Monday, January 30, 2012

The Simple Lens

Knowledge of the action of a simple lens helps us to understand how optical instruments work. We start with the definition of the focal length, f, which is the distance along the optical axis from the plane of a converging lens to the point where rays initially parallel to the axis meet. The path that the light takes is reversable so a ray passing through the focal point will end up traveling parallel to the optical axis. To determine how the lens will produce an image of an object we draw an arrow of height, h, on the diagram a distance, o, to the left of the lens plane. Using the rules for the rays passing through the two focal points we find that they meet at a point which is a distance, i, from the lens plane and a distance, y, below it.

Given that o and h are known we can solve for i and y. On both sides of the lens the two pair of triangles formed by the rays passing through the focal points are similar so their sides are proportional. Using these triangles we get two equations involving the two unknowns, i and y. Solving each for y and equating the results gives an expression which reduces to the thin lens formula, 1/i + 1/o = 1/f. We can use this formula to find i/f in order to simplify the first expression for y. The result tells us that the two ratios i/o and y/h are equal. These ratios give the quantity, m, by which the object is magnified to produce the image. The quantity, m, is referred to as the magnification.

These ratios also tell us that a ray passing through the optical axis at the plane of the lens will not be deflected so a ray from the tip of the object passing through center of the lens will also pass through the tip of the image. This fact was used to measure the altitude of the Sun at the Summer Solstice while using a single object lens from a small telescope.

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