Study sheet: Ray Optics and Optical Instruments

📋 Course Outline

1. Light and Ray Approximation
2. Spherical Mirror Reflection
3. Mirror Equation and Magnification
4. Refraction and Snell’s Law
5. Total Internal Reflection
6. Refraction by Lenses
7. Prisms and Optical Instruments
8. Simple and Compound Microscopes
9. Astronomical and Reflecting Telescopes
10. Ray Optics Formula Summary
11. Conceptual Points on Optical Imaging

📖 1. Light and Ray Approximation

🔑 Key Concepts & Definitions

• Light : Electromagnetic radiation with wavelengths of about 400 nm to 750 nm
• Ray of light : The straight-line path along which light is considered to travel when its wavelength is small compared with ordinary objects
• Beam of light : A bundle of rays of light

📝 Essential Points

• The speed of light in vacuum is c = 2.99792458 × 10^8 m s–1, commonly approximated as 3 × 10^8 m s–1.

💡 Memory Hook

Wave nature versus straight-line ray approximation

📖 2. Spherical Mirror Reflection

🔑 Key Concepts & Definitions

• Principal axis : The line joining its pole to its centre of curvature
• Principal focus : The point where reflected parallel paraxial rays converge, while for a convex mirror it is the point from which the reflected rays appear to diverge

📝 Essential Points

📌 For reflection at a spherical mirror, the incident ray, reflected ray, and normal at the point of incidence lie in the same plane, and the angle of reflection equals the angle of incidence.

📐 Formula — For a spherical mirror, the focal length satisfies $f = \frac{R}{2}$, where R is the radius of curvature.

💡 Memory Hook

Pole → focus → centre of curvature

📖 3. Mirror Equation and Magnification

🔑 Key Concepts & Definitions

• Real image : Formed where reflected or refracted rays actually converge
• Virtual image : Formed where reflected or refracted rays do not actually meet but appear to diverge from that point when extended backward

★ Must-know

• The Cartesian sign convention assigns signs as follows:
  • Distances measured in the direction of incident light are positive
  • Distances measured opposite to incident light are negative
  • Upward heights are positive
  • Downward heights are negative

📐 Formula — The mirror equation is $\frac{1}{v}+\frac{1}{u}=\frac{1}{f}$.

📐 Formula — The linear magnification of a spherical mirror is $m=\frac{h'}{h}=-\frac{v}{u}$.

Further detail

• Covering half of a concave mirror still produces the complete image of the object, but the image intensity is reduced, in this case by half.

📖 4. Refraction and Snell’s Law

🔑 Key Concepts & Definitions

• Refraction : The change in direction of an obliquely incident ray when it enters another transparent medium at an interface

★ Must-know

📐 Formula — Snell’s law is $n_{21}=\frac{\sin i}{\sin r}$, where n21 is the refractive index of medium 2 relative to medium 1.

📌 When n21 is greater than 1, the refracted ray bends toward the normal, whereas when n21 is less than 1, it bends away from the normal.

Further detail

📐 Formula — Reciprocal refractive indices satisfy $n_{12}=\frac{1}{n_{21}}$, and relative indices satisfy $n_{32}=n_{31}n_{12}$.

• For a parallel-sided rectangular slab, the emergent ray is parallel to the incident ray but is laterally displaced.

• For near-normal viewing through water, apparent depth equals real depth divided by the refractive index of water.

💡 Memory Hook

Optically denser: toward the normal; optically rarer: away from the normal

📖 5. Total Internal Reflection

🔑 Key Concepts & Definitions

• Total internal reflection : The complete reflection of light back into an optically denser medium when light attempts to pass into a rarer medium at an incidence angle greater than the critical angle
• Critical angle : The angle of incidence in the denser medium for which the refracted ray in the rarer medium makes an angle of 90° with the normal

★ Must-know

📐 Formula — For a denser medium 1 and rarer medium 2, the critical angle satisfies $\sin i_c=n_{21}$ and $n_{12}=\frac{1}{\sin i_c}$.

• Optical fibres transmit light through repeated total internal reflection between a higher-index core and a lower-index cladding, with no appreciable loss of signal intensity at each reflection.

Further detail

• The critical angles with respect to air are:

  • 48.75° for water
  • 41.14° for crown glass
  • 37.31° for dense flint glass
  • 24.41° for diamond

• In silica glass fibres, more than 95% of the light can be transmitted over a fibre length of 1 km.

💡 Memory Hook

Incidence beyond the critical angle → complete internal reflection

📖 6. Refraction by Lenses

🔑 Key Concepts & Definitions

• Lens : A transparent optical medium bounded by two surfaces, at least one of which is spherical

★ Must-know

📐 Formula — Refraction at a spherical surface satisfies $\frac{n_2}{v}-\frac{n_1}{u}=\frac{n_2-n_1}{R}$.

📐 Formula — The thin lens formula is $\frac{1}{v}-\frac{1}{u}=\frac{1}{f}$.

📐 Formula — The power of a lens is $P=\frac{1}{f}$ when f is measured in metres, and its SI unit is the dioptre, with $1\,\mathrm{D}=1\,\mathrm{m}^{-1}$.

Further detail

📐 Formula — The magnification produced by a lens is $m=\frac{h'}{h}=\frac{v}{u}$.

📐 Formula — The total magnification of a combination of thin lenses is the product of the individual magnifications: $m=m_1m_2m_3\cdots$.

💡 Memory Hook

First surface → second surface → equivalent lens

📖 7. Prisms and Optical Instruments

🔑 Key Concepts & Definitions

• Simple microscope : A converging lens of small focal length used to produce an erect, magnified, virtual image of a nearby object
• Compound microscope : A compound microscope uses an objective to form a real, inverted, magnified image and an eyepiece to magnify that intermediate image into a final virtual image.

★ Must-know

📐 Formula — For a prism, the angle of deviation satisfies $\delta=i+e-A$ and the prism geometry gives $r_1+r_2=A$.

📐 Formula — At minimum deviation, the ray inside the prism is parallel to its base, the incidence and emergence angles are equal, and the refractive index satisfies $n_{21}=\frac{\sin[(A+D_m)/2]}{\sin(A/2)}$.

📐 Formula — For a simple microscope with the final image at the near point, the magnification is $m=1+\frac{D}{f}$, while for the final image at infinity it is $m=\frac{D}{f}$.

Further detail

📐 Formula — For a thin prism, the minimum deviation is approximately $D_m=(n_{21}-1)A$.

📖 8. Simple and Compound Microscopes

★ Must-know

📐 Formula — For a simple microscope with the final image at the near point, the magnifying power is $m=1+\frac{D}{f}$, where $D=25\,\mathrm{cm}$ is the least distance of distinct vision and $f$ is the focal length of the convex lens.

📐 Formula — For a simple microscope with the final image at infinity, the angular magnifying power is $m=\frac{D}{f}$, where $D=25\,\mathrm{cm}$.

📐 Formula — For a compound microscope with the final image at infinity, the total magnifying power is $m=\frac{L}{f_o}\frac{D}{f_e}$, where $L$ is the tube length and $f_o$ and $f_e$ are the focal lengths of the objective and eyepiece.

Further detail

• A compound microscope achieves large magnification by using small focal lengths for both the objective and eyepiece, but in practice making a focal length much smaller than 1 cm is difficult.

💡 Memory Hook

Object → objective → eyepiece → final image

📖 9. Astronomical and Reflecting Telescopes

🔑 Key Concepts & Definitions

• Telescope : Provides angular magnification of distant objects using an objective and an eyepiece; its objective has a large focal length and a much larger aperture than the eyepiece.
• Reflecting telescope : Uses a concave mirror instead of a lens as its objective, avoiding chromatic aberration and allowing support over the mirror's back surface.

★ Must-know

📐 Formula — For a refracting telescope in normal adjustment, the magnifying power is $m=\frac{f_o}{f_e}=\frac{\beta}{\alpha}$ and the telescope tube length is $f_o+f_e$.

• The light-gathering power of an astronomical telescope depends on the area of its objective, while its resolving power also improves when the objective diameter is increased.

Further detail

• A Cassegrain telescope uses a convex secondary mirror to deflect light through a hole in the primary mirror, providing a large focal length in a short telescope.

💡 Memory Hook

Refracting telescope: lens objective; reflecting telescope: mirror objective

📖 10. Ray Optics Formula Summary

🔑 Key Concepts & Definitions

• Critical angle : The angle of incidence in a denser medium for which the refracted ray in the rarer medium makes an angle of 90°; for incidence greater than this angle, total internal reflection occurs.
• Dispersion : The splitting of light into its constituent colours.

★ Must-know

📐 Formula — The mirror equation is $\frac{1}{v}+\frac{1}{u}=\frac{1}{f}$, and the focal length of a spherical mirror is approximately half its radius of curvature, $f=R/2$.

📐 Formula — The thin-lens formula is $\frac{1}{v}-\frac{1}{u}=\frac{1}{f}$, the lens-maker formula is $\frac{1}{f}=(n-1)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)$, and the power of a lens is $P=1/f$ measured in dioptres, with $1\,\mathrm{D}=1\,\mathrm{m}^{-1}$.

Further detail

📐 Formula — For thin lenses in contact, the effective focal length satisfies $\frac{1}{f}=\frac{1}{f_1}+\frac{1}{f_2}+\frac{1}{f_3}+\cdots$ and the total power satisfies $P=P_1+P_2+P_3+\cdots$.

📖 11. Conceptual Points on Optical Imaging

★ Must-know

• A real image can exist in space without a screen because rays from each object point converge at an image point and diverge afterward; the screen only diffuses some rays toward the eye.

📌 Image formation requires regular reflection or refraction so that rays from a given object point reach the same image point.

📌 A magnifying glass can have equal angular sizes for the object and virtual image while still providing angular magnification because the object is viewed closer than the normal near point of 25 cm.

Further detail

• Thick lenses produce coloured images because of dispersion, and the perceived colour of an object depends on the constituent colours of the incident light.

💡 Memory Hook

Regular reflection or refraction → rays meet at a definite image point

📊 Synthesis Tables

Mirror and lens image relations

System	Equation	Magnification
Spherical mirror	$\frac{1}{v}+\frac{1}{u}=\frac{1}{f}$	$m=-\frac{v}{u}$
Thin lens	$\frac{1}{v}-\frac{1}{u}=\frac{1}{f}$	$m=\frac{v}{u}$

Optical Instrument Comparison

Instrument	Objective or lens	Main magnification relation
Simple microscope	Single convex lens	Near point: $1+D/f$; infinity: $D/f$
Compound microscope	Objective plus eyepiece	Approximately $LD/(f_of_e)$
Refracting telescope	Large-focal-length lens objective plus eyepiece	$f_o/f_e$
Reflecting telescope	Concave mirror objective	Large aperture improves light gathering and resolution