Optical Instruments: Cameras, Microscopes, Telescopes

Optical instruments use lenses and mirrors to form images: a camera forms a real, inverted image on a sensor, a microscope magnifies small nearby objects with an objective and an eyepiece, and a telescope forms a magnified image of distant objects, with an angular magnification M = f_objective ÷ f_eyepiece. In brief: the camera focuses by changing the lens-to-sensor distance, and the f-number (focal length ÷ aperture diameter) controls the brightness. In a compound microscope, the total magnification is the product of the objective and eyepiece magnifications. Telescopes collect light, so a larger aperture gives brighter and sharper images.

Cameras, microscopes and telescopes extend human vision to the very small, the very distant and to recording. This guide describes how each instrument works, gives the key formulas, compares the instruments in a table, works through numerical examples and corrects common mistakes.

How does a camera work?

A camera has a converging lens (or a system of lenses), an adjustable aperture, a shutter and a light-sensitive sensor (or film). The lens forms a real, inverted, reduced image of the scene on the sensor. To focus on objects at different distances, the lens is moved: for a distant object, the sensor is at the focal length; for a nearer one, the lens is moved away from the sensor (1/f = 1/d_o + 1/d_i). The aperture, with diameter D, controls how much light enters and the depth of field; the f-number N = f ÷ D (written f/N) gives the brightness of the image: each step of √2 in N (f/2.8, f/4, f/5.6, …) halves the light. The shutter speed sets the exposure time. A longer focal length gives a narrower field of view and a larger image of a distant object (telephoto), and a shorter one gives a wider view.

How does a compound microscope work?

A compound microscope has two convex lenses. The objective (short focal length, a few millimeters) is placed just beyond its focal point from the specimen, and forms a real, inverted, magnified image inside the tube. The eyepiece (focal length of a few centimeters) acts as a magnifying glass for that image, producing a virtual, further magnified image at the near point or infinity. The total magnification M = M_objective × M_eyepiece. For a tube length L between the lenses, M_obj ≈ L ÷ f_obj, and M_eye = 250 mm ÷ f_eye (with the final image at infinity, using the conventional near point of 250 mm). The useful magnification is limited by the wavelength of light: optical microscopes can resolve details of about 200 nm (0.2 µm), so higher magnification does not show more detail.

How does a telescope work?

A refracting telescope uses a long-focus objective lens to form a real image of a distant object at its focal plane, and a short-focus eyepiece to magnify it. In normal adjustment, the final image is at infinity, the lenses are separated by f_o + f_e, and the angular magnification is M = f_o ÷ f_e. A reflecting telescope uses a concave mirror as the objective (Newtonian design: a flat secondary mirror sends the light to an eyepiece at the side), which avoids chromatic aberration and allows much larger apertures. The aperture D determines the light-gathering power (proportional to D²) and the resolution, with the smallest resolvable angle about 1.22 λ ÷ D. Binoculars are two small refracting telescopes with prisms that make the image upright. A “10 × 50” binocular has a magnification of 10 and objective diameter 50 mm.

Instruments compared

Worked example: magnification and exposure

1. Microscope. The objective has f = 4.0 mm, the tube length is 160 mm, and the eyepiece has f = 25 mm. M_obj = 160 ÷ 4.0 = 40. M_eye = 250 ÷ 25 = 10. Total M = 40 × 10 = 400. A cell 0.025 mm across looks like an object of 0.025 × 400 = 10 mm at the near point.
2. Telescope. An objective of f_o = 1,000 mm and an eyepiece of f_e = 25 mm give M = 1,000 ÷ 25 = 40. The tube length is 1,000 + 25 = 1,025 mm. The Moon, with an angular diameter of 0.5°, would appear 20° wide.
3. Light gathering. A telescope with an aperture of 200 mm collects (200 ÷ 7)² = 816 times more light than the dark-adapted pupil of the eye of 7 mm.
4. Camera f-number. A lens with f = 50 mm and aperture diameter 25 mm has N = 2.0 (f/2). Stopping down to f/4 reduces the diameter to 12.5 mm and the area to one quarter, so the exposure time must be 4 times longer for the same brightness (for example 1/250 s instead of 1/1000 s).
5. Field of view. A sensor 36 mm wide with a 50 mm lens: the angle of view = 2 tan⁻¹(18 ÷ 50) = 2 × 19.8° = 39.6°. With a 200 mm lens: 2 tan⁻¹(18 ÷ 200) = 10.3°.

How were these instruments invented?

Dutch spectacle makers built the first telescopes and microscopes about 1590 to 1608. Galileo made improved telescopes in 1609 and discovered the moons of Jupiter and the phases of Venus. Robert Hooke published Micrographia in 1665 with drawings from his microscope, and introduced the term “cell”. Antonie van Leeuwenhoek observed bacteria with simple single-lens microscopes in the 1670s. The first photograph from a camera was taken by Nicéphore Niépce around 1826, and digital sensors replaced film in the 2000s.

Which mistakes are common?

Mistake 1: Thinking that a higher magnification always gives a better image

Resolution is limited by the wavelength of light and the aperture. Beyond this limit, the magnification only enlarges blur.

Mistake 2: Confusing the telescope’s aperture with its magnification

The aperture determines the brightness and resolution. The magnification is f_o ÷ f_e.

Mistake 3: Saying that the camera image is upright

The lens forms an inverted real image. Cameras and phones present it the right way up.

Mistake 4: Using the same formula for angular and linear magnification

Telescopes give angular magnification (the angle seen). Microscopes and cameras are described by linear magnification.