Magnifying glass, microscopes and telescopes

Physics · Optical instruments. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

The closest distance at which a normal eye sees an object clearly without strain is the least distance of distinct vision, d = 25 cm (the near point). An object looks bigger when it subtends a bigger angle at the eye. Angular magnification is the angle subtended by the image seen through the instrument divided by the angle subtended by the object placed at the near point and seen directly.

A simple microscope is one convex lens with the object inside its focal length, giving a virtual, erect, enlarged image. A compound microscope uses a short-focus objective to form a real enlarged image, which the eyepiece magnifies again. The final image is virtual and inverted. Resolving power is the ability to show two close objects as separate.

Rules to remember

Simple microscope (image at near point): M = 1 + d/f, with d = 25 cm
Compound microscope: M = (q/p)(1 + d/f_e) for the objective's image and object distances, or about (L/f_o)(1 + d/f_e), where L is the length of the microscope
Astronomical telescope in normal adjustment: M = f_o/f_e
Length of the telescope in normal adjustment = f_o + f_e; final image at infinity and inverted
Telescope: long-focus, wide objective and short-focus eyepiece. Microscope: both lenses of short focal length
Smallest angle resolved: α_min = 1.22λ/D; resolving power R = 1/α_min, so a wider aperture D and a shorter wavelength resolve better
Resolving power of a grating: R = λ/Δλ = N × m (N lines, order m)

Common mistake

Forgetting the "1 +" in M = 1 + d/f for a magnifying glass with the image at the near point, or turning the telescope formula upside down. The telescope objective has the longer focal length, so f_o/f_e is greater than 1.

Worked example

An astronomical telescope in normal adjustment is 63 cm long and has a magnifying power of 20. Find the focal lengths of its two lenses.

M = f_o/f_e = 20, so f_o = 20f_e.
Length = f_o + f_e = 20f_e + f_e = 21f_e = 63 cm, so f_e = 63 ÷ 21 = 3.0 cm.
f_o = 20 × 3.0 = 60 cm.

Answer: Objective 60 cm, eyepiece 3.0 cm.

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