Example 1.) The positive value for the image distance places it on the far side of the
lens, which means the image is real. These values also confirm the answer to a
question posed in an earlier section, where the nature of the image of an object
between F and 2F was determined using ray tracing.
The formulas in Equation 2 deal with lateral magnification, which is defined in the same
way for lenses as it is for mirrors: the ratio of the image height to object height. It can
also be computed using distances: The lateral magnification equals the negative of the
ratio of image distance to object distance. This is the second equation. For instance, in
the example just discussed above, this ratio is í2.0: the image is twice as tall as the
object. The negative sign means the image is inverted.
The equation in Equation 3 is the lensmaker’s equation. It relates the focal length of the
lens to some of its physical properties, specifically its index of refraction and the radius
of curvature of each surface of the lens. The equation enables a lensmaker (or physics
student) to calculate the focal length for a lens that has differing curvatures on the two
sides of the lens, or to create a lens with a particular focal length.
We show the version of the equation for a lens in air. If the lens is immersed in another
substance, say water, then the variable n has to be replaced by the ratio of the index of
refraction of the lens to the index of the surrounding material (for example,
n = nlens/nwater). The object side of a microscope objective lens is often immersed in a
special optical oil, as with the German “Oel Immersion” objective lens shown below.
The equations can be tricky to apply because various values like image distance can be
either positive or negative. You need to use the table in Concept 1 above, or perform
the more difficult task of memorizing the conventions.
To review perhaps the trickiest sign convention, the radius of curvature R in the
lensmaker’s equation is negative when the center of curvature is on the near (the
object) side of the lens and positive when it is on the far side. If this sounds confusing,
consider each lens surface as part of a sphere. If the sphere’s center of curvature is on
the object side, so the sphere would enclose an object relatively near the lens, then R is
negative.
With a converging lens that is convex on both sides, like the ones used in the
illustrations to the right, the radius of curvature for the near surface, R 1 , is positive, and
the radius of curvature for the far surface, R 2 , is negative. As the diagram illustrates,
with a lens that is convex on both sides, the center of curvature for each surface is on
the opposite side of the lens from the surface. With a diverging lens made up of two
concave surfaces, the signs are reversed and the centers are on the same side as the
lens surfaces. Remember, we said this was tricky!
There are additional conventions: Focal lengths are positive for converging lenses and
negative for diverging lenses. Object distances are positive when the object is real and negative when it is virtual. (Virtual objects can arise
when there are multiple lenses. A virtual object is on the side of a lens opposite to the source of the light.) Image distances are positive when
the image is real and negative when it is virtual. Magnification is positive for an image that is upright relative to the object and negative for one
that is inverted relative to the object.