National Year 13 Physics

Chapters: 8

1. 3.6 Further mechanics and thermal physics

3.6.1 Periodic motion · 3.6.2 Thermal physics

Simple Harmonic Motion (SHM) – A motion that repeats after a fixed time is periodic. If the object goes to and fro about a middle point and the force pulling it back is proportional to its distance from the middle (F = −kx), the motion is simple harmonic. Its position is x = A sin(ωt + φ), with ω = 2π/T = 2πf. Speed is largest in the middle, acceleration is largest at the ends, and total energy ½kA² stays constant.
Kinetic Theory of Gases: Pressure, Temperature and rms Speed – Kinetic theory explains gas behaviour by picturing a gas as tiny, fast, randomly moving molecules that bounce elastically and do not pull on each other. Each hit on a wall reverses the molecule's velocity and hands the wall momentum 2mvₓ. Adding the hits of all molecules gives the pressure P = ⅓ n m v̄² = ⅓ ρ v̄². Comparing with PV = N k T shows that the average kinetic energy of a molecule is (3/2) k T: temperature is a measure of the average kinetic energy of the molecules. The root mean square speed is v_rms = √(3RT/M) = √(3kT/m), so lighter gases move faster at the same temperature.

2. 3.7 Fields and their consequences

3.7.1 Fields · 3.7.2 Gravitational fields · 3.7.3 Electric fields · 3.7.4 Capacitance · 3.7.5 Magnetic fields

Fields and Forces – Some forces need touch, like a push with your hand. Others act across empty space: gravity, the electric force and the magnetic force. We explain these with a field: a region around an object where another object feels a force. Mass makes a gravitational field, charge makes an electric field, and magnets or currents make a magnetic field. We draw fields with field lines: the arrow shows the direction of force, and lines close together mean a strong field. Gravity and electric force get weaker with distance by the inverse-square law: double the distance, one quarter of the force. Fields also store energy. All forces in nature come from four fundamental forces: gravity, electromagnetism, the strong force and the weak force.
Newton's Universal Law of Gravitation – Every mass in the universe pulls every other mass. The pull between two point masses m₁ and m₂ a distance r apart is F = G m₁ m₂ / r². It acts along the line joining them. The two bodies pull each other with equal and opposite forces. G = 6.67 × 10⁻¹¹ N m² kg⁻² is the same everywhere, so it is called the universal gravitational constant. When many masses pull one body, the forces add as vectors (superposition).
Electric Charges and Fields – Charge comes in two kinds, is conserved and comes in whole-number packets of e = 1.6 × 10⁻¹⁹ C. Two point charges push or pull with F = kq₁q₂/r² (k = 9 × 10⁹ N m² C⁻²). Forces and fields from many charges add as vectors (superposition). The field E = F/q₀ of a point charge is kq/r²; field lines show it. A dipole (±q a distance 2a apart) has moment p = q·2a; in a uniform field it feels zero net force but a torque τ = pE sinθ. Electric flux Φ = E·A, and Gauss's law says the flux out of any closed surface is q_enclosed/ε₀, which quickly gives E for a long wire (λ/2πε₀r), a plane sheet (σ/2ε₀) and a thin spherical shell (kq/r² outside, 0 inside).
Electrostatic Potential and Capacitance – Electrostatic potential V at a point is the work done per unit positive charge to bring it from infinity: V = kq/r for a point charge, and potentials of many charges add as plain numbers. Potential difference V_B − V_A = W_AB/q. Equipotential surfaces join equal-V points; E is perpendicular to them and E = −dV/dr. Potential energy of two charges is U = kq₁q₂/r, and of a dipole in a field U = −pE cosθ. Conductors have free charges (E inside = 0); dielectrics have bound charges that polarise and cut the field to E₀/K. A capacitor stores charge Q = CV. A parallel plate capacitor has C = ε₀A/d, and KC with a dielectric. In series 1/C = 1/C₁ + 1/C₂ + …; in parallel C = C₁ + C₂ + …. Energy stored U = ½CV² = Q²/2C = ½QV.
Electromagnetic Induction – When the magnetic flux through a coil changes, an emf is produced in it: ε = −N dΦ/dt (Faraday). The minus sign is Lenz's law: the induced current always opposes the change that made it. Moving a rod in a field gives motional emf ε = Blv. A changing current makes an emf in its own coil (self-induction, L) and in a nearby coil (mutual induction, M).
Force on a Current-Carrying Conductor, Motor and Induction – A wire carrying current inside a magnetic field feels a push. The push is biggest when the wire is at 90° to the field and zero when it is parallel. Fleming's left-hand rule gives its direction. A motor uses this push to spin a coil; a generator does the reverse and makes current by moving a coil or magnet.

3. 3.8 Nuclear physics

3.8.1 Radioactivity

Radioactivity: Alpha, Beta, Gamma and Half-Life – Some atomic nuclei are unstable. They give out radiation at random to become more stable. This is radioactive decay. Alpha (2 protons + 2 neutrons), beta (a fast electron) and gamma (a wave of energy) are the three main kinds. Half-life is the time for half of the unstable nuclei to decay.

4. 3.9 Astrophysics

3.9.1 Telescopes · 3.9.2 Classification of stars · 3.9.3 Cosmology

Optical Instruments: Microscopes and Telescopes – How big a thing looks depends on the angle it makes at the eye. A simple microscope (one convex lens) gives m = 1 + D/f (image at D) or D/f (image at infinity). A compound microscope uses a short-focus objective and an eyepiece: m = mₒ × mₑ ≈ (L/fₒ)(D/fₑ). An astronomical telescope uses a long-focus objective and short-focus eyepiece: m = fₒ/fₑ in normal adjustment, with tube length fₒ + fₑ. Reflecting telescopes use a concave mirror as objective.
Cosmology: The Expanding Universe and the Big Bang – Cosmology is the study of the whole universe: its structure, history and future. Galaxies gather in groups, clusters and filaments around huge voids. Distant galaxies are moving away from us, faster the farther they are (Hubble's law, v = H₀d), because space itself is expanding. Running the expansion backwards leads to a hot, dense beginning about 13.8 billion years ago, the Big Bang. The main evidence is redshift, the cosmic microwave background and the amounts of hydrogen and helium. Most of the universe is dark matter and dark energy.

5. 3.10 Medical physics

3.10.1 Physics of the eye · 3.10.2 Physics of the ear · 3.10.3 Biological measurement · 3.10.4 Non-ionising imaging · 3.10.5 X-ray imaging · 3.10.6 Radionuclide imaging and therapy

Physics of the Eye: How We See and How Lenses Fix Vision – The eye is a converging lens system: the cornea does most of the bending and the lens fine-tunes the focus (accommodation). Light lands on the retina, where rods give sensitive night vision with poor detail and three types of cones give colour and fine detail. A normal eye has a near point of about 25 cm and a far point at infinity. Myopia (eye too long) is corrected with a diverging lens of f = −far point; hypermetropia (eye too short) with a converging lens; astigmatism with a cylindrical lens. Lens power P = 1/f in dioptres (D), and powers of thin lenses in contact add.
Physics of the Ear – The ear turns tiny pressure waves into nerve signals. The eardrum vibrates, three small bones (ossicles) act as a lever, and because the eardrum is much bigger than the oval window the pressure rises about 20 times. In the cochlea the basilar membrane vibrates at a place that depends on frequency: high frequencies near the base, low near the apex. Hair cells there send signals to the brain. A healthy young ear hears about 20 Hz to 20 kHz and is most sensitive at 2–5 kHz. The threshold of hearing at 1 kHz is I₀ = 1.0 × 10⁻¹² W m⁻². Intensity level is L = 10 log₁₀(I/I₀) dB. Hearing gets worse with age (mostly high frequencies) and with loud noise (a dip near 4 kHz).
Biological Measurement: the ECG – An electrocardiogram (ECG) records the tiny voltages made by heart muscle as it depolarises and repolarises. Electrodes on the skin, with conducting gel, pick up about 1 mV. A high-input-resistance amplifier makes the signal bigger (about 1000 times) and it is shown against time. A normal beat has a P wave (atria depolarise), a QRS complex (ventricles depolarise, atria repolarise) and a T wave (ventricles repolarise). The time between two R peaks is one beat; heart rate (beats per minute) = 60 ÷ R–R time in seconds.
Non-Ionising Imaging – Ultrasound, endoscopes and MRI let doctors see inside the body without X-rays or gamma rays, so there is no ionisation damage. Ultrasound sends short pulses (1–15 MHz) from a piezoelectric probe; echoes return from boundaries. Depth = c t ÷ 2 and the fraction reflected depends on acoustic impedance Z = ρc: R = (Z₂ − Z₁)² ÷ (Z₂ + Z₁)². Gel removes air, which would reflect almost everything. An endoscope uses optical fibres: light stays in the core by total internal reflection. An incoherent bundle carries light in; a coherent bundle carries the image out. In MRI a strong magnetic field lines up hydrogen nuclei, which precess at the Larmor frequency; a radio pulse tips them, and the radio signal they give out as they relax is used to build a detailed image of soft tissue.
X-ray Imaging – X-rays are very short-wavelength electromagnetic waves. In an X-ray tube, electrons are sped up by a high voltage and hit a metal target, which gives out X-rays. Different body parts absorb X-rays by different amounts: the intensity falls as I = I₀e^(−μx). Bone absorbs much more than soft tissue, so it shows white on the image. Contrast media, image intensifiers and CT scanners make the pictures clearer and three-dimensional.
Radionuclide Imaging and Therapy – In radionuclide imaging a radioactive tracer is put into the body; it collects in an organ and gives out gamma rays that a gamma camera detects. Technetium-99m is the most used tracer: pure gamma emitter, 140 keV, physical half-life 6 hours, easy to attach to many chemicals. The amount in the body falls by decay (physical half-life Tp) and by the body removing it (biological half-life Tb): 1/Te = 1/Tp + 1/Tb. A gamma camera has a lead collimator, a sodium iodide scintillator crystal, photomultiplier tubes and a computer. In therapy, high-energy (MeV) X-ray beams from a linear accelerator are aimed from many directions to meet at a tumour, or radioactive implants (brachytherapy) give a high dose right where it is needed. Each imaging method (X-ray, CT, ultrasound, MRI, gamma camera, PET) has its own strengths, costs and risks.

6. 3.11 Engineering physics

3.11.1 Rotational dynamics · 3.11.2 Thermodynamics and engines

Moment of Inertia, Radius of Gyration and Rotational Motion – Every idea of straight-line motion has a turning twin: θ for x, ω for v, α for a, I for m, τ for F, L = Iω for p. Moment of inertia I = Σmr² tells how hard it is to change a body's spin; it grows fast when mass sits far from the axis. Radius of gyration k is the distance at which all mass could sit to give the same I: I = Mk². Standard values: ring MR², disc ½MR², solid sphere ⅖MR², rod about centre ML²/12. With constant α: ω = ω₀ + αt, θ = ω₀t + ½αt², ω² = ω₀² + 2αθ, and τ = Iα.
First Law of Thermodynamics – Internal energy U is the total energy of the molecules inside a system. It changes in only two ways: by heat Q (energy that flows because of a temperature difference) and by work W (energy moved by a force, like a moving piston). First law: ΔQ = ΔU + ΔW. Heat given to a gas partly raises its internal energy and partly lets it do work. Work by a gas at constant pressure is PΔV. For an ideal gas Cp − Cv = R.

7. 3.12 Turning points in physics

3.12.1 The discovery of the electron · 3.12.2 Wave-particle duality · 3.12.3 Special relativity

The Discovery of the Electron – In low-pressure discharge tubes, rays came from the cathode. These cathode rays travel in straight lines, carry negative charge and are bent by electric and magnetic fields. In an electron gun a hot filament releases electrons by thermionic emission and an anode voltage V accelerates them: eV = ½mv². J. J. Thomson balanced an electric field against a magnetic field so the beam went straight: v = E/B. Then e/m = v² ÷ 2V ≈ 1.76 × 10¹¹ C kg⁻¹, about 1800 times the value for a hydrogen ion, showing the particle was very light and the same in every metal. Millikan balanced charged oil drops between plates (QV/d = mg) and found every charge was a whole-number multiple of e = 1.6 × 10⁻¹⁹ C: charge is quantised.
Matter Waves and the de Broglie Relation – Light behaves like a wave and a particle. In 1924 Louis de Broglie said matter does the same: every moving particle has a wave with wavelength λ = h/p = h/mv. For a body of kinetic energy K, λ = h/√(2mK); for an electron accelerated through V volts, λ = 1.227/√V nm. Everyday objects have far too tiny a λ to notice, but electrons have λ about the size of atoms.
Special Relativity – Special relativity (Einstein, 1905) rests on two postulates: the laws of physics are the same in all inertial frames, and the speed of light in a vacuum, c ≈ 3.00 × 10⁸ m/s, is the same for every observer. It follows that moving clocks run slow (t = γt₀), moving objects are shorter along their motion (L = L₀/γ), and mass is a form of energy (E = mc²), with γ = 1/√(1 − v²/c²). At everyday speeds γ ≈ 1, so Newton's mechanics works; near c it fails.

8. 3.13 Electronics

3.13.1 Discrete semiconductor devices · 3.13.2 Analogue and digital signals · 3.13.3 Analogue signal processing · 3.13.4 Operational amplifier configurations · 3.13.5 Digital signal processing · 3.13.6 Data communication systems

Semiconductors and the p-n Junction Diode – A semiconductor has a small energy gap (about 1 eV), so a little heat frees some electrons. Pure silicon is intrinsic (electrons = holes). Adding a 5-valence atom makes n-type; a 3-valence atom makes p-type. Joining p and n makes a junction with a depletion layer and a barrier (about 0.7 V for Si). The diode conducts in forward bias, almost not in reverse bias, so it can change AC into one-way DC (rectifier).
Data Representation: How Computers Store Numbers, Text, Images and Sound – Data is raw facts; information is data given meaning; knowledge is information we can use. A computer stores all data as bits (0 or 1). 8 bits make a byte, and 1 kB = 1000 bytes, 1 MB = 1000 kB, 1 GB = 1000 MB, 1 TB = 1000 GB. Numbers are stored in binary, where place values double: 1, 2, 4, 8 and so on. Text uses a character set: in ASCII 'A' is 65; Unicode covers every script. A bitmap image is a grid of pixels; size = width × height × colour depth. Sound is sampled: size = sample rate × bit depth × seconds. Vector images store shapes instead of pixels. Compression makes files smaller: lossless keeps every bit, lossy throws some detail away.
Analogue Signal Processing: LC Resonance Filters and the Ideal Op-Amp – An analogue signal changes smoothly. We process it in two big ways: an LC circuit picks out one frequency (a filter), and an operational amplifier (op-amp) makes a small difference in voltage much bigger. The LC circuit resonates at f₀ = 1/(2π√LC). The quality factor Q = f₀/Δf tells how sharp the peak is. An ideal op-amp has infinite open-loop gain, infinite input resistance, zero output resistance and infinite bandwidth.
Operational Amplifier Configurations: Inverting, Non-Inverting and Summing – An op-amp with negative feedback gives a fixed, useful gain set only by resistors. Inverting amplifier: G = −Rf/Rin (output flipped). Non-inverting amplifier: G = 1 + Rf/Rin (same way up). Summing amplifier: Vout = −Rf(V1/R1 + V2/R2 + …), used in audio mixers and digital-to-analogue converters. Real op-amps have finite gain, finite bandwidth (gain × bandwidth ≈ constant) and their output saturates near the supply voltages.
Digital Signal Processing: Combinational Logic, Sequential Logic and Astables – Digital signals have only two levels: 1 (high) and 0 (low). Combinational logic (gates such as AND, OR, NOT, NAND, NOR, XOR) gives an output that depends only on the inputs now. Sequential logic (flip-flops, counters) also depends on what happened before, so it has memory; it changes on clock pulses. An astable circuit has no stable state and switches by itself, making the clock square wave. Its frequency f = 1/T.
Data Communication Systems – A communication system sends information from a transmitter through a channel to a receiver. The channel can be copper wire, optical fibre or radio waves. To send a message by radio we put it on a fast carrier wave: in AM the carrier's amplitude follows the message, in FM its frequency does. Time-division multiplexing lets many signals share one channel by giving each a short time slot in turn.