How Two Small Lenses Reveal the Cells in an Onion Peel

A single convex lens can magnify small print only a few times. By pairing two lenses — or a curved mirror and a lens — a microscope reveals the cells in an onion peel and a telescope brings the craters of the Moon into view.

This part covers the compound microscope, the refracting telescope, and the reflecting telescope.

**A compound microscope uses an objective of very short focal length to form a real, magnified image, which an eyepiece magnifies further like a simple magnifier, so the total magnifying power is $m=m_o\times m_e$.

How the image forms:

- The object sits just beyond the focus of the objective
- The objective forms a real, inverted, magnified image inside the focal length of the eyepiece
- The eyepiece turns this into a large virtual image

Magnifying power.** With the final image at infinity, $m_e=\frac{D}{f_e}$, where $D=25$ cm is the least distance of distinct vision; with the final image at the near point, $m_e=1+\frac{D}{f_e}$. For tube length $L$,

$$m\approx \frac{L}{f_o}\times \frac{D}{f_e}$$

Worked example. An objective of focal length $2$ cm views an object $2.5$ cm away, so $u=-2.5$ cm:

$$\frac{1}{v}=\frac{1}{2}-\frac{1}{2.5}=\frac{1}{10}\Rightarrow v=10\text{ cm},m_o=\frac{v}{u}=-4$$

An eyepiece of focal length $5$ cm with the final image at infinity gives $m_e=\frac{25}{5}=5$, so $|m|=20$, with the lenses $10+5=15$ cm apart. With the final image at the near point, $m_e=6$ and $|m|=24$.

An everyday example. Viewing onion peel cells in a biology practical relies on exactly this two-stage magnification.

The substance. The two magnifications multiply, not add — which is why both lenses are given short focal lengths.

**A refracting telescope uses an objective of long focal length to form a real image of a distant object at its focus, and an eyepiece of short focal length to magnify that image; in normal adjustment, with the final image at infinity, $m=\frac{f_o}{f_e}$ and the tube length is $f_o+f_e$.

How the image forms:

- Parallel rays from a distant object meet at the focus of the objective, forming a small real, inverted image
- In normal adjustment this image lies at the focus of the eyepiece
- The eyepiece sends out parallel rays, so the final image is at infinity

Magnifying power** is the ratio of the angle the final image subtends at the eye to the angle the object subtends:

$$m=\frac{\beta }{\alpha }=\frac{f_o}{f_e},L=f_o+f_e$$

If the final image is at the near point, $m=\frac{f_o}{f_e}\left(1+\frac{f_e}{D}\right)$.

Worked example. An objective of focal length $100$ cm and an eyepiece of $2.5$ cm:

$$m=\frac{100}{2.5}=40,L=100+2.5=102.5\text{ cm}$$

A distant tower subtending $0.2^{\circ }$ appears to subtend $40\times 0.2^{\circ }=8^{\circ }$. With the final image at the near point, $m=40\times 1.1=44$.

An everyday example. Binoculars are two small refracting telescopes, with prisms that use total internal reflection to shorten the tube and turn the image upright.

The substance. A wider objective does not increase magnifying power — it gathers more light and shows finer detail.

A reflecting telescope collects light with a large concave mirror instead of a lens, which avoids chromatic aberration, can be made far larger and lighter, and gathers much more light; in the Cassegrain design, a small convex mirror reflects the light back through a hole in the main mirror to the eyepiece.

Advantages over refracting telescopes:

- No chromatic aberration — a mirror reflects all colours alike, while a lens splits them
- No spherical aberration when the mirror is parabolic
- Easier support — a mirror can be held from behind, while a heavy lens can only be held at its rim and sags
- Only one surface needs grinding and polishing
- Larger aperture, so more light is gathered and finer detail resolved

Cassegrain arrangement. The concave primary mirror reflects light to a small convex secondary mirror, which sends it back through a central hole to the eyepiece, folding a long focal length into a short tube.

Worked example. A primary mirror with radius of curvature $4$ m has $f_o=\frac{R}{2}=2$ m $=200$ cm. With an eyepiece of focal length $2$ cm,

$$m=\frac{f_o}{f_e}=\frac{200}{2}=100$$

Light gathered depends on the area of the aperture, so a $2$ m mirror collects ${\left(\frac{2}{0.5}\right)}^2=16$ times as much light as a $0.5$ m one.

An everyday example. A satellite TV dish on a rooftop is a concave reflector that focuses weak signals onto a receiver at its focus — the same principle, working with microwaves.

The substance. The secondary mirror blocks a little light but leaves no hole in the image, because every part of the primary mirror contributes to every point of the image.