Wave Motion and Energy Transfer Study Guide

Section A: Temperature

• Definition of Temperature: A measure of the hotness of a body.
• S.I. Unit: The Kelvin ($K$).
• Celsius Scale:
  • Developed by Anders Celsius in 1742.
  • Based on two reference points: the freezing point of water at $0^{\circ}\text{C}$ and the boiling point of water at $100^{\circ}\text{C}$.
  • Common uses: Air temperature, body temperature, oven temperature.
• Kelvin Scale:
  • Developed by Lord Kelvin in 1848.
  • Removes the need for negative temperatures because its lowest value is zero ($0\,\text{K}$).
  • Absolute Zero: This point is where internal energy is at a theoretical minimum value.
• Temperature Conversion Formulas:
  • Temperature in Celsius = Temperature in Kelvin $- 273.15$
  • $T(^{\circ}\text{C}) = T(\text{K}) - 273.15$
  • Example 1(i): To convert $325\,\text{K}$ to Celsius: $t = 325 - 273.15 = 51.85^{\circ}\text{C}$.
  • Example 1(ii): To convert $26^{\circ}\text{C}$ to Kelvin: $26 = T - 273.15 \Rightarrow T = 299.15\,\text{K}$.
• Thermometric Properties: Any physical property that changes measurably with a change in temperature.
  • Length of a column of liquid: Liquids expand when heated (e.g., Mercury-in-glass thermometer). Liquid rises in a narrow column as it expands.
  • Electrical Resistance:
    • In metals, resistance increases with temperature because particles vibrate more, hindering current flow.
    • In thermistors (semiconductors), resistance decreases as temperature increases.
  • E.m.f. of a thermocouple: If two different metals are joined in a circuit and junctions are kept at different temperatures, an electromotive force (voltage) is induced. Greater temperature difference results in greater e.m.f.
  • Volume of a gas at constant pressure: Volume increases when heated ($pV$ is constant).
  • Pressure of a gas at constant volume: Pressure increases when heated.
  • Colour: Certain crystals change colour with temperature, used in plastic strip body thermometers.
• Practical Thermometers:
  • Clinical thermometer: Small scale ($36^{\circ}\text{C} - 37^{\circ}\text{C}$); uses alcohol instead of poisonous mercury; features a constriction to prevent liquid backflow until shaken.
  • Oven thermometer.
  • Temperature gauge in a car.

Section B: Heat, Internal Energy, and Transfers

• Heat: A form of energy that causes temperature rise when added and temperature fall when removed.
• Internal Energy: The sum of the potential and kinetic energies of the molecules in a body.
• States of Matter: Generally three states: solid, liquid, and gas. Adding heat causes transitions from solid to liquid to gas.
• Thermal Equilibrium: Heat moves from hot to cold until temperatures are identical, at which point flow stops.
• Conductors and Insulators:
  • Conductor: Substance allowing easy heat flow (e.g., metal spoon).
  • Insulator: Substance resisting heat flow (e.g., polystyrene cup).
• Methods of Heat Transfer:
  • Conduction: Transfer of heat through a solid without net movement of the particles. Energy passes via particle vibrations.
    • Demonstration: Conductivity ring with wax and ball bearings on strips of different metals. Copper allows the bearing to fall first, proving it is the best conductor.
  • Convection: Transfer of heat through a fluid (liquid or gas) due to particle movement. Heated particles become less dense and rise, creating a convection current.
    • Example: In a hot water immersion tank, the "sink" setting heats the top element (small quantity), while the "bath" setting heats the bottom element (entire tank).
  • Radiation: Transfer of heat as electromagnetic (usually infrared) radiation without needing matter. Heat from the sun travels through the vacuum of space this way.
    • Applications: Night-vision goggles, security camera motion sensors.
• Greenhouse Effect: Greenhouse gases trap escaping thermal energy reflected from the Earth's surface, acting like a blanket. Pollution increases these gases, causing global warming.
• Solar Constant: The amount of energy falling normally on $1\,\text{m}^{2}$ of the earth's atmosphere per second at mean distance from the sun.
  • Value: $\approx 1360\,\text{Wm}^{-2}$.
• Efficiency and U-values:
  • U-value: Amount of heat energy conducted per second through $1\,\text{m}^{2}$ of a substance when a temperature difference of $1^{\circ}\text{C}$ ($1\,\text{K}$) is maintained. Unit: $\text{Wm}^{-2}\text{K}^{-1}$. Lower U-values mean better insulation.
  • Percentage Efficiency Formula:
    • $\text{Percentage efficiency} = \frac{\text{Energy}_{\text{out}}}{\text{Energy}_{\text{in}}} \times 100$
    • $\text{Percentage efficiency} = \frac{\text{Power}_{\text{out}}}{\text{Power}_{\text{in}}} \times 100$
  • Example calculation: A kettle with $2.5\,\text{kJ}$ input and $1800\,\text{J}$ useful output has an efficiency of $\frac{1800}{2500} \times 100 = 72\%$.

Section C: Heat Transfers and Latent Heat

• Heat Capacity: Heat energy needed to change a substance's temperature by $1\,\text{K}$.
  • Formula: $\Delta E = \text{Heat Capacity} \times \Delta \theta$
• Specific Heat Capacity (SHC): Amount of energy needed to change the temperature of $1\,\text{kg}$ of a substance by $1\,\text{K}$. Unit: $\text{J kg}^{-1}\text{K}^{-1}$.
  • Formula: $\Delta E = m c \Delta \theta$
  • Common SHC Values:
    • Water: $4180$
    • Aluminium: $897$
    • Steel: $466$
    • Copper: $385$
  • Applications of SHC:
    • Storage Heaters: High SHC bricks heat up at night and release heat during the day.
    • Car Coolants: Water's high SHC allows it to absorb significant engine heat without boiling quickly.
    • Cookware: Low SHC metals (copper/aluminium) heat up fast.
• Conservation of Energy in Heat: Energy lost by one body = Energy gained by the other body.
• Latent Heat: Heat taken in or released during a change of state without a change in temperature.
• Specific Latent Heat (SLH): Energy needed to change the state of $1\,\text{kg}$ of a substance. Unit: $\text{J kg}^{-1}$.
  • Formula: $\Delta E = m l$
  • Specific Latent Heat of Fusion: Change from solid to liquid.
  • Specific Latent Heat of Vaporisation: Change from liquid to gas.
  • Example: Energy to condense $500\,\text{g}$ of steam ($100^{\circ}\text{C}$) and cool to $50^{\circ}\text{C}$.       $\text{Total Energy} = (m l) + (m c \Delta \theta) = (0.5 \times 2.26 \times 10^{6}) + (0.5 \times 4180 \times 50) = 1,234,500\,\text{J}$.
• The Heat Pump: Transfers energy from a cold body to a warm body using a refrigerant with high SLH and a low boiling point.
  • Evaporation Phase: Pressure is reduced, liquid vaporises and absorbs latent heat (cooling the surroundings).
  • Compression Phase: Vapour is compressed into a liquid, releasing latent heat (heating the surroundings).

Section D: Optics – Reflection & Refraction

• Reflection: The bouncing of light off a surface.
  • Laws of Reflection:
    1. Angle of incidence ($i$) = Angle of reflection ($r$).
    2. The incident ray, normal, and reflected ray lie in the same plane.
  • Types: Regular reflection (smooth surfaces) vs. Diffuse reflection (rough surfaces).
  • Virtual Image: Formed by the apparent intersection of light rays; same size, upright, and located behind the mirror.
• Refraction: Bending of light as it passes between media of different densities due to speed changes.
  • Speed comparison: Light travels faster in air/vacuum and slower in glass/water.
  • Rules of Bending:
    • Rare to Denser: Bends towards the normal ($i > r$).
    • Denser to Rare: Bends away from the normal ($i < r$).
  • Snell’s Law: $\frac{\sin(i)}{\sin(r)} = n$ (where $n$ is the refractive index).
  • Refractive Index ($n$): Ratio of the sine of $i$ to the sine of $r$ when traveling from a vacuum into a medium.
    • $n = \frac{c_{1}}{c_{2}}$ (ratio of speeds).
    • $n = \frac{\text{Real depth}}{\text{Apparent depth}}$.
• Total Internal Reflection (TIR): Occurs when light travels from a denser to a rarer medium at an angle of incidence greater than the critical angle ($c$).
  • Critical Angle Formula: $n = \frac{1}{\sin(c)}$.
  • Applications of TIR:
    • Prism reflectors: Used in bicycles and road signs.
    • Mirages: Caused by light refracting through air layers of different temperatures near the ground.
    • Snell’s Window: The circle of light visible to an underwater observer.
    • Optical Fibres: Thin glass rods where light is trapped via TIR. Used in telecommunications and medical endoscopes.

Section E: Optics – Lenses

• Converging (Convex) Lens: Focuses parallel rays at a focal point ($F$).
  • Images: Can be real or virtual depending on object position ($u$).
  • Magnifying glass: Object placed between $F$ and the pole produces a virtual, enlarged, upright image.
• Diverging (Concave) Lens: Spreads out parallel rays. Always produces a virtual, upright, diminished image.
• Lens Formula: $\frac{1}{f} = \frac{1}{u} + \frac{1}{v}$
  • Real is positive rule ($f$ and $v$ are negative for diverging/virtual setups).
• The Human Eye:
  • Long-sightedness (Hypermetropia): Image focuses behind the retina. Corrected with a converging lens.
  • Short-sightedness (Myopia): Image focuses in front of the retina. Corrected with a diverging lens.

Section F: Waves and Wave Motion

• Definition of a Wave: A means of transferring energy through a medium without net movement of the medium itself.
• Types of Waves:
  • Transverse: Vibration is perpendicular to the direction of travel (e.g., light, water waves, rope waves).
  • Longitudinal: Vibration is parallel to the direction of travel (e.g., sound, ultrasound, compression on a spring).
• Wave Terminology:
  • Wavelength ($\lambda$): Distance between corresponding points on consecutive cycles.
  • Amplitude ($A$): Maximum displacement from rest. Energy $\propto A^{2}$.
  • Frequency ($f$): Cycles per second ($\text{Hz}$).
  • Period ($T$): Time for one complete oscillation. $T = \frac{1}{f}$.
  • Wave Speed Formula: $v = f \lambda$.
• Wave Phenomena:
  • Reflection: Bouncing off obstacles.
  • Refraction: Speed and wavelength change when entering a new medium, but frequency remains constant.
  • Diffraction: Spreading of waves through gaps. More pronounced when wavelength is similar to the gap width. Sound diffracts more than light around doors because its wavelength is longer.
  • Interference: Combining amplitudes of overlapping waves.
    • Constructive: Amplitudes add up.
    • Destructive: Amplitudes cancel. Total destructive interference occurs if waves are $180^{\circ}$ out of phase.
  • Polarisation: Only applies to transverse waves. Restricts vibration to one plane.
    • Applications: Polaroid sunglasses (reduce glare); stress testing in engineering (photoelasticity).
• Doppler Effect: Apparent shift in frequency due to the relative motion between source and observer.
  • Moving Towards: $f' = \frac{fc}{c - u}$ (Higher frequency).
  • Moving Away: $f' = \frac{fc}{c + u}$ (Lower frequency).
  • Applications: Red shift of stars (determining galactic speeds), radar guns for speed traps.

Section G: Sound

• Nature of Sound: Longitudinal wave produced by vibrations; requires a medium.
• Speed of Sound: Varies with density and elasticity. Air ($0^{\circ}\text{C}$) is $331\,\text{m/s}$; Steel is $4800\,\text{m/s}$.
• Limits of Audibility: $20\,\text{Hz}$ to $20\,\text{kHz}$.
• Ultrasonics: Frequencies $> 20\,\text{kHz}$. Used in medicine (scans), industrial cleaning (NDT), and navigation (SONAR).
• Stationary (Standing) Waves: Formed when two identical waves travel in opposite directions and meet.
  • Nodes: Zero movement points.
  • Antinodes: Maximum movement points.
  • Distance between two nodes = $\frac{\lambda}{2}$.
• Resonance: Transfer of energy between bodies with the same natural frequency.
  • Examples: Barton’s Pendulums, singers shattering glass, microwave ovens heating water molecules.
• Stretched Strings: Natural frequency depends on length ($l$), tension ($T$), and mass per unit length ($\mu$).
  • Formula: $f = \frac{n}{2l} \sqrt{\frac{T}{\mu}}$

Section H: Wave Nature of Light

• Theories of Light: Newton (corpuscular/particle theory) vs. Wave theory. Einstein unified these through the dual nature (photons acting as waves).
• Diffraction Grating: Slide with many parallel lines.
  • Formula: $n \lambda = d \sin(\theta)$.
  • Grating constant ($d$) = $\frac{1 \times 10^{-3}}{\text{lines per mm}}$.
• Dispersion: Breaking white light into constituent colours (spectrum). Short wavelengths (violet) refract/bend more than long wavelengths (red).
• Colours of Light:
  • Primary: Red, Green, Blue.
  • Secondary: Yellow ($R+G$), Cyan ($G+B$), Magenta ($B+R$).
• Electromagnetic Spectrum: All travel at $3 \times 10^{8}\,\text{ms}^{-1}$ in a vacuum.
  • Order (Long to Short $\lambda$): Radio, Microwaves, Infrared, Visible, Ultraviolet, X-rays, Gamma rays.
• Ionising Radiation: Radiation with energy to remove electrons from atoms (Alpha $\alpha$, Beta $\beta$, Gamma $\gamma$).
• Solar Irradiance ($I$): Solar power received per unit area.
  • Formula: $I = \frac{P}{A}$. Units: $\text{Wm}^{-2}$.

Questions & Discussion

• Question on Efficiency: Danny's toaster uses $650\,\text{J}$ useful energy from $800\,\text{J}$ input.
  • Efficiency = $\frac{650}{800} \times 100 = 81.25\%$.
• Question on Specific Heat: Find energy to heat $250\,\text{g}$ of water from $20^{\circ}\text{C}$ to $80^{\circ}\text{C}$.
  • $\Delta E = 0.25 \times 4180 \times 60 = 62700\,\text{J}$.
• Question on Refractive Index: Pond appears $2.7\,\text{m}$ deep, $n = \frac{4}{3}$.
  • Actual depth = $n \times \text{Apparent depth} = \frac{4}{3} \times 2.7 = 3.6\,\text{m}$.
• Question on Doppler Effect: Train moving at $25\,\text{ms}^{-1}$ towards observer, horn is $1000\,\text{Hz}$.
  • $f' = \frac{1000 \times 340}{340 - 25} = 1079.37\,\text{Hz}$.
• Question on Diffraction Grating: Grating has $280\,\text{lines/mm}$.
  • $d = \frac{1 \times 10^{-3}}{280} = 3.57 \times 10^{-6}\,\text{m}$.