In this section, we discuss the magnifying power of microscopes. We set aside the definition used with simple magnifiers and telescopes, which was based on the ratio of the angular size of the image created by the optical instrument to the angular size of the object viewed without assistance. We do so because the naked eye angular size of an object under a microscope is near zero.

The definition used for the magnifying power of a microscope is the product of the lateral magnification associated with the objective lens and the angular magnification of the eyepiece. Microscopes often have three objective lenses on a rotating stage and a set of interchangeable eyepieces. The manufacturer labels the power of the objectives and the eyepieces. Multiplying an objective and eyepiece value yields the magnifying power of that combination.

A microscope’s objective lens produces a real image whose lateral height is greater than the lateral height of the object by a certain factor. The ratio of these lateral heights defines the magnification provided by this lens. The image produced by the objective lens is further magnified by the eyepiece. Since the virtual image is essentially at infinity, the eyepiece magnification is measured in terms of an increase in angular size. This definition of magnification for a microscope is shown as the first equation on the right. To stress a point: it is the product of a magnification measured in terms of height (the objective lens factor) and magnification measured in terms of angular size (the eyepiece lens factor).

The second equation on the right provides a formula that allows you to approximate the overall magnification of a microscope based on physical characteristics of its lenses. This definition relates to the ray-tracing diagram in Equation 1, where the strength of each lens is defined by its focal point instead of its magnification. As you can see, the object is placed just outside the focal point of the objective lens, and the initial real image appears far from the objective, slightly inside the focal point of the eyepiece. The distances away from the two focal points are exaggerated in the diagram.

In the second equation, L is the distance between the two lenses and N is the near point of the human eye, with 25 cm a common value for this quantity. As the equation indicates, the magnification of the microscope increases with L, and is greater when the focal lengths of the lenses are short.

The following derivation shows that the computational formula on the right is equivalent to the definition.

Variables

Strategy

1. Use the lateral magnification formula for the objective lens, and two rough equalities from the diagram to get an approximation for m_ob.
2. Combine a simple-magnifier equation with the thin lens equation to state an approximation for M_ey.
3. Multiply the two approximations obtained in previous steps to obtain a formula for the overall magnification.

Physics principles and equations

The equation for lateral magnification is

The equation for the angular magnification of a simple magnifier is

The distance between the lenses of a microscope is L = d_i + D_o.

We will use the thin lens equation

Step-by-step derivation

The example to the right assumes a near point of 25 cm. The microscope has a magnification of −1500. The negative sign indicates the image is inverted.