---
title: "Optical Instruments and the Eye"
book: "University Physics — Year 1"
subject: physics
language: en
chapter: 4
exercises: 12
source: https://one-course.com/books/physics/3/en/chapter/4-optical-instruments-and-the-eye
---

# Chapter 4 — Optical Instruments and the Eye

From the back row of a concert hall a pair of binoculars pulls the singer’s face to within arm’s reach; a [microscope](#def-b1-optical-instruments-microscope) turns a drop of pond water into a zoo; a phone the thickness of a finger photographs the Milky Way. Every one of these instruments is a few thin lenses arranged for one purpose: to present to the [eye](#def-b1-optical-instruments-eye), or to a sensor, an image larger, brighter or sharper than the [eye](#def-b1-optical-instruments-eye) could form alone. So the chapter begins with the [eye](#def-b1-optical-instruments-eye) itself — what it can and cannot do — and then builds the [magnifier](#def-b1-optical-instruments-G), the [microscope](#def-b1-optical-instruments-microscope), the telescope and the [camera](#def-b1-optical-instruments-camera) out of the conjugation relation of the previous chapter.

![The 40-inch refractor of Yerkes Observatory (1897), still the largest lens telescope ever built: an objective of one metre and a tube of nineteen. Beyond this size a lens sags under its own weight, and astronomy turned to mirrors.](https://one-course.com/images/onecourse/chapters/physics-3/b1-optical-instruments/img-4e4f7ec86ae8.jpg)

*The 40-inch refractor of Yerkes Observatory (1897), still the largest [lens](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#def-b1-mirrors-thin-lenses-lens) telescope ever built: an [objective](#def-b1-optical-instruments-microscope) of one metre and a tube of nineteen. Beyond this size a [lens](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#def-b1-mirrors-thin-lenses-lens) sags under its own weight, and astronomy turned to mirrors.*

![A small refractor at dusk: an objective of long focal length and an eyepiece of short one — the afocal telescope of this chapter, aimed at the Moon of its weekend problem.](https://one-course.com/images/onecourse/chapters/physics-3/b1-optical-instruments/img-ccb52c3300bb.jpg)

*A small refractor at dusk: an [objective](#def-b1-optical-instruments-microscope) of long [focal length](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#def-b1-mirrors-thin-lenses-lens) and an [eyepiece](#def-b1-optical-instruments-microscope) of short one — the afocal telescope of this chapter, aimed at the Moon of its weekend problem.*

## 4.1 The eye

**Definition 4.1 (Reduced eye).**

The *reduced eye* models the cornea and the crystalline [lens](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#def-b1-mirrors-thin-lenses-lens) as a single thin [converging lens](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#def-b1-mirrors-thin-lenses-lens) of *variable* [focal length](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#def-b1-mirrors-thin-lenses-lens), and the *retina* as a screen at a fixed distance $d \approx 17\,\mathrm{mm}$ behind it. Muscles change the [lens](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#def-b1-mirrors-thin-lenses-lens)’s curvature to keep the image of the object looked at on the retina: this is *accommodation*. The iris in front of the [lens](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#def-b1-mirrors-thin-lenses-lens) is a variable aperture of diameter $2\,$ to $8\,\mathrm{mm}$, the *pupil*.

**Definition 4.2 (Far point, near point).**

The *far point* (punctum remotum) is the object point seen sharply without accommodating; the *near point* (punctum proximum) the closest point seen sharply with maximal [accommodation](#def-b1-optical-instruments-eye). For a normal (emmetropic) [eye](#def-b1-optical-instruments-eye) the far point is at infinity and the near point at $d_m \approx 25\,\mathrm{cm}$, the conventional *distance of distinct vision*. The difference of [vergences](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#def-b1-mirrors-thin-lenses-lens) needed for the two is the *amplitude of [accommodation](#def-b1-optical-instruments-eye)*: about $4\,\delta$ for a young adult, falling with age.

**Example 4.3 (The eye’s vergence).**

Distant object: image at $d = 17\,\mathrm{mm}$ means $f' = 17\,\mathrm{mm}$, $V = 1/0.017 = 59\,\delta$. Object at $25\,\mathrm{cm}$: $V = 1/0.017 + 1/0.25 = 59 + 4 = 63\,\delta$. Four diopters of [accommodation](#def-b1-optical-instruments-eye) out of sixty: the crystalline [lens](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#def-b1-mirrors-thin-lenses-lens) adds only a few percent to the cornea’s job, but those few percent are everything for reading.

**Proposition 4.4 (Defects and corrections).**

A *myopic* [eye](#def-b1-optical-instruments-eye) is too convergent: its [far point](#def-b1-optical-instruments-punctum) is at a finite distance $D_r$; a [diverging lens](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#def-b1-mirrors-thin-lenses-lens) of [focal length](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#def-b1-mirrors-thin-lenses-lens) $f' = -D_r$ (worn close to the [eye](#def-b1-optical-instruments-eye)) images infinity at the [far point](#def-b1-optical-instruments-punctum) and restores distant vision. A *hyperopic* [eye](#def-b1-optical-instruments-eye) is not convergent enough: its [far point](#def-b1-optical-instruments-punctum) is virtual (behind the [eye](#def-b1-optical-instruments-eye)) and its [near point](#def-b1-optical-instruments-punctum) too far; a [converging lens](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#def-b1-mirrors-thin-lenses-lens) corrects it. *Presbyopia* is the loss of [accommodation](#def-b1-optical-instruments-eye) with age: the [near point](#def-b1-optical-instruments-punctum) recedes and reading needs converging lenses, whatever the [far point](#def-b1-optical-instruments-punctum).

**Proof.** A [lens](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#def-b1-mirrors-thin-lenses-lens) close to the [eye](#def-b1-optical-instruments-eye) and the [eye](#def-b1-optical-instruments-eye) form a system whose object must be what the [eye](#def-b1-optical-instruments-eye) can see: the [lens](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#def-b1-mirrors-thin-lenses-lens) must turn an object at infinity into a [virtual image](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#def-b1-mirrors-thin-lenses-image) at the [far point](#def-b1-optical-instruments-punctum), $\overline{OA'} = -D_r$ for $\overline{OA} = -\infty$, so $f' = \overline{OA'} = -D_r$. The other cases are the same argument with the relevant points. ∎

![The reduced eye: a thin lens of variable focal length and a retina 17\, mm behind it. Relaxed, it focuses distant objects; accommodating, it shortens f' to bring a near object’s image onto the retina. The image is inverted — the brain turns it back.](https://one-course.com/images/onecourse/chapters/physics-3/b1-optical-instruments/fig-80cdfde028ba.svg)

*The [reduced eye](#def-b1-optical-instruments-eye): a [thin lens](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#def-b1-mirrors-thin-lenses-lens) of variable [focal length](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#def-b1-mirrors-thin-lenses-lens) and a [retina](#def-b1-optical-instruments-eye) $17\,\mathrm{mm}$ behind it. Relaxed, it focuses distant objects; accommodating, it shortens $f'$ to bring a near object’s image onto the [retina](#def-b1-optical-instruments-eye). The image is inverted — the brain turns it back.*

**Definition 4.5 (Angular size and resolution).**

The *angular size* of an object of height $h$ at distance $D$ is $\theta \approx h/D$ (small angles). The [eye](#def-b1-optical-instruments-eye) distinguishes two points only if their angular separation exceeds its *angular resolution*, about $\epsilon \approx 3 \times 10^{-4}\,\mathrm{rad}$ (one arcminute): the spacing of the cone cells and diffraction at the pupil both set this limit.

**Remark 4.6 (What an instrument is for).**

Brought to the [near point](#def-b1-optical-instruments-punctum), a $1\,\mathrm{mm}$ detail subtends $1/250 =
4 \times 10^{-3}\,\mathrm{rad}$: thirteen times the resolution limit; a $0.05\,\mathrm{mm}$ detail subtends $2 \times 10^{-4}\,\mathrm{rad}$ and is invisible. Instruments that look at near objects ([magnifier](#def-b1-optical-instruments-G), [microscope](#def-b1-optical-instruments-microscope)) increase the [angular size](#def-b1-optical-instruments-angular) beyond what bringing the object closer can do; instruments for distant objects (telescope) increase the [angular size](#def-b1-optical-instruments-angular) of things one cannot approach; the [camera](#def-b1-optical-instruments-camera) replaces the [retina](#def-b1-optical-instruments-eye) by a sensor that can integrate light for seconds and be enlarged afterwards.

## 4.2 The magnifier

**Definition 4.7 (Angular magnification).**

For an instrument used by the [eye](#def-b1-optical-instruments-eye), the *angular magnification* is $G = \theta'/\theta$, where $\theta'$ is the [angular size](#def-b1-optical-instruments-angular) of the image seen through the instrument and $\theta$ the [angular size](#def-b1-optical-instruments-angular) of the object seen with the naked [eye](#def-b1-optical-instruments-eye) — at the [near point](#def-b1-optical-instruments-punctum) $d_m = 25\,\mathrm{cm}$ for a near object, in place for a distant one.

**Proposition 4.8 (The magnifier).**

A [converging lens](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#def-b1-mirrors-thin-lenses-lens) of [focal length](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#def-b1-mirrors-thin-lenses-lens) $f' < d_m$ with the object in its object focal plane gives an image at infinity, seen without [accommodation](#def-b1-optical-instruments-eye), with

$$
G = \frac{d_m}{f'} = d_m V .
$$

**Proof.** Object $AB$ of height $h$ in the focal plane: rays from $B$ leave parallel, at the angle of the ray through $O$, $\theta' = h/f'$. Naked [eye](#def-b1-optical-instruments-eye) at best: $\theta = h/d_m$. Ratio $d_m/f'$. ∎

**Example 4.9 (A watchmaker’s loupe).**

$f' = 25\,\mathrm{mm}$: $G = 250/25 = 10$, sold as “$10\times$”. A $0.05\,\mathrm{mm}$ detail then subtends $2 \times 10^{-3}\,\mathrm{rad}$, comfortably resolved. Pushing $G$ higher means $f'$ of a few millimeters, a tiny [lens](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#def-b1-mirrors-thin-lenses-lens) held against the [eye](#def-b1-optical-instruments-eye) — the limit of the single [lens](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#def-b1-mirrors-thin-lenses-lens), and the reason the [microscope](#def-b1-optical-instruments-microscope) exists.

## 4.3 The microscope

**Definition 4.10 (Compound microscope).**

A *microscope* is an *objective* $L_1$ of short [focal length](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#def-b1-mirrors-thin-lenses-lens) $f_1'$ forming a real, enlarged intermediate image of a nearby object in the object focal plane of an *eyepiece* $L_2$ ([focal length](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#def-b1-mirrors-thin-lenses-lens) $f_2'$), which acts as a [magnifier](#def-b1-optical-instruments-G). The distance $\Delta =
\overline{F_1'F_2}$ between the objective’s image focus and the eyepiece’s object focus is the *optical interval*, a standardized $160\,\mathrm{mm}$ on classical instruments.

**Proposition 4.11 (Magnification of a microscope).**

With the final image at infinity,

$$
\gamma_1 = -\frac{\Delta}{f_1'}, \qquad G_2 = \frac{d_m}{f_2'},
\qquad G = \abs{\gamma_1}\,G_2 = \frac{\Delta\,d_m}{f_1' f_2'} .
$$

**Proof.** The intermediate image $A_1B_1$ lies in the [eyepiece](#def-b1-optical-instruments-microscope)’s object focal plane, i.e. at $\overline{F_1'A_1} = \Delta$; Newton’s relation gives $\gamma_1 = -\overline{F_1'A_1}/f_1' = -\Delta/f_1'$. The [eyepiece](#def-b1-optical-instruments-microscope) then shows $A_1B_1$ (height $\abs{\gamma_1} h$) at infinity under the angle $\theta' = \abs{\gamma_1} h/f_2'$, against $\theta = h/d_m$ for the naked [eye](#def-b1-optical-instruments-eye). ∎

![The microscope: the objective forms a real, enlarged, inverted intermediate image A_1B_1 in the object focal plane of the eyepiece; the eyepiece sends it to infinity, where the relaxed eye views it under a large angle.](https://one-course.com/images/onecourse/chapters/physics-3/b1-optical-instruments/fig-448a0b7dfcf9.svg)

*The [microscope](#def-b1-optical-instruments-microscope): the [objective](#def-b1-optical-instruments-microscope) forms a real, enlarged, inverted intermediate image $A_1B_1$ in the object focal plane of the [eyepiece](#def-b1-optical-instruments-microscope); the [eyepiece](#def-b1-optical-instruments-microscope) sends it to infinity, where the relaxed [eye](#def-b1-optical-instruments-eye) views it under a large angle.*

**Example 4.12 (A student microscope).**

[Objective](#def-b1-optical-instruments-microscope) $f_1' = 4.0\,\mathrm{mm}$, [eyepiece](#def-b1-optical-instruments-microscope) $f_2' = 25\,\mathrm{mm}$, $\Delta = 160\,\mathrm{mm}$: $\gamma_1 = -40$, $G_2 = 10$, $G = 400$. The object sits at $\overline{F_1A} = -f_1'^2/\Delta = -16/160 =
-0.10\,\mathrm{mm}$ from $F_1$: $4.1\,\mathrm{mm}$ from the [objective](#def-b1-optical-instruments-microscope). Focusing means moving the whole tube by fractions of a millimeter.

**Remark 4.13 (The resolution limit).**

Magnification without limit would be useless: light is a wave, and an [objective](#def-b1-optical-instruments-microscope) accepting rays within a half-angle $u$ (in a medium of index $n$) cannot separate two points closer than about $\lambda/(2n\sin u)$ — the wave treatment of the Year 2 volume derives it. With $n\sin u \approx 1.4$ (oil immersion) and $\lambda =
0.5\,\text{µ}\mathrm{m}$, the limit is near $0.2\,\text{µ}\mathrm{m}$. To make that detail visible to the [eye](#def-b1-optical-instruments-eye) ($\epsilon = 3 \times 10^{-4}\,\mathrm{rad}$ at $25\,\mathrm{cm}$, i.e. $75\,\text{µ}\mathrm{m}$), a magnification of about $400$ suffices; beyond $1000$ the image is only larger, not richer: *empty magnification*.

## 4.4 The astronomical telescope

**Proposition 4.14 (Refracting telescope).**

An [objective](#def-b1-optical-instruments-microscope) $L_1$ and an [eyepiece](#def-b1-optical-instruments-microscope) $L_2$ with $F_1' = F_2$ form an [afocal system](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#prop-b1-mirrors-thin-lenses-doublet): a star (at infinity) is seen at infinity, the relaxed [eye](#def-b1-optical-instruments-eye) views it, and

$$
G = -\frac{f_1'}{f_2'} .
$$

The image of the [objective](#def-b1-optical-instruments-microscope)’s rim through the [eyepiece](#def-b1-optical-instruments-microscope) is the *[exit pupil](#prop-b1-optical-instruments-telescope)*: a bright disk of diameter $D/\abs G$ located $\overline{O_2P'} = f_2'(f_1' + f_2')/f_1'$ behind the [eyepiece](#def-b1-optical-instruments-microscope), where the [eye](#def-b1-optical-instruments-eye)’s pupil must be placed to receive all the light.

**Proof.** $G$ was derived in [Proposition 3.18](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#prop-b1-mirrors-thin-lenses-doublet). The [objective](#def-b1-optical-instruments-microscope), seen from $L_2$, is an object at $\overline{O_2O_1} = -(f_1'
+ f_2')$; Descartes gives $1/\overline{O_2P'} = 1/f_2' - 1/(f_1' + f_2')
= f_1'/[f_2'(f_1' + f_2')]$, and the magnification $\overline{O_2P'}/
\overline{O_2O_1} = -f_2'/f_1' = 1/G$ scales the diameter $D$ to $D/\abs G$. Every ray entering the [objective](#def-b1-optical-instruments-microscope) leaves through the [exit pupil](#prop-b1-optical-instruments-telescope), since each point of the [objective](#def-b1-optical-instruments-microscope) is imaged there. ∎

![The astronomical telescope: a parallel beam from a star is focused by the objective in the common focal plane and sent back to infinity by the eyepiece, at an angle G times larger. All the light collected by the objective of diameter D passes through the exit pupil, of diameter D/ G.](https://one-course.com/images/onecourse/chapters/physics-3/b1-optical-instruments/fig-ed2a25f0bc6a.svg)

*The astronomical telescope: a parallel beam from a star is focused by the [objective](#def-b1-optical-instruments-microscope) in the common focal plane and sent back to infinity by the [eyepiece](#def-b1-optical-instruments-microscope), at an angle $\abs G$ times larger. All the light collected by the [objective](#def-b1-optical-instruments-microscope) of diameter $D$ passes through the [exit pupil](#prop-b1-optical-instruments-telescope), of diameter $D/\abs G$.*

**Proposition 4.15 (What a telescope gains).**

Compared with the naked [eye](#def-b1-optical-instruments-eye) (pupil $d_p$):

- *[angular size](#def-b1-optical-instruments-angular)* : multiplied by $\abs G$ ;
- *light gathered* from a point source: multiplied by $(D/d_p)^2$ , provided the [exit pupil](#prop-b1-optical-instruments-telescope) is not larger than the [eye](#def-b1-optical-instruments-eye) ’s pupil, i.e. $\abs G \geq D/d_p$ ;
- *[angular resolution](#def-b1-optical-instruments-angular)* : diffraction at the [objective](#def-b1-optical-instruments-microscope) limits it to $\theta_{\min} \approx 1.22\,\lambda/D$ (Rayleigh criterion), far below the [eye](#def-b1-optical-instruments-eye) ’s $\epsilon$ ; the magnification that makes this limit visible is $G_r = \epsilon/\theta_{\min}$ , and larger magnifications only enlarge the blur.

**Proof.** *Admitted at this level.* ∎

**Example 4.16 (An amateur’s 150-millimeter refractor).**

$D = 150\,\mathrm{mm}$, $f_1' = 1200\,\mathrm{mm}$, [eyepiece](#def-b1-optical-instruments-microscope) $25\,\mathrm{mm}$: $G = -48$; [exit pupil](#prop-b1-optical-instruments-telescope) $3.1\,\mathrm{mm}$; light gain $(150/7)^2 \approx
460$ against a dark-adapted pupil of $7\,\mathrm{mm}$; resolution $1.22 \times
550\,\mathrm{nm}/0.15\,\mathrm{m} = 4.5 \times 10^{-6}\,\mathrm{rad} = 0.9''$, so craters $1.7\,\mathrm{km}$ wide on the Moon; $G_r = 3\times10^{-4}/4.5\times
10^{-6} \approx 67$ — the $6\,\mathrm{mm}$ [eyepiece](#def-b1-optical-instruments-microscope) ($G = 200$) shows no more detail, only a dimmer, bigger disk. The weekend problem designs this instrument.

**Remark 4.17 (Reflectors, binoculars, spyglasses).**

A [concave mirror](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#def-b1-mirrors-thin-lenses-mirror) of [focal length](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#def-b1-mirrors-thin-lenses-lens) $f_1'$ replaces the [objective](#def-b1-optical-instruments-microscope) [lens](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#def-b1-mirrors-thin-lenses-lens) in the *reflecting* telescope (no chromatic dispersion, and a mirror can be made meters wide and supported from behind): everything above holds with $f_1'$ the mirror’s [focal length](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#def-b1-mirrors-thin-lenses-lens). The image is inverted — harmless for stars; for terrestrial use, binoculars insert a pair of total-reflection prisms to erect it, and Galileo’s spyglass uses a diverging [eyepiece](#def-b1-optical-instruments-microscope) ($f_2' < 0$, $G > 0$), shorter but with a narrow field.

## 4.5 The camera

**Definition 4.18 (Camera; f-number).**

A *camera* is a [converging lens](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#def-b1-mirrors-thin-lenses-lens) of [focal length](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#def-b1-mirrors-thin-lenses-lens) $f'$ forming a [real image](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#def-b1-mirrors-thin-lenses-image) on a sensor, at $\overline{OA'} \approx f'$ for distant subjects; focusing moves the [lens](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#def-b1-mirrors-thin-lenses-lens). A diaphragm of diameter $D$ limits the beam; the *f-number* is $N = f'/D$ (written $f/N$). The *angle of view* is $2\arctan(\ell/2f')$ for a sensor of width $\ell$.

**Proposition 4.19 (Exposure and depth of field).**

The irradiance on the sensor scales as $1/N^2$, so the exposure time for a given image brightness scales as $N^2$. A point at a distance other than the focused one forms a blur disk; requiring it to stay below a tolerated diameter $c$, a [lens](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#def-b1-mirrors-thin-lenses-lens) focused at the *[hyperfocal distance](#prop-b1-optical-instruments-dof)*

$$
H = \frac{f'^2}{N c}
$$

renders everything from $H/2$ to infinity acceptably sharp.

**Proof.** Irradiance $=$ power/area $\propto D^2/f'^2 = 1/N^2$ (the image of a given subject has a size fixed by $f'$, and the collected power grows as $D^2$). Focus at distance $p$ with the image at $q \approx f'$; a point at infinity focuses at $f'$, i.e. $q - f'$ before the sensor; its cone of rays, of base $D$ at the [lens](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#def-b1-mirrors-thin-lenses-lens), has diameter $D(q - f')/q$ on the sensor (similar triangles). With $1/q = 1/f' - 1/p$ paraxially, $q - f' \approx f'^2/p$ for $p \gg f'$, so the blur is $Df'/p = f'^2/(Np)$; it equals $c$ for $p = H$. The same geometry on the near side gives the limit $H/2$. ∎

![Depth of field: the lens is focused on a point at distance p (image on the sensor); a point at infinity focuses at F', before the sensor, and leaves a blur disk of diameter c = Df'/p — smaller for a small aperture (large N), hence the trade-off between sharpness and exposure time.](https://one-course.com/images/onecourse/chapters/physics-3/b1-optical-instruments/fig-92bf796987e2.svg)

*[Depth of field](#def-b1-optical-instruments-camera): the [lens](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#def-b1-mirrors-thin-lenses-lens) is focused on a point at distance $p$ (image on the sensor); a point at infinity focuses at $F'$, before the sensor, and leaves a blur disk of diameter $c = Df'/p$ — smaller for a small aperture (large $N$), hence the trade-off between sharpness and exposure time.*

**Example 4.20 (Full frame and phone).**

A $50\,\mathrm{mm}$ [lens](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#def-b1-mirrors-thin-lenses-lens) on a $36\,\mathrm{mm}$ wide sensor: angle of view $2\arctan(18/50) = 40^\circ$; at $f/8$ with $c = 0.03\,\mathrm{mm}$, $H = 2500/(8 \times 0.03) = 10\,\mathrm{m}$: focused at $10\,\mathrm{m}$, everything beyond $5\,\mathrm{m}$ is sharp. A phone [lens](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#def-b1-mirrors-thin-lenses-lens), $f' = 4.3\,\mathrm{mm}$ at $f/1.8$ on a $6\,\mathrm{mm}$ sensor with $c = 5\,\text{µ}\mathrm{m}$: same angle of view ($70{}^{\circ}$ wide), $H = 18.5/(1.8 \times 0.005) =
2.1\,\mathrm{m}$ — almost everything is in focus at once, the reason phones fake background blur in software.

![An optometrist’s trial lenses and trial frame: a converging or diverging lens of the right vergence in front of the eye moves its far point back to infinity.](https://one-course.com/images/onecourse/chapters/physics-3/b1-optical-instruments/img-ee6212e4f9ea.jpg)

*An optometrist’s trial lenses and trial frame: a converging or [diverging lens](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#def-b1-mirrors-thin-lenses-lens) of the right [vergence](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#def-b1-mirrors-thin-lenses-lens) in front of the [eye](#def-b1-optical-instruments-eye) moves its [far point](#def-b1-optical-instruments-punctum) back to infinity.*

## 4.6 Exercises

**Exercise 4.1 ★.**

With the [retina](#def-b1-optical-instruments-eye) $17\,\mathrm{mm}$ behind the [lens](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#def-b1-mirrors-thin-lenses-lens), compute the [vergence](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#def-b1-mirrors-thin-lenses-lens) of a normal [eye](#def-b1-optical-instruments-eye) looking at infinity, then at $25\,\mathrm{cm}$; deduce the [amplitude of accommodation](#def-b1-optical-instruments-punctum). What [focal lengths](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#def-b1-mirrors-thin-lenses-lens) are these?

**Solution of Exercise 4.1.**

Infinity: $f' = 17\,\mathrm{mm}$, $V = 1/0.017 = 58.8\,\delta$. $25\,\mathrm{cm}$: $V = 58.8 + 1/0.25 = 62.8\,\delta$, $f' = 15.9\,\mathrm{mm}$. Amplitude $4.0\,\delta$.

**Exercise 4.2 ★.**

A myopic [eye](#def-b1-optical-instruments-eye) has its [far point](#def-b1-optical-instruments-punctum) at $40\,\mathrm{cm}$. What [lens](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#def-b1-mirrors-thin-lenses-lens) corrects it ([focal length](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#def-b1-mirrors-thin-lenses-lens), [vergence](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#def-b1-mirrors-thin-lenses-lens))? A hyperopic [eye](#def-b1-optical-instruments-eye) can accommodate on nothing closer than $1.0\,\mathrm{m}$: what [lens](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#def-b1-mirrors-thin-lenses-lens) lets it read at $25\,\mathrm{cm}$?

**Solution of Exercise 4.2.**

Myope: infinity must be imaged at the [far point](#def-b1-optical-instruments-punctum), $f' = -40\,\mathrm{cm}$, $V = -2.5\,\delta$. Hyperope: an object at $25\,\mathrm{cm}$ must be imaged (virtually) at $1.0\,\mathrm{m}$: $V = 1/\overline{OA'} - 1/\overline{OA}
= -1 + 4 = +3.0\,\delta$.

**Exercise 4.3 ★.**

A [magnifier](#def-b1-optical-instruments-G) has $f' = 4.0\,\mathrm{cm}$. Give its [angular magnification](#def-b1-optical-instruments-G), where the stamp must be held, and the [angular size](#def-b1-optical-instruments-angular) of a $2\,\mathrm{mm}$ detail through it, compared with the naked [eye](#def-b1-optical-instruments-eye) at $25\,\mathrm{cm}$.

**Solution of Exercise 4.3.**

$G = 25/4.0 = 6.3$; stamp in the focal plane, $4.0\,\mathrm{cm}$ from the [lens](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#def-b1-mirrors-thin-lenses-lens). Detail of $2\,\mathrm{mm}$: $\theta' = 2/40 = 0.05\,\mathrm{rad}$, against $2/250 = 0.008\,\mathrm{rad}$ at $25\,\mathrm{cm}$.

**Exercise 4.4 ★.**

A refractor has $f_1' = 900\,\mathrm{mm}$. Compute $G$ with a $25\,\mathrm{mm}$ and with a $9\,\mathrm{mm}$ [eyepiece](#def-b1-optical-instruments-microscope), the tube length in each case, and the apparent size of the Moon ($0.52^\circ$).

**Solution of Exercise 4.4.**

$G = 900/25 = 36$ and $900/9 = 100$; tube lengths $f_1' + f_2' =
925\,\mathrm{mm}$ and $909\,\mathrm{mm}$; Moon: $36 \times 0.52^\circ = 19^\circ$ and $52^\circ$.

**Exercise 4.5 ★★.**

[Microscope](#def-b1-optical-instruments-microscope): $f_1' = 4.0\,\mathrm{mm}$, $f_2' = 20\,\mathrm{mm}$, $\Delta = 160\,\mathrm{mm}$. Compute $\gamma_1$, $G_2$, $G$, and the object–objective distance. By how much must the tube move to bring the object $10\,\text{µ}\mathrm{m}$ deeper into focus?

**Solution of Exercise 4.5.**

$\gamma_1 = -160/4 = -40$; $G_2 = 250/20 = 12.5$; $G = 500$. $\overline{F_1A} = -f_1'^2/\Delta = -16/160 = -0.10\,\mathrm{mm}$: object $4.1\,\mathrm{mm}$ from the [objective](#def-b1-optical-instruments-microscope). The object must stay at that distance from the [objective](#def-b1-optical-instruments-microscope), so the whole tube moves by $10\,\text{µ}\mathrm{m}$ — the fine-focus knob.

**Exercise 4.6 ★★.**

Two headlights are $1.5\,\mathrm{m}$ apart. From what distance can the naked [eye](#def-b1-optical-instruments-eye) ($\epsilon = 3 \times 10^{-4}\,\mathrm{rad}$) tell them apart? A telescope with $D = 100\,\mathrm{mm}$: compute its diffraction limit ($\lambda =
550\,\mathrm{nm}$), the distance at which it separates the headlights, and the magnification $G_r$ beyond which it shows nothing new.

**Solution of Exercise 4.6.**

Naked [eye](#def-b1-optical-instruments-eye): $1.5/3\times10^{-4} = 5\,\mathrm{km}$. Telescope: $\theta_{\min}
= 1.22 \times 5.5\times10^{-7}/0.100 = 6.7 \times 10^{-6}\,\mathrm{rad}$; headlights separated up to $1.5/6.7\times10^{-6} \approx 220\,\mathrm{km}$ (in principle — air and curvature intervene first). $G_r = 3\times10^{-4}/
6.7\times10^{-6} = 45$.

**Exercise 4.7 ★★.**

A $50\,\mathrm{mm}$ [lens](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#def-b1-mirrors-thin-lenses-lens) at $f/2$: what is the aperture diameter? A scene is correctly exposed at $f/2$ in $1/500$ $\mathrm{s}$; what exposure time at $f/8$? What does the smaller aperture buy?

**Solution of Exercise 4.7.**

$D = 50/2 = 25\,\mathrm{mm}$. Exposure $\propto N^2$: $(8/2)^2 = 16$ times longer, $16/500 \approx 1/30$ $\mathrm{s}$. It buys [depth of field](#def-b1-optical-instruments-camera) (and fewer aberrations) at the price of motion blur or a tripod.

**Exercise 4.8 ★★.**

Compute the [hyperfocal distance](#prop-b1-optical-instruments-dof) of a $35\,\mathrm{mm}$ [lens](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#def-b1-mirrors-thin-lenses-lens) at $f/11$ with $c = 0.03\,\mathrm{mm}$, and the range of sharp distances when it is focused at $H$. Street photographers preset this: why?

**Solution of Exercise 4.8.**

$H = 35^2/(11 \times 0.03) = 3.7\,\mathrm{m}$; sharp from $H/2 =
1.9\,\mathrm{m}$ to infinity. Preset at $H$, the [camera](#def-b1-optical-instruments-camera) needs no focusing: the picture can be taken the instant the scene happens.

**Exercise 4.9 ★★.**

Binoculars “$8\times30$” have $G = 8$ and $D = 30\,\mathrm{mm}$. Compute the [exit pupil](#prop-b1-optical-instruments-telescope); is all the light used by a $7\,\mathrm{mm}$ night pupil? For “$7\times50$”? Which pair is better at dusk, and why does “$20\times30$” make a poor choice?

**Solution of Exercise 4.9.**

$8\times30$: [exit pupil](#prop-b1-optical-instruments-telescope) $30/8 = 3.8\,\mathrm{mm} < 7\,\mathrm{mm}$, all light enters the [eye](#def-b1-optical-instruments-eye). $7\times50$: $7.1\,\mathrm{mm}$, matching the night pupil, and $(50/30)^2 = 2.8$ times more light collected: the dusk pair. $20\times30$: [exit pupil](#prop-b1-optical-instruments-telescope) $1.5\,\mathrm{mm}$, dim image, narrow field and hand shake magnified twenty times.

**Exercise 4.10 ★★★.**

A myope’s [far point](#def-b1-optical-instruments-punctum) is $25\,\mathrm{cm}$ from the [eye](#def-b1-optical-instruments-eye). Compute the [vergence](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#def-b1-mirrors-thin-lenses-lens) of the correcting contact [lens](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#def-b1-mirrors-thin-lenses-lens), then of spectacle lenses worn $15\,\mathrm{mm}$ in front of the [eye](#def-b1-optical-instruments-eye). Which correction is stronger, and why do the two differ?

**Solution of Exercise 4.10.**

Contact [lens](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#def-b1-mirrors-thin-lenses-lens): $f' = -25\,\mathrm{cm}$, $V = -4.0\,\delta$. Spectacles: the [far point](#def-b1-optical-instruments-punctum) is $25 - 1.5 = 23.5\,\mathrm{cm}$ from the [lens](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#def-b1-mirrors-thin-lenses-lens), so $f' =
-23.5\,\mathrm{cm}$, $V = -4.3\,\delta$ — stronger. The [lens](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#def-b1-mirrors-thin-lenses-lens) must form the image at the [eye](#def-b1-optical-instruments-eye)’s [far point](#def-b1-optical-instruments-punctum), whose distance from the [lens](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#def-b1-mirrors-thin-lenses-lens) depends on where the [lens](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#def-b1-mirrors-thin-lenses-lens) sits.

**Exercise 4.11 ★★★.**

Galileo’s spyglass: $f_1' = 30\,\mathrm{cm}$, $f_2' = -10\,\mathrm{cm}$. Find the separation of the lenses for an [afocal system](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#prop-b1-mirrors-thin-lenses-doublet), $G$, and the orientation of the image. Locate the [exit pupil](#prop-b1-optical-instruments-telescope) and explain why the field of view is narrow.

**Solution of Exercise 4.11.**

Afocal: $F_1' = F_2$, separation $f_1' + f_2' = 20\,\mathrm{cm}$ (shorter than the $40\,\mathrm{cm}$ of a Keplerian of the same $G$). $G = -30/(-10) =
+3$: upright. [Exit pupil](#prop-b1-optical-instruments-telescope) $\overline{O_2P'} = f_2'(f_1' + f_2')/f_1' =
-10 \times 20/30 = -6.7\,\mathrm{cm}$: virtual, inside the tube, where the [eye](#def-b1-optical-instruments-eye) cannot be; the [eye](#def-b1-optical-instruments-eye) behind the [eyepiece](#def-b1-optical-instruments-microscope) catches only the beams that happen to reach it — a narrow field, the classic complaint about opera glasses.

**Exercise 4.12 ★★★.**

A $200\,\mathrm{mm}$ telescope and a $7\,\mathrm{mm}$ pupil. By what factor is the light from a star increased? Astronomers grade brightness by magnitudes, $m_2 - m_1 = 2.5\log_{10}(F_1/F_2)$; the [eye](#def-b1-optical-instruments-eye) reaches magnitude $6$: what limiting magnitude does the telescope give? How many times farther can a given lamp be seen?

**Solution of Exercise 4.12.**

$(200/7)^2 = 820$; $\Delta m = 2.5\log_{10}820 = 7.3$: limiting magnitude $13.3$. Flux $\propto 1/r^2$, so the lamp is seen $\sqrt{820} = 29$ times farther.

![The near side of the Moon (Lunar Reconnaissance Orbiter, NASA): the target of the weekend problem’s telescope, 3476\, km across at 384\,000\, km.](https://one-course.com/images/onecourse/chapters/physics-3/b1-optical-instruments/img-ce53ac82ea4e.jpg)

*The near side of the Moon (Lunar Reconnaissance Orbiter, NASA): the target of the weekend problem’s telescope, $3476\,\mathrm{km}$ across at $384\,000\,\mathrm{km}$.*

## 4.7 Problem: Designing a telescope for the Moon

**Problem 4.1.**

Weekend problem — a 150-millimeter refractor on a garden lawn: choose the eyepieces, place the eye, count the light, and find out how small a crater it can show

The [objective](#def-b1-optical-instruments-microscope) is a [converging lens](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#def-b1-mirrors-thin-lenses-lens) $L_1$ of [focal length](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#def-b1-mirrors-thin-lenses-lens) $f_1' = 1200\,\mathrm{mm}$ and diameter $D = 150\,\mathrm{mm}$; [eyepieces](#def-b1-optical-instruments-microscope) $L_2$ of [focal lengths](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#def-b1-mirrors-thin-lenses-lens) $25\,\mathrm{mm}$ and $6\,\mathrm{mm}$ are available. The Moon is $3.84 \times 10^{5}\,\mathrm{km}$ away and subtends $0.52^\circ$; the [eye](#def-b1-optical-instruments-eye)’s night pupil is $7\,\mathrm{mm}$, its resolution $\epsilon = 3.0 \times 10^{-4}\,\mathrm{rad}$; $\lambda = 550\,\mathrm{nm}$.

**Part I — The afocal arrangement.**

1. Where must the [eyepiece](#def-b1-optical-instruments-microscope) be placed for a star to be seen without [accommodation](#def-b1-optical-instruments-eye) ? Give the objective–eyepiece distance for each [eyepiece](#def-b1-optical-instruments-microscope) .
2. Check with Descartes’s relation, applied to each [lens](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#def-b1-mirrors-thin-lenses-lens) in turn, that a point at infinity on the axis is imaged at infinity.
3. Show that a beam from a star at angle $\alpha$ from the axis emerges as a parallel beam at angle $\alpha' = -(f_1'/f_2')\alpha$ .
4. Compute $G$ for each [eyepiece](#def-b1-optical-instruments-microscope) and the apparent diameter of the Moon through each.
5. The image is inverted. Why is that acceptable here, and what would a terrestrial spotting scope add?
6. Where is the real intermediate image of the Moon, and what is its diameter? (It can be photographed there.)

**Part II — Where the [eye](#def-b1-optical-instruments-eye) goes.**

7. Show that the [eyepiece](#def-b1-optical-instruments-microscope) forms a [real image](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#def-b1-mirrors-thin-lenses-image) of the [objective](#def-b1-optical-instruments-microscope) ’s rim at $\overline{O_2P'} = f_2'(f_1' + f_2')/f_1'$ behind it, of diameter $D/\abs G$ (the [exit pupil](#prop-b1-optical-instruments-telescope) ).
8. Compute the position and diameter of the [exit pupil](#prop-b1-optical-instruments-telescope) for each [eyepiece](#def-b1-optical-instruments-microscope) .
9. Why must the [eye](#def-b1-optical-instruments-eye) ’s pupil be placed at the [exit pupil](#prop-b1-optical-instruments-telescope) ? What happens to the field if it is placed farther back?
10. For the light of a star to be fully used, the [exit pupil](#prop-b1-optical-instruments-telescope) must not exceed the [eye](#def-b1-optical-instruments-eye) ’s pupil: deduce the minimum useful magnification of this telescope at night.

**Part III — Resolution.**

11. Compute the diffraction limit $\theta_{\min} = 1.22\lambda/D$ in radians and arcseconds.
12. What is the smallest crater this angle corresponds to on the Moon? And for the naked [eye](#def-b1-optical-instruments-eye) ?
13. Define and compute the resolving magnification $G_r =  \epsilon/\theta_{\min}$ . Which [eyepiece](#def-b1-optical-instruments-microscope) is closest to it?
14. With the $6\,\mathrm{mm}$ [eyepiece](#def-b1-optical-instruments-microscope) the image looks blurrier and dimmer than with the $25\,\mathrm{mm}$ . Explain both effects.
15. Atmospheric turbulence smears star images to about $1''$ . For which [objective](#def-b1-optical-instruments-microscope) diameters does the Rayleigh limit stop mattering from the ground, and what does a larger $D$ still buy?

**Part IV — Light.**

16. By what factor does the [objective](#def-b1-optical-instruments-microscope) collect more light than the night pupil?
17. Convert this to magnitudes ( $\Delta m = 2.5\log_{10}$ of the ratio); if the [eye](#def-b1-optical-instruments-eye) reaches magnitude $6$ , what magnitude does the telescope reach?
18. The full Moon delivers an irradiance of about $2 \times 10^{-3}\,\mathrm{W}/\mathrm{m}^{2}$ at the ground. What power enters the [objective](#def-b1-optical-instruments-microscope) , and how is it spread over the [retina](#def-b1-optical-instruments-eye) compared with [naked-eye](#def-b1-optical-instruments-eye) viewing (same power per unit solid angle argument: compare surface brightness through the two [eyepieces](#def-b1-optical-instruments-microscope) )?
19. Why does a telescope not make the Moon’s surface brighter per unit area than the naked [eye](#def-b1-optical-instruments-eye) , whereas it does make stars brighter?

**Part V — Field of view.** The [eyepiece](#def-b1-optical-instruments-microscope) carries, in its focal plane, a circular field stop of diameter $20\,\mathrm{mm}$ ($25\,\mathrm{mm}$ [eyepiece](#def-b1-optical-instruments-microscope)) or $5\,\mathrm{mm}$ ($6\,\mathrm{mm}$ [eyepiece](#def-b1-optical-instruments-microscope)).

20. Show that the true field of view is $2\arctan(\phi/2f_1')  \approx \phi/f_1'$ for a stop of diameter $\phi$ .
21. Compute it for both [eyepieces](#def-b1-optical-instruments-microscope) and say whether the whole Moon fits in the view.
22. Compute the apparent field (the angle under which the [eye](#def-b1-optical-instruments-eye) sees the stop through the [eyepiece](#def-b1-optical-instruments-microscope) ) for both.
23. The Earth turns at $15''$ per second of time: how long does the Moon take to cross the field of each [eyepiece](#def-b1-optical-instruments-microscope) without tracking?
24. Summarize the design: which [eyepiece](#def-b1-optical-instruments-microscope) for the whole Moon, which for craters, and the smallest crater size the instrument can reveal — its one number.
25. The same [objective](#def-b1-optical-instruments-microscope) is replaced by a $150\,\mathrm{mm}$ [concave mirror](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#def-b1-mirrors-thin-lenses-mirror) of the same [focal length](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#def-b1-mirrors-thin-lenses-lens) . Which of the above results change?

**Solution of Problem 4.1.**

**1.** $F_2 = F_1'$: separation $f_1' + f_2' = 1225\,\mathrm{mm}$ ($25\,\mathrm{mm}$ [eyepiece](#def-b1-optical-instruments-microscope)) or $1206\,\mathrm{mm}$ ($6\,\mathrm{mm}$).

**2.** $L_1$: $\overline{O_1A_1} = f_1'$ (image at $F_1'$). $L_2$: $\overline{O_2A_1} = -f_2'$, so $1/\overline{O_2A'} = 1/f_2' + 1/(-f_2')
= 0$: image at infinity.

**3.** The beam focuses in the common focal plane at height $h =
f_1'\alpha$ (ray through $O_1$); from there the ray through $O_2$ sets the emerging direction, $\alpha' = -h/f_2' = -(f_1'/f_2')\alpha$.

**4.** $G = -48$ and $-200$; Moon: $25^\circ$ and $104^\circ$.

**5.** Up and down have no meaning in the sky; a spotting scope adds an erecting prism pair (or [lens](https://one-course.com/books/physics/3/en/chapter/3-mirrors-and-thin-lenses#def-b1-mirrors-thin-lenses-lens) relay).

**6.** In the focal plane of $L_1$, diameter $f_1'\theta = 1200 \times
9.1\times10^{-3} = 11\,\mathrm{mm}$.

**7.** Object $O_1$ at $\overline{O_2O_1} = -(f_1' + f_2')$: $1/\overline{O_2P'} = 1/f_2' - 1/(f_1' + f_2') = f_1'/[f_2'(f_1' + f_2')]$; magnification $\overline{O_2P'}/\overline{O_2O_1} = -f_2'/f_1' = 1/G$, diameter $D/\abs G$.

**8.** $25\,\mathrm{mm}$: $\overline{O_2P'} = 25 \times 1225/1200 =
25.5\,\mathrm{mm}$, diameter $150/48 = 3.1\,\mathrm{mm}$. $6\,\mathrm{mm}$: $6.0\,\mathrm{mm}$ behind, diameter $0.75\,\mathrm{mm}$.

**9.** Every ray that entered the [objective](#def-b1-optical-instruments-microscope) passes through the [exit pupil](#prop-b1-optical-instruments-telescope): an [eye](#def-b1-optical-instruments-eye) placed there receives the whole field. Farther back it intercepts only the central beams: the field shrinks to a keyhole.

**10.** $D/\abs G \leq 7\,\mathrm{mm}$: $\abs G \geq 150/7 \approx 21$.

**11.** $\theta_{\min} = 1.22 \times 5.5\times10^{-7}/0.150 =
4.5 \times 10^{-6}\,\mathrm{rad} = 0.92''$.

**12.** $4.5\times10^{-6} \times 3.84\times10^5 = 1.7\,\mathrm{km}$; naked [eye](#def-b1-optical-instruments-eye) $3\times10^{-4} \times 3.84\times10^5 = 115\,\mathrm{km}$.

**13.** $G_r = 3\times10^{-4}/4.5\times10^{-6} = 67$; the $25\,\mathrm{mm}$ [eyepiece](#def-b1-optical-instruments-microscope) ($48$) is closest, slightly under; the $6\,\mathrm{mm}$ ($200$) is three times over.

**14.** At $G = 200$ the diffraction blur subtends $200 \times 4.5
\times10^{-6} = 9 \times 10^{-4}\,\mathrm{rad} > \epsilon$: visibly fuzzy. The Moon’s light is spread over an area $(200/48)^2 = 17$ times larger on the [retina](#def-b1-optical-instruments-eye): dimmer.

**15.** $1'' = 4.85 \times 10^{-6}\,\mathrm{rad}$: $D = 1.22\lambda/\theta =
140\,\mathrm{mm}$. Beyond that, seeing limits resolution from the ground (barring adaptive optics); a larger $D$ still collects more light, reaching fainter objects.

**16.** $(150/7)^2 = 460$.

**17.** $\Delta m = 2.5\log_{10}460 = 6.7$: magnitude $12.7$.

**18.** $P = 2\times10^{-3} \times \pi(0.075)^2 = 3.5 \times 10^{-5}\,\mathrm{W}$, $460$ times the naked [eye](#def-b1-optical-instruments-eye)’s. The retinal image is $\abs G^2$ times larger in area: surface brightness ratio $460/G^2 = 0.20$ ($G = 48$), $0.012$ ($G = 200$) — equal to $(D/\abs G)^2/d_p^2$, the squared ratio of [exit pupil](#prop-b1-optical-instruments-telescope) to [eye](#def-b1-optical-instruments-eye) pupil.

**19.** For an extended object the collected light grows as $D^2$ but the image area as $G^2 \geq (D/d_p)^2$: the surface brightness can at best equal the [naked-eye](#def-b1-optical-instruments-eye) value ([exit pupil](#prop-b1-optical-instruments-telescope) $=$ [eye](#def-b1-optical-instruments-eye) pupil). A star is unresolved: all its light lands in one diffraction spot, so its brightness grows as $(D/d_p)^2$.

**20.** A point at angle $\beta$ images at height $f_1'\beta$ in the focal plane; the stop passes $\abs{f_1'\beta} \leq \phi/2$: full field $\phi/f_1'$ (small angles).

**21.** $20/1200 = 0.0167\,\mathrm{rad} = 0.95^\circ$: the whole Moon fits. $5/1200 = 0.24^\circ$: it does not.

**22.** Apparent field $\phi/f_2'$: $20/25 = 0.8\,\mathrm{rad} = 46^\circ$; $5/6 = 48^\circ$.

**23.** $0.95^\circ = 3420''$: $3420/15 = 230\,\mathrm{s}$, nearly four minutes; $0.24^\circ = 864''$: $58\,\mathrm{s}$.

**24.** $25\,\mathrm{mm}$ [eyepiece](#def-b1-optical-instruments-microscope) ($G = 48$, near $G_r$, bright, whole Moon) for the disk; $6\,\mathrm{mm}$ ($G = 200$) for a closer look at craters, dimmer and fuzzier. The instrument’s number: craters down to $1.7\,\mathrm{km}$.

**25.** Same $f_1'$ and $D$: same $G$, fields, [exit pupils](#prop-b1-optical-instruments-telescope), resolution and light (minus any central obstruction). Gains: no chromatic aberration. Practical change: the focus lies in front of the mirror, so a small secondary mirror must fold the beam out to the [eyepiece](#def-b1-optical-instruments-microscope).