Flashcards · H556 · Topic 6

Particles and medical physics flashcards

60 cards on Topic 6, subtopics 6.1 to 6.5. Pick a subtopic or study them all.

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6.1 Capacitors

Capacitance: C = Q/V, the charge stored per unit p.d. (Q is the charge on one plate). 1 F = 1 C V⁻¹.
Capacitors in series vs in parallel: Series: same charge on each, p.d.s add, 1/C = 1/C₁ + 1/C₂ + … (below the smallest). Parallel: same p.d., charges add, C = C₁ + C₂ + … (above the largest).
Energy stored by a capacitor: The area under the p.d.–charge graph: W = ½QV = ½Q²/C = ½V²C.
A battery charges a capacitor to its e.m.f. V. How much energy does it supply, and how much is stored?: Supplied: QV. Stored: ½QV. The other half is transferred as thermal energy in the circuit.
Why is a capacitor useful as an energy store (flash, defibrillator)?: It stores far less energy than a battery of the same mass, but can release it very quickly.
Time constant and the discharge equation: τ = CR (seconds). Q, V and I all decay as x = x₀e^(−t/CR).
Fraction left after 1 CR, 2 CR and 3 CR of discharge: 1 CR: 36.8%. 2 CR: 13.5%. 3 CR: 5.0%. (When charging, Q and V reach 63.2%, 86.5% and 95.0%.)
Graph of ln V against t for a discharging capacitor: A straight line: ln V = ln V₀ − t/CR. Gradient = −1/CR; intercept = ln V₀.
Half-time of a capacitor discharge: t½ = CR ln 2 = 0.693 CR, the same wherever you start on the curve.
Charging through a resistor: how do V_C and the current change?: V_C and Q rise from zero as V₀(1 − e^(−t/CR)). The current starts at V₀/R and decays as I₀e^(−t/CR).
Iterative model of a discharge: Choose Δt much smaller than CR. ΔQ = −(Q/CR)Δt, then Q_new = Q_old + ΔQ, repeated. A smaller Δt brings the model closer to the exact curve.

6.2 Electric fields

Electric field strength: Force per unit positive charge, E = F/q, in N C⁻¹ (= V m⁻¹). A vector, in the direction of the force on a positive charge.
Coulomb's law and the field of a point charge: F = Qq/(4πε₀r²). E = Q/(4πε₀r²). A uniformly charged sphere acts, outside, as if its charge were at its centre.
Uniform field between parallel plates: E = V/d: parallel, evenly spaced field lines; the same field strength everywhere in the gap.
Capacitance of parallel plates and the role of permittivity: C = εA/d with ε = εᵣε₀ (C = ε₀A/d in a vacuum or air). The relative permittivity εᵣ is at least 1.
Electric potential at a point: The work done in bringing unit positive charge from infinity to that point. V = Q/(4πε₀r). Zero at infinity; a scalar.
Capacitance of an isolated sphere, and electric potential energy of two charges: Sphere: C = 4πε₀R. Potential energy: Ep = Vq = Qq/(4πε₀r).
Gravitational vs electric fields: similarities and differences: Both inverse-square, radial for a point source, potential zero at infinity. Gravity is always attractive (force per unit mass); electric forces can attract or repel (force per unit charge).
Charged particle entering a uniform field at right angles: Horizontal speed constant; constant acceleration a = qE/m across the field; sideways displacement y = ½at² with t = plate length ÷ speed. The path is a parabola.

6.3 Electromagnetism

Magnetic field patterns of a long straight wire and a long solenoid: Wire: concentric circles in planes perpendicular to the wire, weaker with distance (right-hand grip rule). Solenoid: uniform parallel lines inside; bar-magnet pattern outside.
Fleming's left-hand rule: Thumb: force (motion). First finger: field (north to south). Second finger: conventional current. All at right angles.
Force on a current-carrying wire, and the tesla: F = BIL sin θ, θ between wire and field. 1 T: a 1 m conductor carrying 1 A at right angles to the field feels a force of 1 N.
Measuring B with a current balance: Magnets on a digital balance, wire through the field. Force on the wire F = Δm g. Plot Δm against I: gradient = BL/g, so B = gradient × g/L (gradient in kg A⁻¹).
Charged particle moving at right angles to a magnetic field: F = BQv, perpendicular to the velocity, so no work is done and speed is constant. Circular path: r = mv/(BQ); period T = 2πm/(BQ), independent of speed.
Velocity selector: Electric and magnetic fields at right angles, with opposing forces. Particles are undeflected when QE = BQv, so v = E/B, independent of charge and mass.
Magnetic flux and the weber: Φ = BA cos θ, θ between B and the normal to the area. 1 Wb is the flux through 1 m² perpendicular to a uniform field of 1 T.
Faraday's law and Lenz's law: Faraday: induced e.m.f. is directly proportional to the rate of change of magnetic flux linkage, E = −Δ(NΦ)/Δt. Lenz: the induced e.m.f. acts to oppose the change that causes it.
How does Lenz's law follow from conservation of energy?: Pushing a magnet into a coil needs a force because the induced current opposes the motion; the work done becomes electrical energy. If the induced effect helped the change, energy would be created from nothing.
a.c. generator: e.m.f. equation and when it is greatest: E = NBAω sin ωt; peak E₀ = NBAω. The e.m.f. is greatest when the coil is parallel to the field (flux linkage zero) and zero when the coil is perpendicular to it.
Ideal transformer equations, and why the core is laminated: Ns/Np = Vs/Vp = Ip/Is (VpIp = VsIs for no losses). Lamination reduces eddy currents and the energy wasted as heat.
Why does a transformer need an alternating input?: A steady current gives a constant flux, so there is no changing flux linkage in the secondary and no induced e.m.f.

6.4 Nuclear and particle physics

Alpha scattering: observations and conclusions: Most pass straight through: the atom is mostly empty space. A few deflect through large angles: positive charge is concentrated in a small nucleus. Very few rebound: the nucleus is massive.
Nuclear radius and why nuclear density is constant: R = r₀A^(1/3), with r₀ ≈ 1.2 fm. Volume ∝ A and mass ∝ A, so density is the same for all nuclei, about 2.3 × 10¹⁷ kg m⁻³.
Strong nuclear force: range and behaviour: Repulsive below about 0.5 fm, attractive from about 0.5 fm to 3 fm, negligible beyond a few fm. It beats proton repulsion but stops the nucleus collapsing.
Hadron vs lepton: Hadron: feels both the strong and the weak force; made of quarks (proton, neutron). Lepton: feels the weak force but not the strong force (electron, neutrino).
Quark charges, and the quark make-up of the proton and neutron: u +⅔e, d −⅓e, s −⅓e (antiquarks opposite). Proton uud (+1); neutron udd (0).
β⁻ and β⁺ decay in terms of quarks: β⁻: d → u + e⁻ + ν̄ (n → p + e⁻ + ν̄). β⁺: u → d + e⁺ + ν (p → n + e⁺ + ν). Both are weak interactions.
Radioactive decay is spontaneous and random. What does each word mean?: Spontaneous: no trigger; unaffected by temperature, pressure or chemical state. Random: you cannot predict which nucleus will decay, or when.
Activity, decay constant and half-life: Activity A = λN (Bq). λ: the probability per second that a given nucleus decays. Half-life: time for the number of undecayed nuclei (or the activity) to halve. λt½ = ln 2.
Exponential decay equations: N = N₀e^(−λt); A = A₀e^(−λt). ln N = ln N₀ − λt, so a graph of ln N against t has gradient −λ.
Carbon dating: the principle and the assumptions: Living things keep a constant carbon-14 activity per gram of carbon; after death it decays (half-life 5730 years). t = (1/λ) ln(A₀/A). Assumes the past ratio was the same and there is no contamination.
Mass defect and binding energy: Mass defect: total mass of the free protons and neutrons minus the mass of the nucleus. Binding energy: energy needed to separate the nucleus into free nucleons = mass defect × c².
Why do both fission and fusion release energy?: Binding energy per nucleon peaks near A = 56 (iron). Fusion of light nuclei and fission of heavy nuclei both produce nuclei with higher binding energy per nucleon.
Minimum photon energy for electron–positron pair creation, and annihilation at rest: Pair creation: at least 2 × 0.511 = 1.02 MeV. Annihilation at rest: two photons in opposite directions, each of about 0.511 MeV, so momentum stays zero.
Roles of the moderator and control rods in a fission reactor: Moderator (water, graphite): slows fast neutrons so they are likely to cause further fission. Control rods (boron, cadmium): absorb neutrons to control the rate; one neutron per fission should cause another.
Why does fusion need a very high temperature?: Nuclei repel electrostatically and must get within about 10⁻¹⁵ m for the strong force to act. They need very high kinetic energy, mean (3/2)kT, so the fuel is a plasma at millions of kelvin.

6.5 Medical imaging

X-ray tube: what do the tube voltage and the heater (tube) current control?: Voltage: the maximum photon energy and penetration. Current: the number of electrons, so the beam intensity. Target: tungsten (high atomic number, high melting point); about 1% of the energy becomes X-rays.
Minimum wavelength from an X-ray tube at 120 kV: E_max = eV = 1.60 × 10⁻¹⁹ × 120 000 = 1.92 × 10⁻¹⁴ J. λ_min = hc/E_max = (6.63 × 10⁻³⁴ × 3.00 × 10⁸) ÷ 1.92 × 10⁻¹⁴ = 1.04 × 10⁻¹¹ m.
Simple scatter vs Compton scattering: Simple scatter: the photon changes direction with the same energy; the atom is not ionised (low photon energies). Compton: the photon is scattered with reduced energy and an outer electron is ejected.
Photoelectric effect and pair production as X-ray attenuation mechanisms: Photoelectric: the photon is absorbed and an inner-shell electron ejected; much more likely at high atomic number, giving bone–tissue contrast. Pair production: photon of at least 1.02 MeV becomes an electron and a positron near a nucleus.
Attenuation of X-rays: I = I₀e^(−μx), μ the attenuation coefficient (m⁻¹). Half-value thickness x½ = ln 2/μ.
Barium and iodine contrast media: High atomic number, so strong absorption by the photoelectric effect. Barium sulfate is swallowed to outline the gut; iodine compounds are injected for blood vessels and the urinary tract.
CAT scan: how it works and its advantages over an X-ray image: A rotating tube sends a thin fan beam through one slice to a ring of detectors; a computer works out the attenuation of each voxel. No overlapping structures, better soft-tissue contrast, 3D views; but a higher dose and costly.
Technetium-99m vs fluorine-18 as tracers: Tc-99m: gamma only (about 140 keV), half-life about 6 h, used with a gamma camera. F-18: positron emitter, half-life about 110 minutes, used in fluorodeoxyglucose for PET.
Gamma camera: the job of each part: Collimator: lead with parallel holes, passing only photons travelling along them. Scintillator: turns each gamma photon into a flash of light. Photomultiplier tubes: turn light into an electrical pulse. Computer: finds positions and builds the image.
PET scanning: A positron annihilates with an electron, giving two 0.511 MeV gamma photons in opposite directions. A ring of detectors records pairs arriving together; each pair defines a line, and the computer finds where many lines cross. No collimator is needed.
Ultrasound frequency: the trade-off: Ultrasound is above 20 kHz; scanners use a few MHz. Higher frequency gives a shorter wavelength and more detail, but more absorption, so less depth.
Acoustic impedance and the reflected fraction at a boundary: Z = ρc. Ir/I₀ = (Z₂ − Z₁)²/(Z₂ + Z₁)². A big difference in Z means strong reflection. Gel removes the air gap and has Z close to skin.
A-scan vs B-scan: A-scan: echo strength against time along one line; depth d = ct/2. B-scan: many A-scans combined into a 2D image; brightness shows echo strength.
Doppler ultrasound for blood flow: Δf/f = 2v cos θ/c, θ the angle between the beam and the blood flow. The 2 arises because the blood acts first as a receiver, then as a source. At 90° there is no shift.