Optical Instruments

For a simple microscope in normal adjustment (image at infinity): $m = \frac{D}{f}$

Where:

• D is the least distance of distinct vision (typically 25 cm)
• f is the focal length of the microscope lens

Key insight: Magnifying power increases as focal length decreases, which is why microscopes use lenses with very short focal lengths.

Note: This represents angular magnification - the ratio of the angle subtended by the image when viewed through the microscope to the angle subtended by the object at the near point.

Normal adjustment (image at infinity):

• Length = $L = f_o + f_e$
• First image forms at focal point of objective
• Eyepiece positioned so its focal point coincides with this image

Near point adjustment:

• Using lens equation for eyepiece: $\frac{1}{P'E} = \frac{1}{f_e} + \frac{1}{D}$
• Solving: $P'E = \frac{f_e D}{f_e + D}$
• Total length: $L = f_o + \frac{f_e D}{f_e + D}$

Relationship: Length for near point is always shorter than normal adjustment by factor: $L_{near} = f_o + \frac{f_e D}{f_e + D} < f_o + f_e = L_{normal}$

This occurs because the eyepiece must move closer to the objective to form an image at finite distance D rather than at infinity.

Microscope resolving power:

• Formula: $R = \frac{1}{\Delta d} = \frac{2\mu\sin\theta}{\lambda}$
• Factors:
  • λ = wavelength of light (shorter = better resolution)
  • μ = refractive index of medium between object and objective
  • θ = angle subtended by objective radius on object
• Maximizing resolution:
  • Use shorter wavelengths (blue/UV light)
  • Immersion oil to increase μ
  • Higher numerical aperture objectives

Telescope resolving power:

• Formula: $R = \frac{1}{\Delta\theta} = \frac{a}{1.22\lambda}$
• Factors:
  • a = diameter of objective
  • λ = wavelength of light
• Maximizing resolution:
  • Larger objective diameter (why astronomical telescopes are massive)
  • Shorter wavelengths
  • Adaptive optics to counter atmospheric distortion

The fundamental difference: microscope resolution depends on medium properties; telescope resolution on aperture size.

A compound microscope achieves magnification in two stages:

1. The objective lens creates a real, inverted, magnified image of the object
2. The eyepiece further magnifies this first image

For normal adjustment (final image at infinity):

• Magnifying power: m = (v/u)(D/fe)
• Simplified as: m = -(l/fo)(D/fe)

Where:

• v = distance of first image from objective
• u = distance of object from objective
• D = least distance of distinct vision (typically 25 cm)
• fe = focal length of eyepiece
• l = tube length (separation between lenses)
• fo = focal length of objective

This two-stage magnification allows for much greater total magnification than a simple microscope.

• The magnifying power formula is: $m = -\frac{f_o}{f_e}$

Where:

• $f_o$ = focal length of the objective lens
• $f_e$ = focal length of the eyepiece
• The negative sign indicates the image is inverted

This represents the ratio of the angle subtended by the final image to the angle subtended by the object when viewed by unaided eye.

Example: A telescope with 100 cm objective focal length and 2 cm eyepiece focal length has magnifying power of -50, meaning objects appear 50 times larger but inverted.

• For normal adjustment (image at infinity):

  • Length = $f_o + f_e$
  • First image forms at objective's focal plane
  • Eyepiece positioned so its focal plane coincides with objective's focal plane

• For near point vision (image at distance D):

  • Length = $f_o + \frac{f_e D}{f_e + D}$
  • Eyepiece must be moved closer to objective
  • Distance between first image and eyepiece becomes $\frac{f_e D}{f_e + D}$

• Factors determining length:

  • Focal lengths of both lenses
  • Desired image position (infinity vs. near point)
  • Near point distance D (typically 25 cm for normal eyes)

Note: The near point adjustment provides greater magnification but causes more eye strain.

• Light rays follow this sequence:

  1. Objective lens (Lo) creates a real, inverted image (P'Q') in its focal plane
  2. Inverting lens (L) with focal length f is positioned 2f away from P'Q'
  3. The inverting lens creates a second real image (P''Q'') that is inverted relative to P'Q' (thus upright relative to object)
  4. Eyepiece magnifies P''Q'' to create the final virtual image viewed by the eye

• Key distances:

  • Distance between objective and inverting lens: fo + 2f
  • Distance between inverting lens and eyepiece: 2f + eyepiece adjustments
  • Total length: fo + 4f + fe for normal adjustment

• The angular magnification remains m = fo/fe, but with upright image orientation

• Uses a converging lens as objective and a diverging lens as eyepiece
• Eyepiece intercepts converging rays before they form a real image
• Produces an erect (not inverted) final image
• Has a shorter tube length compared to astronomical telescopes
• Angular magnification: m = -fo/fe (where fe is negative)
• Light path: objective begins to form image, but eyepiece intercepts and diverges rays before real image forms
• Primary advantage: compact design and upright image ideal for terrestrial viewing
• Formula for normal adjustment: m = -fo/fe
• Length of telescope: L = fo + fe (where fe is negative, so L < fo)

Step 1: Determine what the lens must accomplish

• Without correction: Near point at 75 cm
• With correction: Near point should be at 25 cm
• The lens must make an object at 25 cm appear as if it's at 75 cm

Step 2: Apply the virtual image formula

• Object distance: $u = 25$ cm
• Desired virtual image distance: $v = -75$ cm (negative because it's a virtual image)
• Using lens formula: $\frac{1}{f} = \frac{1}{u} + \frac{1}{v} = \frac{1}{25} + \frac{1}{-75} = \frac{3-1}{75} = \frac{2}{75}$

Step 3: Calculate focal length and power

• $f = \frac{75}{2} = 37.5$ cm = 0.375 m
• $P = \frac{1}{f} = \frac{1}{0.375} = +2.67$ D

The person needs a convergent lens with power +2.67 diopters.

Normal Eye:

• Far point: Infinity
• Near point: ~25 cm
• Functionality: Full range from reading to distant viewing without correction

Myopic Eye:

• Far point: Finite distance (e.g., 1-2 meters)
• Near point: Often closer than 25 cm
• Functionality: Excellent close vision (may read at closer distances), but struggles with driving, recognizing people at distance, viewing presentations

Hyperopic Eye:

• Far point: Infinity (with accommodation strain)
• Near point: Further than 25 cm (e.g., 50-100 cm)
• Functionality: May see distant objects clearly (with muscle strain), but struggles with reading, phone use, detailed close work, leading to eye fatigue and headaches

In practical terms, myopia affects navigation and recognition at distance, while hyperopia impacts reading, writing, and precision tasks at normal working distances.