34-5 Optical Instruments

34-5 Optical Instruments#

Prompts

  • What is the near point of the eye? Why can’t you see clearly when an object is closer than the near point?

  • How does a simple magnifying lens work? Where is the object placed? What is angular magnification \(m_\theta\)?

  • Describe a compound microscope: objective and eyepiece. How does the objective’s image become the eyepiece’s object? What contributes to the overall magnification?

  • Describe a refracting telescope. How does its lens arrangement differ from a microscope? Where do the focal points of the objective and eyepiece coincide?

  • State the formulas for angular magnification of (a) a simple magnifier, (b) a compound microscope, and (c) a refracting telescope.

Lecture Notes#

Overview#

  • Optical instruments extend the eye’s range: simple magnifier, compound microscope, refracting telescope.

  • The near point (~25 cm) is the closest distance at which the eye can focus sharply.

  • Angular magnification \(m_\theta\) compares the angle subtended by the image (with the instrument) to the angle subtended by the object at the near point (without it).

  • All three instruments use converging lenses; the microscope and telescope combine an objective and an eyepiece.

The near point and angular size#

The near point \(P_n\) is the closest distance at which the eye can form a sharp image on the retina. Closer objects appear blurry. A typical reference value is 25 cm.

The apparent size of an object depends on the angle \(\theta\) it subtends at the eye. At the near point, \(\theta \approx h/25\,\text{cm}\) for object height \(h\). Moving the object closer would increase \(\theta\) but blur the image.

The eye and corrective lenses#

  • Farsightedness (hyperopia): near point farther than 25 cm; need converging lenses to bring near objects into focus.

  • Nearsightedness (myopia): far point closer than infinity; need diverging lenses to bring distant objects into focus.

  • Corrective lens principle: Glasses create a virtual image at the person’s actual near/far point when viewing an object at the standard 25 cm / infinity.

Poll: Corrective lenses—object and image locations

Grandma Shotwell’s near point is at 100 cm. She buys glasses so she can take an object ___ and have its image appear at ___.

  • (A) at infinity, 25 cm

  • (B) at her near point, 25 cm

  • (C) at 25 cm, infinity

  • (D) at 25 cm, her near point

Poll: Corrective lens power

For Grandma Shotwell (near point 100 cm), which glasses prescription is the best option?

  • (A) +1.50 D

  • (B) +2.00 D

  • (C) +2.50 D

  • (D) +3.00 D

Example: Nearsighted correction

A person can see clearly up close but cannot focus on objects beyond 75 cm. She opts for contact lenses. (a) Is she nearsighted or farsighted? (b) What type of lens is needed? (c) What focal length and power (in diopters)?

Solution: (a) Nearsighted—far point at 75 cm. (b) Diverging lens. (c) Lens must take object at infinity and form image at 75 cm (her far point). \(1/f = 1/\infty - 1/0.75 = -4/3\) \(\Rightarrow\) \(f = -0.75\) m, \(P = -1.33\) D.

Simple magnifying lens#

A converging lens placed between the object and the eye, with the object at or just inside the focal point, produces a virtual image far enough away for the eye to focus. The image subtends a larger angle \(\theta'\) than the object would at the near point.

Angular magnification (object at focal point):

(262)#\[ m_\theta = \frac{\theta'}{\theta} \approx \frac{25\,\text{cm}}{f} \]

where \(f\) is the focal length. Shorter \(f\) gives greater magnification.

Compound microscope#

A compound microscope has two lenses:

  • Objective (focal length \(f_{\text{ob}}\)): Object placed just outside its focal point; produces a real, inverted image \(I\).

  • Eyepiece (focal length \(f_{\text{ey}}\)): Image \(I\) is placed just inside its focal point; acts as a simple magnifier on \(I\), producing a virtual final image.

Tube length \(s\): distance from objective to image \(I\) (typically \(s \gg f_{\text{ob}}\)).

Lateral magnification of objective: \(m \approx -s/f_{\text{ob}}\).

Overall magnification:

(263)#\[ M = m \cdot m_\theta = -\frac{s}{f_{\text{ob}}} \cdot \frac{25\,\text{cm}}{f_{\text{ey}}} \]

The minus sign indicates the final image is inverted.

Refracting telescope#

A refracting telescope views distant objects:

  • Objective (\(f_{\text{ob}}\)): Parallel rays from a distant object form a real image \(I\) at its focal point.

  • Eyepiece (\(f_{\text{ey}}\)): The second focal point of the objective coincides with the first focal point of the eyepiece; image \(I\) lies there and acts as object for the eyepiece, which produces a virtual final image.

Note

Unlike the microscope, the telescope has no tube length between the focal points—they coincide. The telescope is designed for distant objects; the microscope for nearby ones.

Angular magnification:

(264)#\[ m_\theta = -\frac{f_{\text{ob}}}{f_{\text{ey}}} \]

Longer \(f_{\text{ob}}\) and shorter \(f_{\text{ey}}\) increase magnification. The minus sign indicates an inverted image.

Cameras and exposure#

  • Distant objects (\(s \gg f\)): \(s' \approx f\), \(m \approx -f/s\); magnification \(\propto f\).

  • f-number: \(f\)-number \(= f/D\); intensity \(I \propto D^2/f^2 = 1/(f\text{-number})^2\).

  • Exposure \(\propto I \cdot \Delta t_{\text{shutter}}\); zoom changes \(f\) and thus \(I\).

Poll: Camera zoom—exposure

A camera’s exposure is correct at focal length 8.0 mm. You zoom to 16.0 mm without changing the aperture diameter or shutter speed. Is the picture overexposed, underexposed, or still correct?

  • (A) Overexposed

  • (B) Underexposed

  • (C) Still correct

Poll: Fixing exposure when zoomed

After zooming (as in the previous poll), to fix the exposure you should:

  • (A) Increase aperture diameter \(D\) and/or increase shutter time \(\Delta t\)

  • (B) Increase \(D\) and/or decrease \(\Delta t\)

  • (C) Decrease \(D\) and/or increase \(\Delta t\)

  • (D) Decrease \(D\) and/or decrease \(\Delta t\)

Design considerations#

Real instruments involve more than magnification:

  • Light-gathering power: Larger objective diameter \(\to\) brighter image (important for faint objects).

  • Resolving power: Ability to distinguish closely spaced objects; limited by diffraction.

  • Field of view: Angular extent of the scene.

  • Aberrations: Spherical aberration (non-sharp focus with spherical surfaces); chromatic aberration (different wavelengths focus at different points).

Poll: Camera refocus—object moves closer

A photographer focuses her camera on an object. The object then moves closer to the camera. To refocus, should the lens move closer to or farther from the detector?

  • (A) Closer to the detector

  • (B) Farther from the detector

Summary#

  • Near point ~25 cm: closest sharp-focus distance.

  • Simple magnifier: \(m_\theta \approx 25\,\text{cm}/f\); object at focal point.

  • Compound microscope: \(M = -(s/f_{\text{ob}})(25\,\text{cm}/f_{\text{ey}})\); objective forms real image, eyepiece magnifies it.

  • Refracting telescope: \(m_\theta = -f_{\text{ob}}/f_{\text{ey}}\); objective and eyepiece focal points coincide.