portion of the image to "bloom".

Our first few data points came when p was small. We had looked for the appearance of an image as the lens moved just away from its focal point distance to the light source. In this region the image was present, but not sharply defined for any lens-film distance. As p increased, the image quality improved.

The rail apparatus was in two parts, which created an abrupt transition in the look of the image as the film plate moved from one rail to the other. This occurred within the region where small changes in p were associated with large changes in i. A second abrupt transition came as the lens moved from one rail to the other. This is associated with a wrinkle in the i vs. p curve when i was about 16.7 ( [1 / i ] was about 0.06).

The most fun of all was trying to make a transient telescope by holding two lenses in the air. With some practice and careful choice of object, it became possible to compare the virtual image size to the image viewed by unaided eye.

We had less success on that account, when using the laboratory telescope. Any size object we could fit into the eyepiece, was still too small to be compared accurately - the unaided eye's image size was one millimeter or less with both choices of eyepiece.

But it was a clear day for viewing Mt. Tam.