Hoja de repaso: Fundamentals of Wave Optics and Cosmology

📋 Course Outline

1. Travelling waves and wave speed relations
2. Reflection and refraction of plane waves
3. Lenses, ray diagrams and image formation
4. Electromagnetic spectrum and cosmology basics
5. Solar system, stars and stellar evolution
6. Total internal reflection and optical instruments
7. Forces, motion, momentum and impulse
8. Magnetism and electromagnetism
9. Electric circuits, transformers and rectification

📖 1. Travelling waves and wave speed relations

🔑 Key Concepts & Definitions

• Travelling wave : A travelling wave is a disturbance that propagates through space, carrying energy and momentum without transporting matter overall.
• Transverse wave : A transverse wave is a wave where particle displacement is perpendicular to the direction of propagation.
• Longitudinal wave : A longitudinal wave is a wave where particle displacement is parallel to the direction of propagation.
• Wavelength : Wavelength is the spatial period, measured as the distance between two successive points in phase.
• Frequency : Frequency is the number of oscillations per second, measured in hertz (Hz).

📝 Essential Points

• Wave speed relates to wavelength and frequency through $v=f\lambda$.
• For a fixed medium, changing frequency changes wavelength if wave speed stays constant.
• Light is concluded to behave as a transverse wave in the context of wave behaviour.
• Travelling waves can be described using wavefronts and propagation direction in diagrams.
• Speed–frequency–wavelength relations are used to solve problems involving the speed of light.

💡 Memory Hook

Think $v=f\lambda$: speed equals frequency times wavelength.

📖 2. Reflection and refraction of plane waves

🔑 Key Concepts & Definitions

• Plane wave reflection : Plane wave reflection is the bouncing of wavefronts from a flat boundary while preserving the wavefront structure.
• Plane wave refraction : Plane wave refraction is the bending of wavefronts when a wave crosses into a different medium.
• Equal angle law : The equal angle law states that the incident and reflected rays make equal angles with the normal.
• Refractive index : Refractive index is a dimensionless quantity that links wave speed in a medium to wave speed in vacuum.
• Critical angle : Critical angle is the incidence angle above which total internal reflection occurs at a boundary.

📝 Essential Points

• Reflection at a plane reflector uses the equal angle law for the incident and reflected angles.
• Wave frequency is not changed when a wave reflects at a boundary.
• Refraction is explained by a change in propagation speed across the boundary.
• Snell–Descartes law of sines is used to relate angles to refractive indices.
• The refractive index relates to propagation speeds on each side of the boundary via $n=\frac{c}{v}$ (and ratios between media).
• For total internal reflection, the critical angle depends on refractive index through the boundary condition for the onset of refraction.

💡 Memory Hook

Reflection keeps frequency; refraction changes speed and bends the wavefronts.

📖 3. Lenses, ray diagrams and image formation

🔑 Key Concepts & Definitions

• Principal axis : The principal axis is the symmetry line of a lens along which principal rays are drawn.
• Principal focus : The principal focus is the point on the principal axis where rays parallel to the axis converge after refraction (for a converging lens).
• Optical center : The optical center is the point of a lens where a ray passes undeviated in direction (for thin-lens ray diagrams).
• Focal length : Focal length is the distance from the optical center to the principal focus.
• Thin convex lens : A thin convex lens is a converging lens that can form real images when object distance is suitable.

📝 Essential Points

• Rays parallel to the principal axis emerge passing through the principal focus (for a thin convex lens).
• Rays aimed towards a principal focus emerge parallel to the principal axis (for a thin lens).
• A ray through the optical center continues straight without changing direction in the thin-lens model.
• Using the sign convention, real images are positive and virtual images are negative for distances.
• Thin-lens equation is $\frac{1}{u}+\frac{1}{v}=\frac{1}{f}$ and magnification is used alongside it to predict image size.
• Scale drawings and experiments are used to test the thin-lens formulae and magnification predictions.

💡 Memory Hook

Three principal rays: parallel→focus, focus→parallel, center→straight.

📖 4. Electromagnetic spectrum and cosmology basics

🔑 Key Concepts & Definitions

• Electromagnetic spectrum : The electromagnetic spectrum is the range of electromagnetic radiation types ordered by frequency and wavelength.
• Vacuum speed of light : Vacuum speed of light is the constant propagation speed $c$ shared by electromagnetic waves in vacuum.
• Astronomical Unit (AU) : The astronomical unit is a distance unit used for solar-system scales, defined in the course as AU.
• Light year : A light year is a distance unit defined by how far light travels in one year.
• Parsec : A parsec is a distance unit used for interstellar scales, related to parallax limitations.

📝 Essential Points

• Visible light is only a small part of the electromagnetic spectrum.
• All electromagnetic waves share the same speed in vacuum.
• Major components of the e-m spectrum are listed with their characteristic properties.
• Gravity is identified as the physical reason for orbital motion and for forming planets and stars.
• Cosmological red-shift is used as evidence for an expanding universe and is analysed using line spectra.
• The Hubble constant can be found from a plot of recessional velocity versus distance, and $1/H_0$ is used as an approximate age of the universe.

💡 Memory Hook

EM waves differ by frequency; in vacuum they all share the same speed.

📖 5. Solar system, stars and stellar evolution

🔑 Key Concepts & Definitions

• Heliocentric model : The heliocentric model places the Sun at the center of the solar system with planets orbiting it.
• Sidereal day : A sidereal day is the time for Earth to rotate once relative to distant stars.
• Nuclear fusion : Nuclear fusion is the process where stars convert mass into energy.
• Stellar evolution path : A stellar evolution path is the sequence of stages a star follows over time.
• Mass of a star : Mass is the key stellar property that determines the star’s evolutionary path and final state.

📝 Essential Points

• The solar system includes major features, and planet distances are very large compared with planet sizes.
• Evidence for heliocentrism is recognised and described as being applied to support the model.
• Roles of Copernicus, Galileo, Kepler and Newton are recalled as part of the development of orbital understanding.
• Kepler’s laws are recalled and applied to orbital motion problems.
• The definition of AU is recalled and unit conversions for astronomical measurement are required.
• Constellations are described as non-scientific groupings used to identify stars and patterns.

💡 Memory Hook

Star fate is mass-driven: mass → evolution path → final state.

📖 6. Total internal reflection and optical instruments

🔑 Key Concepts & Definitions

• Total internal reflection : Total internal reflection is the complete reflection of a wave at a boundary when the incidence angle exceeds the critical angle.
• Mirage formation : Mirage formation is an optical effect explained using total internal reflection in the atmosphere.
• Prismatic periscope : A prismatic periscope is an optical device that uses total internal reflection to redirect light.
• Fibre optics : Fibre optics uses guided light in fibres, relying on internal reflection to transmit signals.
• Endoscope : An endoscope is an optical instrument that uses fibre optics to view internal parts of the body.

📝 Essential Points

• Total internal reflection is linked to the critical angle and refractive index conditions at a boundary.
• Mirage formation is explained as a consequence of total internal reflection in air layers.
• Prismatic periscope operation is described as using total internal reflection to turn light paths.
• Bicycle reflectors are identified as practical applications of total internal reflection.
• Fibre optics are used for communication and for endoscopes to transmit light efficiently.

💡 Memory Hook

If the angle is too big (beyond critical), light refuses to escape: it reflects.

📖 7. Forces, motion, momentum and impulse

🔑 Key Concepts & Definitions

• Newton’s three laws of motion : Newton’s three laws describe how forces relate to motion, inertia, and action–reaction pairs.
• SUVAT equations : SUVAT equations are kinematic relations connecting displacement, initial velocity, final velocity, acceleration, and time in 1D motion.
• Momentum : Momentum is the quantity of movement defined as mass times velocity.
• Impulse : Impulse is the effect of a force acting over a time interval, equal to the change in momentum.
• Reaction time : Reaction time is the delay between a stimulus and the start of braking or corrective action.

📝 Essential Points

• The three laws of motion are applied to controlled acceleration/deceleration and to impacts.
• SUVAT equations in one dimension are used, including verifying the acceleration due to gravity.
• For road safety, relevant forces include gravity, engine/braking forces, friction between tyres and road, and forces on the driver/passenger.
• Stopping distance is linked to braking distance plus the distance travelled during reacting, and reaction time can increase with tiredness or certain drugs.
• Kinetic energy change is related quantitatively to the distance over which a force acts during stopping.
• Momentum is defined as $p=mv$ and conservation is used in 1D systems to solve problems.

💡 Memory Hook

Impulse = force × time → gives the momentum change.

📖 8. Magnetism and electromagnetism

🔑 Key Concepts & Definitions

• Ferromagnetic elements : Ferromagnetic elements are materials that strongly respond to magnetic fields, listed in the course as cobalt, nickel and iron.
• Magnetic field lines : Magnetic field lines are a graphical representation of magnetic field direction and relative strength.
• Neutral point : A neutral point is a location where magnetic effects cancel so the net force on a test pole is zero.
• Right-hand rule : The right-hand rule is a convention that links current direction in a conductor to the direction of the magnetic field.
• Laplace’s force : Laplace’s force is the force on a current-carrying wire in a magnetic field, used to predict direction and existence of turning effects.

📝 Essential Points

• Magnets have properties that include attraction/repulsion behaviour between poles.
• Cobalt, nickel and iron are the three (ferro) magnetic elements listed.
• Magnetic attraction and repulsion rules are recalled and used to predict interactions.
• A plotting compass can be used to map magnetic field directions.
• Field patterns for simple magnet arrangements can be interpreted to identify neutral points.
• The direction of a field line is taken as the direction of force on a north-seeking pole.

💡 Memory Hook

Field line direction tells the force on a north pole.

📖 9. Electric circuits, transformers and rectification

🔑 Key Concepts & Definitions

• Series resistors : Series resistors are connected end-to-end so the same current flows through each component.
• Parallel resistors : Parallel resistors are connected across the same two nodes so the voltage across each is the same.
• Electromagnet : An electromagnet is a magnet produced by current in a coil, with strength depending on circuit conditions.
• AC and DC supplies : AC and DC supplies differ in how the electrical quantity varies with time, with AC alternating and DC remaining steady.
• RMS value : RMS value is the effective AC voltage level used for power-equivalent comparisons.

📝 Essential Points

• Combined resistance calculations are performed for resistors in series and in parallel.
• A simple electromagnet is constructed and compared with a permanent magnet in terms of behaviour.
• A moving-coil loudspeaker is interpreted using the interaction between current and magnetic field to explain its action.
• A changing magnetic field induces an e.m.f in a circuit, with size and direction depending on qualitative factors.
• AC and DC are distinguished in nature, and 230 V ac is stated as the RMS value with $V_{rms}=\frac{V_{peak}}{\sqrt{2}}$.
• A basic iron-cored transformer operates by electromagnetic induction to transform voltage and current using the turns ratio, and high-voltage/low-current transmission reduces energy loss.

💡 Memory Hook

Transformer: turns ratio controls voltage/current; RMS is the effective AC voltage.

📊 Synthesis Tables

Reflection vs refraction at boundaries

⚠️ Common Pitfalls & Confusions

1. Mixing up transverse and longitudinal displacement directions relative to propagation.
2. Using $v=f\lambda$ with the wrong assumption that speed stays constant across media.
3. Forgetting that reflection keeps frequency unchanged while refraction involves speed change and bending.
4. Applying the equal angle law to refraction instead of reflection.
5. Using the thin-lens sign convention incorrectly (real positive, virtual negative).
6. Confusing critical angle with the angle of incidence itself rather than the threshold for total internal reflection.
7. Treating stopping distance as only braking distance and forgetting the reacting distance component.

✅ Exam Checklist

1. Define travelling waves and distinguish transverse from longitudinal waves.
2. Use $v=f\lambda$ to solve speed–frequency–wavelength problems, including light speed applications.
3. State and apply the equal angle law for reflection of plane waves.
4. State that reflection does not change wave frequency and use it in reasoning.
5. Explain refraction qualitatively and quantitatively using speed change and Snell–Descartes law of sines.
6. Define refractive index and relate it to propagation speeds on each side of a boundary.
7. Draw wavefront and ray diagrams for refraction across media.
8. Define critical angle and relate it to refractive index for total internal reflection.
9. Apply total internal reflection to mirages, prismatic periscopes, and bicycle reflectors.
10. Describe fibre optics uses in communication and endoscopes.
11. Define principal axis, principal focus, optical center and focal length for thin lenses.
12. Use principal rays to predict ray paths through thin convex and concave lenses.
13. Construct scale ray diagrams to determine image location, nature and magnification.
14. Apply $\frac{1}{u}+\frac{1}{v}=\frac{1}{f}$ with the sign convention real positive, virtual negative and use magnification formulas to solve lens problems; verify with experiments conceptually/qualitatively as required.