Why Large Telescopes Use Curved Mirrors Instead of Lenses

A glass prism splits white light into colours, a microscope shows the cells in a drop of blood, and a telescope brings the craters of the Moon into view. Each works by refracting or reflecting light in a way that a few derivations make precise.

This lesson covers minimum deviation and dispersion, simple and compound microscopes, and refracting and reflecting telescopes.

**At minimum deviation the ray passes symmetrically through the prism, giving $n=\frac{\sin \left(\frac{A+D_m}{2}\right)}{\sin \left(\frac{A}{2}\right)}$, and dispersion happens because n is larger for violet light than for red.

The derivation.** For any ray, $r_1+r_2=A$ and the deviation is $\delta =i+e-A$. At minimum deviation, $i=e$ and $r_1=r_2=A/2$, so $i=(A+D_m)/2$. Snell's law $n=\sin i/\sin r$ then gives the formula.

Worked example 1. A prism of angle $60^{\circ }$ gives $D_m=38^{\circ }$:

$$n=\frac{\sin 49^{\circ }}{\sin 30^{\circ }}=\frac{0.755}{0.500}=1.51$$

Thin prisms and dispersion. For a small angle A, $\delta =(n-1)A$. Since $n_v>n_r$, violet deviates more, and the dispersive power is $\omega =\frac{n_v-n_r}{n-1}$.

Worked example 2. A $6.0^{\circ }$ thin prism has $n_v=1.532$, $n_r=1.514$ and mean $n=1.523$:

$${\delta }_v-{\delta }_r=(1.532-1.514)\times 6.0^{\circ }=0.108^{\circ }\omega =\frac{0.018}{0.523}=0.034$$

An everyday example. The glass pieces of a chandelier throw small patches of rainbow colours onto the walls, each acting as a prism.

The substance. At minimum deviation, a slightly larger or smaller angle of incidence both increase the deviation — which is why the minimum can be found reliably by rotating the prism.

**A simple microscope has magnifying power $M=1+\frac{D}{f}$ with the image at the near point, where $D=25$ cm, and a compound microscope has $M=m_o\times m_e=\frac{v_o}{u_o}\left(1+\frac{D}{f_e}\right)$.

Simple microscope.** Magnifying power is the ratio of the angle the image subtends to the angle the object subtends at the near point. With the image at $v=-D$, the lens formula gives the object distance, and $M=D/|u|=1+D/f$. With the image at infinity, $M=D/f$.

Worked example 1. A lens of $f=5.0$ cm gives $M=1+25/5.0=6$ at the near point and $M=5$ at infinity.

Compound microscope. The objective forms a real, magnified image, which the eyepiece then magnifies like a simple microscope.

Worked example 2. An objective of $f_o=1.0$ cm, an object at $u_o=-1.2$ cm, and an eyepiece of $f_e=5.0$ cm:

$$\frac{1}{v_o}=\frac{1}{1.0}-\frac{1}{1.2}=\frac{1}{6.0}\Rightarrow v_o=6.0\text{ cm}{m}_o=\frac{6.0}{-1.2}=-5$$

$$M=-5\times \left(1+\frac{25}{5.0}\right)=-30$$

The final image is 30 times larger in angle, and inverted.

An everyday example. A pathology lab technician uses a compound microscope to count blood cells that no simple lens could separate.

The substance. Both lenses of a compound microscope should have short focal lengths, with the objective's even shorter, since M grows as $f_o$ and $f_e$ shrink.

**In normal adjustment a telescope has magnifying power $M=\frac{f_o}{f_e}$ and length $f_o+f_e$, whether its objective is a lens or a concave mirror; reflecting telescopes avoid chromatic aberration and can be built far larger.

Refracting telescope.** A long-focus objective lens forms a real image of a distant object at its focus, and a short-focus eyepiece magnifies it. With the final image at the near point, $M=\frac{f_o}{f_e}\left(1+\frac{f_e}{D}\right)$.

Worked example 1. $f_o=100$ cm and $f_e=5.0$ cm:

$$M=\frac{100}{5.0}=20L=105\text{ cm}{M}_{\text{near point}}=20\left(1+\frac{5.0}{25}\right)=24$$

Reflecting telescope. A concave mirror of focal length $R/2$ collects the light, and a small secondary mirror sends it to the eyepiece.

Worked example 2. A mirror of radius 4.0 m, so $f_o=2.0$ m, with a 2.5 cm eyepiece gives $M=\frac{200}{2.5}=80$.

Comparing the two:

- Reflectors — no chromatic aberration; a parabolic mirror removes spherical aberration; a mirror can be supported from behind, so it can be made very large
- Refractors — no secondary mirror blocking part of the light, but large lenses sag under their weight and suffer chromatic aberration

An everyday example. At a planetarium's sky-watching evening, the telescopes that show Saturn's rings are often reflectors.

The substance. A larger objective does not raise magnification — it gathers more light and resolves finer detail, while magnification depends on the focal-length ratio.