Physics · Optics · Grade 9

Optical instruments

How can a microscope with two lenses magnify a cell 400 times?

The situation

In a biology lesson the students looked at onion-skin cells under a microscope: cells invisible to the naked eye appeared as big “bricks”. In the evening, the school telescope showed the craters of the Moon. Both instruments have two lenses each — so why does one magnify a tiny thing close by, and the other something huge far away? And why does an ordinary lens make the edges of an image blurred and coloured?

Topic

Optical instruments. Magnifying glass, microscope, telescope and projector: how an image is enlarged

Key ideas

Magnifying glass and microscope
Telescope
Aberration

Formulas

Γ_{mag} = \frac{d_{0}}{F}, d_{0} = 25 cm

Angular magnification of a magnifying glass (image at infinity, eye relaxed).

Γ_{micro} = \frac{Δ}{F_{obj}} \cdot \frac{d_{0}}{F_{ep}}

Microscope: the product of the magnifications of the objective and the eyepiece; $Δ = 16$ cm — the optical tube length.

Γ_{tel} = \frac{F_{obj}}{F_{ep}}, d_{exit} = \frac{D}{Γ}

Telescope: angular magnification and exit pupil. Kepler — inverted image, Galileo ( $F_{ep} < 0$ ) — upright image.

\frac{1}{F} = \frac{1}{d} + \frac{1}{f}, δ = \frac{D | f - f_{0} |}{f}

Camera and projector: the thin-lens formula and the blur circle of a point away from the focus ( $D = F / N$ — the aperture opening).

n = \frac{d}{h}

Glass plate: the true thickness $d$ and the shift $h$ when the microscope focus is moved from the upper mark to the lower mark (apparent thickness).

Theory

The eye and angular magnification

The eye is an optical system: the cornea and the lens form a real, reduced image on the retina (for more, see the “The eye and spectacles” experiment). How large an object looks is determined by its visual angle. If we bring an object closer to the eye than 25 cm, the eye cannot focus; a magnifying glass places the object near its focus and gives a large virtual image: $Γ = 25 cm / F$ . A magnifying glass of focal length 5 cm magnifies ≈ 5 times. The dashed lines on the stage are the extensions of the rays that lead to the virtual image.

Microscope

The objective (small $F_{obj}$ ) forms an enlarged, real, inverted image of the object inside the tube; the eyepiece magnifies it again like a magnifying glass: $Γ = (Δ / F_{obj}) (25 cm / F_{ep})$ . In the laboratory the magnification is determined with a stage micrometre (1 mm = 100 divisions): one eye looks into the microscope, the other at a ruler at 25 cm. The “10× · 10×” on the rim is a nominal value: the tube length and focal lengths differ by a few per cent in manufacture, so the true $Γ$ is measured. The microscope can also measure the refractive index of a glass plate: the lower mark is seen through the plate at a depth $d / n$ . Because of the wave nature of light, a microscope cannot resolve anything smaller than ≈ $0.61 λ / A$ (≈ 0.2 µm) — this is the resolving power ( $A$ is the numerical aperture).

Telescopes

In the Kepler telescope both lenses are converging and the image is inverted; in the Galileo telescope the eyepiece is diverging, the image is upright and the telescope is short (theatre binoculars). The magnification is $Γ = F_{obj} / F_{ep}$ , but the main characteristic is the aperture $D$ : the light it collects depends on $D^{2}$ , and the angle it can resolve on $1.22 λ / D$ . Large telescopes use mirrors (no chromatic aberration); they stand on an altazimuth or equatorial mount. Solar telescopes work by filtering the light, and adaptive optics corrects the wavefront distorted by the atmosphere with a flexible mirror. Radio telescopes and radio interferometers observe in radio waves, and space, X-ray and gamma-ray telescopes observe in the rays the atmosphere does not transmit.

Camera, projector and technical instruments

The lens of a camera casts a real, reduced image onto the sensor; focusing means moving the lens. The f-number is $N = F / D$ : the larger $N$ , the less light, but the greater the depth of field. The lens of a projector casts an enlarged, inverted image of a slide, placed slightly beyond the focus, onto the screen — which is why the slide is put in upside down. Binoculars (prism telescopes), the periscope, the endoscope and spectral instruments (prism and diffraction — they split light into a spectrum; their resolving power is $R = λ / Δ λ$ ) are all assembled from these same lenses and mirrors. On the stage the camera and projector lenses are taken to be ideal.

Aberrations and how to reduce them

A lens with spherical surfaces refracts the rays at the edge more strongly than those near the centre — spherical aberration (calculated exactly with Snell's law on the stage): a diaphragm reduces it, and an aspheric (hyperboloid) surface removes it. In the laboratory it is measured with zone diaphragms: a central opening lets through only paraxial rays, and a ring lets through the zone at height $h$ ; in each case the screen is moved to where the spot is smallest, and the longitudinal spherical aberration $Δ s = x_{centre} - x_{zone}$ is found — it is roughly proportional to $h^{2}$ . The $n$ of glass depends on colour — chromatic aberration: blue light is focused nearer. An achromat made of crown and flint glass brings the red and blue foci together. In an oblique beam coma (a comma-shaped spot) appears, as well as astigmatism and distortion (bending of straight lines); multi-element lenses correct them. In telescopes a mirror is free from chromatic aberration, and a paraboloid mirror from spherical aberration.

The atmosphere and astronomical observation

The atmosphere is transparent only in “windows” — visible light, part of the infrared and radio waves; starlight is more attenuated the closer it comes to the horizon. Turbulence in the air makes a star twinkle and smears the image — astronomers call this seeing; the astroclimate of a site is judged by it. For the photoelectric measurement of luminous flux (photoelectric photometry), the filters of the UBV system and spectroscopy, see the “Luminous flux and illuminance” and “Atomic structure and spectra” experiments.

Did you know?

The Maidanak Observatory in Uzbekistan (Kashkadarya region) is one of the places with the best astroclimate in the world: according to measurements, the median blurring of a star image by the atmosphere (seeing) there is about 0.69 arcseconds.
The 2.4-metre mirror of the Hubble Space Telescope, launched in 1990, was ground wrongly at its edge by about 2 micrometres — the images came out blurred because of spherical aberration. In 1993 astronauts installed corrective optics.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

More experiments in this section

Car mirrors and the periscope — Grade 7
Laser, water and optical fibre — Grade 7
The eye and spectacles — Grade 7
Shadow and eclipses — Grade 7
Measuring the speed of light — Grade 9
Interference and diffraction — Grade 11
Polarisation of light — Grade 11
Laser physics and holography — Grade 11
Luminous flux and illuminance — Grade 11
Infrared, UV and X-rays — Grade 11