Optical Instruments notes: ISC Class 12 Physics

1. Magnifying power and the simple microscope

Magnifying power $M = \dfrac{\beta}{\alpha}$ : β is the angle subtended at the eye by the final image; α is the angle subtended by the object without the instrument (for a microscope, the object held at the near point, D = 25 cm). Small angles: angle = height/distance.
Simple microscope (a convex lens held close to the eye, object inside F): virtual, erect, enlarged image.
Image at D (derivation): v = −D in $\dfrac{1}{v} - \dfrac{1}{u} = \dfrac{1}{f}$ gives $\dfrac{1}{|u|} = \dfrac{1}{f} + \dfrac{1}{D}$ . With the eye at the lens, β = h/|u| and α = h/D, so $M = \dfrac{D}{|u|} = 1 + \dfrac{D}{f}$ (equal to the linear magnification v/u).
Image at infinity (derivation): object at F, rays leave parallel; β = h/f, α = h/D, so $M = \dfrac{D}{f}$ (relaxed eye; one less than at D).
A shorter focal length magnifies more, but very short lenses are tiny, need the object almost touching them and suffer aberrations: in practice M up to about 10.

image at D: 1 + D/f
image at infinity: D/f

Shorter focal length, larger magnifying power; image at D always gives one more.

Formulas

Simple microscope, image at D
$M = 1 + \dfrac{D}{f}$
Simple microscope, image at infinity
$M = \dfrac{D}{f}$

Key terms

Near point (D): The least distance of distinct vision, about 25 cm for a normal eye: the closest an object can be seen sharply.
Magnifying power: The ratio of the angle subtended at the eye by the final image to the angle subtended by the object viewed without the instrument.

2. The compound microscope

Construction: objective of very short focal length f_o and small aperture; eyepiece of short focal length f_e. Object just beyond F_o; the objective forms a real, inverted, enlarged image A′B′ just inside F_e; the eyepiece forms the final virtual image A″B″ (inverted relative to the object, greatly enlarged).
Final image at D (derivation): β = h′/|u_e| (A″B″ subtends the same angle as A′B′ at the eye), α = h/D, so $M = \dfrac{h'}{h}\cdot\dfrac{D}{|u_e|} = m_o m_e$ with $m_o = \dfrac{v_o}{|u_o|}$ and $m_e = 1 + \dfrac{D}{f_e}$ : $M = \dfrac{v_o}{|u_o|}\left(1 + \dfrac{D}{f_e}\right)$ . Lens separation $v_o + |u_e|$ .
Approximation: $\dfrac{v_o}{|u_o|} = \dfrac{v_o}{f_o} - 1 \approx \dfrac{L}{f_o}$ (object near F_o, A′B′ a distance L ≈ tube length from the objective).
Final image at infinity (expression only, derivation not required): A′B′ at F_e; $M = \dfrac{v_o}{|u_o|}\cdot\dfrac{D}{f_e} \approx \dfrac{L}{f_o}\cdot\dfrac{D}{f_e}$ ; separation $v_o + f_e$ .
High M needs short f_o and short f_e; an object at 2F_o would give m_o = 1.

Formulas

Compound microscope, image at D
$M = \dfrac{v_o}{|u_o|}\left(1 + \dfrac{D}{f_e}\right)$
Compound microscope, image at infinity
$M = \dfrac{v_o}{|u_o|}\cdot\dfrac{D}{f_e}$

Key terms

Objective: The lens nearer the object; in a microscope it has a very short focal length and forms a real, enlarged image.
Eyepiece: The lens next to the eye; it acts as a simple microscope on the image formed by the objective.

3. The refracting astronomical telescope

Construction: objective of long focal length f_o and large aperture (collects light, forms a real image of the distant object at F_o); eyepiece of short focal length f_e. α is the angle the object subtends where it is.
Normal adjustment (derivation): the image I (height h) lies at F_o and at F_e. α = h/f_o, β = h/f_e, so $M = \dfrac{f_o}{f_e}$ ; length $L = f_o + f_e$ ; final image at infinity, inverted.
Final image at D (derivation): I lies |u_e| from the eyepiece with $\dfrac{1}{|u_e|} = \dfrac{1}{f_e} + \dfrac{1}{D}$ ; β = h/|u_e|, α = h/f_o, so $M = \dfrac{f_o}{|u_e|} = \dfrac{f_o}{f_e}\left(1 + \dfrac{f_e}{D}\right)$ ; length $f_o + |u_e|$ (slightly shorter).
The objective’s diameter does not change M; the light collected grows with its area (brighter images, finer detail). Terrestrial telescopes add an erecting system.

Formulas

Telescope, normal adjustment
$M = \dfrac{f_o}{f_e},\quad L = f_o + f_e$
Telescope, image at D
$M = \dfrac{f_o}{f_e}\left(1 + \dfrac{f_e}{D}\right)$

Key terms

Normal adjustment: The setting of a telescope in which the final image is at infinity, the image formed by the objective lying at the focus of the eyepiece.

4. Reflecting telescopes

Cassegrain type: a large concave paraboloidal primary mirror reflects the light towards its focus; a small convex secondary mirror inside that focus reflects it back through a hole in the primary to a focus behind it, viewed with an eyepiece (or recorded by a camera). Image at infinity: $M = \dfrac{F}{f_e}$ , F being the focal length of the mirror system (the convex secondary makes F several times the primary’s focal length while the tube stays short).
Advantages over refractors: no chromatic aberration (reflection does not depend on n); no spherical aberration with a paraboloid; a mirror is supported over its whole back, so very large apertures are possible (a lens sags); only one surface to figure and no light absorbed in glass, so cheaper and brighter; greater light-gathering and resolving power; short tube.
Drawbacks: the secondary blocks a little light (the image stays complete, slightly dimmer); mirrors need re-coating and careful alignment.

Formulas

Reflecting telescope, image at infinity
$M = \dfrac{F}{f_e}$

Key terms

Chromatic aberration: Coloured fringes in an image because a lens focuses different colours at different distances, its refractive index depending on wavelength.
Spherical aberration: Blurring because rays far from the axis are focused at a different point from paraxial rays.

5. Resolving power; uses, advantages and drawbacks

Resolving power of a microscope (formula only): two points closer than $d = \dfrac{\lambda}{2n\sin\beta}$ cannot be seen as separate; resolving power $= \dfrac{1}{d} = \dfrac{2n\sin\beta}{\lambda}$ . β is the half-angle of the cone of light from the object accepted by the objective; n sin β is the numerical aperture.
Factors: wavelength (shorter is better), refractive index of the medium between object and objective, and β. To increase it: blue, violet or ultraviolet light (or electrons in an electron microscope); oil immersion (n ≈ 1.5); a wide-aperture objective close to the object. A stronger eyepiece gives only empty magnification.
Uses: simple microscope — watch and jewellery repair, reading fine print, field biology; compound microscope — cells, bacteria, blood tests, metallurgy, forensic work; telescopes — astronomy, navigation, surveying.
Advantages and drawbacks: simple microscope cheap and erect but M ≤ about 10; compound microscope high M but inverted image, small field, needs strong light; refractors sealed and simple but chromatic aberration and size limits; reflectors large and colour-free but a central obstruction and upkeep.

green light, 550 nm
violet light, 400 nm

d = λ/(2 n sin β): larger numerical aperture and shorter wavelength resolve finer detail.

Formulas

Resolving power of a microscope
$\dfrac{1}{d} = \dfrac{2n\sin\beta}{\lambda}$

Key terms

Resolving power: The reciprocal of the smallest separation between two points that an instrument can show as separate.
Numerical aperture: The product n sin β for a microscope objective, where β is the half-angle of the accepted cone of light and n the index of the medium in front of the objective.

Exam tips

Always say what α and β are; D = 25 cm; magnifying powers are quoted as magnitudes.
Microscopes: simple 1 + D/f (D) and D/f (infinity); compound (v_o/|u_o|)(1 + D/f_e) at D, (v_o/|u_o|)(D/f_e) at infinity (expression only).
Telescopes: f_o/f_e with L = f_o + f_e; (f_o/f_e)(1 + f_e/D) at D.
Reflectors: concave primary, convex secondary, hole; no chromatic aberration. Resolving power 2n sin β/λ, not changed by the eyepiece.

1. Magnifying power and the simple microscope

Formulas

Key terms

2. The compound microscope

Formulas

Key terms

3. The refracting astronomical telescope

Formulas

Key terms

4. Reflecting telescopes

Formulas

Key terms

5. Resolving power; uses, advantages and drawbacks

Formulas

Key terms

Exam tips

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