**Object Distance vs. Image Distance**

The relationship between the focal length of a thin lens, its distance to an object and its distance to the image formed by the lens is given by the thin lens equation:

1/s’ = 1/s +1/f

where “s” is the object distance, “s'” the image distance and “f” the focal length. The derivation of this equation and related equations which follow can be found in basic text books on geometrical optics, e.g., Ref.1.

The set of ray diagrams below illustrates the relationship between an object and its image for a wide range of object distances for both a positive and a negative lens. Object and image distances are expressed in terms of a unity focal length lens and extend from zero to infinity.The magnification between object and image i.e., the ratio of the image distance to object distance, is also given.

Rays travel from left to right in all cases. Distances to the left of the lens are negative; those to the right are positive. Real objects lie to the left of the lens; virtual objects to the right. Real images lie to the right of the lens; virtual images to the left. Erect images are formed when the magnification is positive; inverted images result when the magnification is negative.

**Object size vs. Image Size**

The object and image sizes are related as follows:

m = y’/y = s’/s

where m is the magnification, y’ is the image height, y is the object height, and s’ and s are the image and object distances, respectively.

For an infinite object distance, the image size is determined by:

y’ = f tan α

where f is the focal length of the lens and α is the angle subtended by the object to the axis of the lens.

**Change in Image Distance vs. Change in Object Distance: Longitudinal Magnification**

The change in image distance for a change in object distance can be determined from the derivative of the thin lens equation given above:

ds’ = m² ds

where ds’ is the change in location of the image and ds the change in location of the object. The term m² is the* longitudinal *magnification of the lens and is the squared value of the lateral magnification m previously presented.

**Alternate Thin Lens Equation**

An alternate equation relating lens object and image distance can be derived from the above thin lens equation by expressing the object and image distances in terms of the lens focal length:

s = x – f and s’ = x’ + f

where the object and image distance components are taken with respect to the lens focal points in lieu of the lens. Values for x and x’ are negative when they lie to the left of the focal points; positive on the right. The relationships are illustrated below:

Substitution of the above terms into the thin lens equation yields an alternate thin lens equation:

x x’ = -f²

Substitution of these terms into the magnification equation yields:

m = f/x or m = – x’/f

An example appears below: