The focal length of the lens used in a camera affects two aspects of the image: the field of view and the magnification.
The field of view is the size of the object that can be contained on the film and the magnification is the ratio of the size of the image to the size of the object that produced it. These two quantities are related to each other – increasing the magnification while leaving the size of the image unchanged inevitably decreases the field of view (since the increase in the magnification implies that the same sized image will be produced by a smaller object), and vice-versa.
The field of view is usually expressed as an angle. There are a number of different ways of estimating this quantity, but all of them yield about the same value. Since we are going to use the approximation that the camera lens is a simple single thin lens, the calculation will not be exactly correct anyway. As with previous calculations, the agreement between this simple model and a real-lens is surprisingly good.
If the object is not too close to the lens, then the image is always formed approximately at the focal distance back from the lens. The size of the image on the image plane is usually limited by the size of the film, and the angular size of the field of view is given by those rays that enter the lens at its center (and so are not deviated) and which then strike the edges of the film as shown in the following figure:
The angles on the object and image side of the lens are the same, so that the tangent of one-half of the angle of the field of view is approximately the ratio of the half-width of the film to the focal length of the lens. For example, the half-width of 35 mm film is 18 mm. The horizontal field of view of a 50 mm lens is then twice the arc-tangent of 18/50 or about 40 degrees. The film is not square, so that the field of view in the vertical direction is not the same. The half-height of the film is only 12 mm, so that the vertical field of view is twice the arc tangent of 12/50 or about 27 degrees.
(Another method for calculating the field of view would be to consider those rays that pass through the first focal point on the object side of the lens. These rays are refracted so that they emerge parallel to the axis on the image side of the lens. The angular spread of these rays provides an alternate method for estimating the field of view. This estimate is essentially the same as the estimate using the method described above unless the object is quite close to the lens. The two estimates differ in this situation because the lens must be moved away from the film plane in this situation so that the distance from the lens to the film in the first figure is somewhat greater than the focal length.) See the following figure, which compares the field of view calculated in this way with the calculation above when the object is far away.
The field of view angle decreases with increasing focal length, but the relationship is through the arc tangent function and is not exactly linear (except at very small angles). The horizontal field of view of a telephoto lens whose focal length was 135 mm would be twice the arc tangent of 18/135 or about 15 degrees.
The magnification of a lens can be estimated using the same sort of analysis. The following figure shows the image produced by an object.
The two shaded triangles are similar to each other, so that the corresponding sides are proportional. The magnification is the ratio of the size of the image to the size of the object that produced it. Since the triangles are similar, that ratio is also equal to the ratio of the distance between the lens and the image (“I” in the figure) and the distance between the lens and the object (“O” in the figure). Since the distance “I” is approximately the focal distance, the magnification of a lens for a fixed object depends directly on the focal length and is equal to the ratio of the focal length to the distance of the object from the lens.
The magnification increases directly with the focal length when the object, the position of the camera, etc. are unchanged. Thus the same object appears larger when imaged by a camera with a telephoto lens because the focal length of that lens is larger than the focal length of a normal lens.
Although a film image is purely two-dimensional, the variation in the magnification of a lens with object distance can distort perspective – that is, how we estimate the size of something in an image by comparing it to other things in the same image. For example, if an object is sufficiently large so that the distances between different parts of the object and the lens are significantly different, then the parts that are closer to the lens will have a greater magnification than the parts that are further away. This effect can be seen when photographing a tall building from a point near its base. The bottom of the building is significantly closer to the camera than the top, is magnified by a larger amount and appears disproportionately too large as a result.