Magnifiers, Microscopes and Telescopes

A microscope uses an intermediate image. Its objective forms a real, enlarged, inverted image of the specimen. The eyepiece then treats that image as its object. In the simplified two-converging-lens arrangement, placing the intermediate image at the eyepiece's object-side focal plane gives a final image at infinity. Suppose the objective's linear magnification is -16 and the eyepiece's angular magnification is 8. The total microscope factor is their product, -128. This is an angular comparison with unaided viewing of the specimen at the chosen reference near point. It is not the height ratio of two finite images. The negative sign records inversion. For a telescope viewing a distant object, the comparison is instead between that object's angles with and without the telescope. In the two-converging-lens infinity arrangement, objective and eyepiece focal lengths of 120 cm and 4 cm give a magnification magnitude of 30. A reflecting telescope replaces the lens objective with a curved mirror. It still forms an intermediate image that an eyepiece can view.

Before calculating magnifier angular magnification, choose the comparison distance D and the final image position. D is the reference near-point distance for unaided viewing. Using D = 25 cm is a convention; actual near points differ. For a final image at infinity, place the object at the converging lens's focal plane. Rays leaving from each object point are parallel, and the small-angle formula is M = D/f. For a finite virtual image, place the object inside the focal length. If the lens is close to the eye and the virtual image is at the near point, the formula becomes M = 1 + D/f. For example, a 10 cm focal-length lens gives 2.5x in the infinity arrangement when D = 25 cm. Under the stated near-point conditions, it gives 3.5x. The lens has not changed; its viewing arrangement has. A microscope eyepiece uses these same principles on the objective's intermediate image. State the image position before choosing its magnification formula.

Light entering the eye is focused by the cornea and lens onto the retina. The retinal image is real and inverted. The eye changes its lens shape to focus objects at different distances; this adjustment is called accommodation. The near point is the closest distance at which the eye can focus clearly. It varies between people and can change with age. A stated reference of 25 cm lets an optics calculation compare magnifiers on the same basis without claiming that every eye has that near point. Moving an object closer increases the angle it occupies at the eye, but unaided focusing has a limit. A magnifier allows a closer object while presenting an image the eye can focus on. Its angular magnification is compared with unaided viewing of the object at the reference near point. A telescope uses a different reference: the same distant object viewed without the telescope. Its angular magnification does not require a near-point distance.