Angle at the eye and the near point
The size of the image on your retina depends on the visual angle — the angle the object makes at your eye. Bring a coin closer and the angle grows, so it looks bigger. But a normal eye cannot focus closer than the near point, D = 25 cm. So without help, the biggest clear angle is when the object is at 25 cm.
An optical instrument makes this angle bigger. Its magnifying power is
m = (angle with the instrument) ÷ (angle without it, object at D or very far)
Simple microscope
A simple microscope (magnifying glass) is one convex lens of short focal length f. The object is placed inside F, close to the lens, and the eye is close to the lens on the other side.
Image at the near point (D)
v = −D. From the lens formula, m = v/u = 1 − v/f = 1 + D/f. This gives the largest magnification but the eye strains a little.
Image at infinity (relaxed eye)
The object is at F. The angle with the lens is h/f; without the lens (at D) it is h/D. So m = D/f.
A lens with f = 5 cm gives m = 6 (image at D) or 5 (image at ∞). Very short f lenses blur, so a single lens rarely goes beyond about 10×.
Compound microscope
A compound microscope has two convex lenses in a tube:
- Objective: very short focal length fₒ, close to the object. The object is just beyond Fₒ, so it forms a big, real, inverted image.
- Eyepiece: short focal length fₑ. It works like a simple microscope on that first image.
Magnifying power
m = mₒ × mₑ. The objective's magnification mₒ = vₒ/uₒ ≈ L/fₒ, where L is the tube length (distance between the second focus of the objective and the first focus of the eyepiece). For the final image at infinity, mₑ = D/fₑ. So
m ≈ (L/fₒ) × (D/fₑ) (image at infinity)
For the final image at D, mₑ = 1 + D/fₑ. The final image is inverted compared with the object. For high magnification both fₒ and fₑ must be small; fₒ < fₑ.
Astronomical (refracting) telescope
A telescope gives a bigger angle for far objects like the Moon. The objective has a large focal length fₒ and a large aperture (to collect more light). The eyepiece has a small focal length fₑ.
Parallel rays from the far object make an angle α. The objective forms a real, inverted image at its focus. In normal adjustment this image also sits at the focus of the eyepiece, so the final rays leave parallel at a bigger angle β, and the final image is at infinity.
m = β/α = fₒ/fₑ, tube length L = fₒ + fₑ.
If the final image is at D: m = (fₒ/fₑ)(1 + fₑ/D). A large objective aperture also gives a brighter image and finer detail (better resolving power).
Reflecting telescope
Very big lenses are heavy, sag under their own weight and split colours (chromatic aberration). So large telescopes use a concave mirror as the objective. In the Cassegrain design, a big concave primary mirror reflects light to a small convex secondary mirror, which sends it back through a hole in the primary to the eyepiece.
Advantages: no chromatic aberration, a parabolic mirror removes spherical aberration, and a mirror can be supported from behind, so it can be made very large. Magnifying power is still fₒ/fₑ, with fₒ = R/2 of the mirror.
Try it: a telescope from two lenses
Take a weak convex lens (like a +2 D reading-glass lens, f = 50 cm) and a strong magnifier (f = 5 cm). Hold the weak lens towards a far building, and look through the strong one behind it. Slide them apart until the view is sharp: the gap is about fₒ + fₑ = 55 cm, and the building looks about 10 times bigger and upside down. Never point it at the Sun. In the 3D, test which focal length you should change to get more magnification.
Key formulas and definitions
- Simple microscope: m = 1 + D/f (image at D); m = D/f (image at ∞)
- Compound microscope: m = mₒ × mₑ ≈ (L/fₒ)(D/fₑ)
- Compound microscope, image at D: m ≈ (L/fₒ)(1 + D/fₑ)
- Telescope (normal adjustment): m = fₒ/fₑ, L = fₒ + fₑ
- Telescope, image at D: m = (fₒ/fₑ)(1 + fₑ/D)
- D = 25 cm (least distance of distinct vision)
Worked examples
1. A magnifying glass has f = 5 cm. Find its magnifying power when the image is at the near point and at infinity.
Step 1: at D, m = 1 + D/f = 1 + 25/5 = 6. Step 2: at ∞, m = D/f = 25/5 = 5.
2. A simple microscope gives m = 11 with the image at D = 25 cm. Find its focal length.
Step 1: 1 + 25/f = 11. Step 2: 25/f = 10. Step 3: f = 2.5 cm.
3. A compound microscope has fₒ = 1 cm, fₑ = 5 cm and tube length 20 cm. Find m for the final image at infinity.
Step 1: mₒ ≈ L/fₒ = 20/1 = 20. Step 2: mₑ = D/fₑ = 25/5 = 5. Step 3: m = 20 × 5 = 100.
4. In a compound microscope, the object is 1.5 cm from an objective of fₒ = 1.25 cm. The eyepiece has fₑ = 5 cm and the final image is at 25 cm. Find the total magnification.
Step 1: objective: 1/vₒ = 1/1.25 − 1/1.5 = 0.8 − 0.667 = 0.133, vₒ = 7.5 cm. Step 2: mₒ = vₒ/uₒ = 7.5/−1.5 = −5. Step 3: mₑ = 1 + 25/5 = 6. Step 4: m = −5 × 6 = −30 (magnitude 30, inverted).
5. A telescope has an objective of focal length 100 cm and an eyepiece of 5 cm. Find the magnifying power and tube length in normal adjustment.
Step 1: m = fₒ/fₑ = 100/5 = 20. Step 2: L = fₒ + fₑ = 105 cm.
6. The same telescope (fₒ = 100 cm, fₑ = 5 cm) forms the final image at 25 cm. Find the magnifying power.
Step 1: m = (fₒ/fₑ)(1 + fₑ/D). Step 2: = 20 × (1 + 5/25) = 20 × 1.2. Step 3: m = 24.
7. A telescope with fₒ = 15 m and fₑ = 1 cm looks at the Moon (diameter 3.48 × 10⁶ m, distance 3.8 × 10⁸ m). Find the diameter of the Moon's image formed by the objective.
Step 1: angle of the Moon α = 3.48 × 10⁶ / 3.8 × 10⁸ ≈ 9.16 × 10⁻³ rad. Step 2: image size = α × fₒ = 9.16 × 10⁻³ × 15 m. Step 3: ≈ 0.137 m = 13.7 cm. Magnifying power is 15/0.01 = 1500.
8. You have lenses of focal lengths 100 cm, 4 cm and 1 cm. Which two would you use for a telescope and which for a microscope?
Step 1: a telescope needs a long fₒ and short fₑ: use 100 cm as objective and 1 cm as eyepiece, m = 100. Step 2: a microscope needs both short, fₒ smallest: 1 cm objective, 4 cm eyepiece.
Common mistakes
- Swapping the lenses of a telescope. The objective has the LARGE focal length; in a microscope the objective has the SMALLER one.
- Using m = D/f when the image is at the near point. At D it is 1 + D/f.
- Writing the telescope tube length as fₒ − fₑ. In normal adjustment it is fₒ + fₑ.
- Thinking a telescope makes stars look bigger in size. Stars stay points; the telescope collects more light and separates close stars.