Mini-Lesson

Particles & Medical Physics

Module 6 is the synoptic finale: capacitors, electric fields, electromagnetism, nuclear and particle physics — and then it turns all of it into a hospital, with X-rays, CT, ultrasound, MRI and PET.

C = Q/V · W = ½QV · F = kQ₁Q₂/r² · F = BQv · E = mc² · I = I₀e^(−µx) · Z = ρcthe physics of an MRI scanner is the physics of a capacitor and a magnet

What makes Module 6 special: every medical imaging technique here is an application of physics you have already met. PET is annihilation. MRI is resonance. Ultrasound is impedance matching. X-ray attenuation is exponential decay, exactly like capacitor discharge.

Work through each screen, answer the questions as you go (several are full A-level calculations) and collect ⭐ stars. Press Start when you're ready.

Module 6 · capacitors

Capacitance, energy and exponential discharge

C = Q/V · W = ½QV = ½CV² = ½Q²/Cthe ½ is the area under the charge–pd graph, which is a triangle

Parallel: C = C₁ + C₂ (capacitances add). Series: 1/C = 1/C₁ + 1/C₂. Exactly the opposite way round from resistors.
Discharge: Q = Q₀e^(−t/RC), and V and I decay with the same exponential.
Time constant τ = RC (seconds): after one τ, the charge has fallen to 1/e ≈ 37%. After 5τ, the capacitor is effectively empty.
Take logs: ln Q = ln Q₀ − t/RC, so a graph of ln Q against t is a straight line of gradient −1/RC.

Where you meet this in a hospital: a defibrillator charges a large capacitor to several kilovolts and then dumps its stored energy (typically 200–360 J) through the patient in a few milliseconds. The energy comes straight from W = ½CV².

Calculate

Your turn — energy stored

1A 4700 µF capacitor is charged to 9.0 V. Calculate the energy stored. Give your answer in mJ to 3 significant figures.

mJ

Hint: W = ½CV² = 0.5 × 4700 × 10⁻⁶ × 81 = 0.190 J. Multiply by 1000 for mJ.

Calculate

Your turn — time constant

2That 4700 µF capacitor discharges through a 10 kΩ resistor. Calculate the time constant. Give your answer in s.

s

Hint: τ = RC = 10 × 10³ × 4700 × 10⁻⁶.

Quick check

After one time constant

?A capacitor discharges through a resistor. After a time equal to one time constant (RC), what fraction of the original charge remains?

Module 6 · fields

Electric fields and Coulomb's law

F = (1/4πε₀) Q₁Q₂/r² · E = F/Q = kQ/r² · E = V/d (uniform field)k = 1/4πε₀ = 8.99 × 10⁹ N m² C⁻²

Coulomb's law is an inverse-square law with exactly the same mathematical form as Newton's law of gravitation — but the electric force can repel as well as attract, and it is about 10³⁶ times stronger.
Radial field (point charge): lines point away from + and towards −, and E ∝ 1/r².
Uniform field (parallel plates): straight, parallel, evenly spaced lines and E = V/d.

Worked example — the electron in a hydrogen atom

Proton and electron separated by 5.3 × 10⁻¹¹ m.

F = (8.99 × 10⁹ × (1.60 × 10⁻¹⁹)²) ÷ (5.3 × 10⁻¹¹)² = 2.30 × 10⁻²⁸ ÷ 2.81 × 10⁻²¹ = 8.19 × 10⁻⁸ N

Calculate

Your turn — Coulomb's law

3Calculate the electrostatic force between a proton and an electron separated by 5.3 × 10⁻¹¹ m. Give your answer as a multiple of 10⁻⁸ N to 3 significant figures. (k = 8.99 × 10⁹ N m² C⁻², e = 1.60 × 10⁻¹⁹ C)

× 10⁻⁸ N

Hint: F = kQ₁Q₂/r². Numerator = 8.99 × 10⁹ × (1.60 × 10⁻¹⁹)² = 2.30 × 10⁻²⁸. r² = 2.81 × 10⁻²¹.

Module 6 · electromagnetism

Magnetic fields, F = BIL, F = BQv and induction

F = BIL sin θ · F = BQv sin θ · r = mv/(BQ)maximum force when perpendicular; ZERO force when parallel to the field

A charged particle moving perpendicular to a magnetic field experiences a force perpendicular to both B and v. Because that force is always perpendicular to the velocity it does no work, so the particle moves in a circle of radius r = mv/(BQ). That is how a mass spectrometer and a cyclotron work.

Φ = BA · ε = −N dΦ/dtFaraday's law with Lenz's minus sign — the induced effect always OPPOSES the change causing it

Lenz's law is conservation of energy. If the induced current helped the change along, you would get free energy forever. Instead you must do work against the induced effect — and that work becomes the electrical energy.

Worked example — a proton in a field

Proton (m = 1.67 × 10⁻²⁷ kg, Q = 1.60 × 10⁻¹⁹ C) at 5.0 × 10⁶ m s⁻¹, perpendicular to B = 0.30 T.

F = BQv = 0.30 × 1.60 × 10⁻¹⁹ × 5.0 × 10⁶ = 2.4 × 10⁻¹³ N

r = mv/(BQ) = (1.67 × 10⁻²⁷ × 5.0 × 10⁶) ÷ (0.30 × 1.60 × 10⁻¹⁹) = 8.35 × 10⁻²¹ ÷ 4.8 × 10⁻²⁰ = 0.174 m

Calculate

Your turn — radius of a circular path

4A proton (m = 1.67 × 10⁻²⁷ kg, Q = 1.60 × 10⁻¹⁹ C) travels at 5.0 × 10⁶ m s⁻¹ perpendicular to a magnetic field of 0.30 T. Calculate the radius of its circular path. Give your answer in m to 3 significant figures.

m

Hint: r = mv/(BQ) = (1.67 × 10⁻²⁷ × 5.0 × 10⁶) ÷ (0.30 × 1.60 × 10⁻¹⁹) = 8.35 × 10⁻²¹ ÷ 4.8 × 10⁻²⁰.

Module 6 · nuclear physics

Binding energy, fission and fusion

ΔE = Δm c² · 1 u = 931.5 MeVthe mass defect Δm is the mass 'missing' when nucleons bind together

Binding energy per nucleon peaks at iron-56 (about 8.8 MeV), the most stable nucleus of all.
Fission of a heavy nucleus and fusion of light nuclei both move the products up the curve towards iron — and both therefore release energy.
Radioactive decay: A = λN, N = N₀e^(−λt), t½ = ln2/λ — mathematically identical to capacitor discharge.
Rutherford scattering showed the nucleus is tiny, dense and positive; R = r₀A^(1/3) gives a constant nuclear density.

Calculate

Your turn — binding energy

5A helium-4 nucleus has a mass defect of 0.0304 u. Calculate its total binding energy. Give your answer in MeV to 3 significant figures. (1 u = 931.5 MeV)

MeV

Hint: Binding energy = Δm × 931.5 = 0.0304 × 931.5.

Module 6 · X-rays & CT

X-ray imaging, attenuation and CT

X-ray production: electrons are accelerated through 30–100 kV and slam into a metal (usually tungsten) target. Most energy becomes heat; a small fraction becomes X-ray photons (bremsstrahlung plus sharp characteristic lines).

I = I₀ e^(−µx)µ = attenuation coefficient (m⁻¹ or mm⁻¹) — another exponential, exactly like capacitor discharge

Contrast comes from differing attenuation. Bone (high Z, calcium) absorbs far more than soft tissue, so bone shows white.
Contrast media (barium, iodine) have a high atomic number and are used to make soft tissue visible.
CT takes many X-ray images from many angles and reconstructs a 3-D image slice by slice. Far more information — but a considerably higher radiation dose.

Worked example

µ = 0.25 mm⁻¹, thickness x = 8.0 mm.

I/I₀ = e^(−0.25 × 8.0) = e^(−2.0) = 0.135 → 13.5% transmitted

Calculate

Your turn — X-ray attenuation

6X-rays pass through 8.0 mm of a material with attenuation coefficient 0.25 mm⁻¹. Calculate the percentage of the intensity that is transmitted. Give your answer in % to 3 significant figures.

%

Hint: I/I₀ = e^(−µx) = e^(−0.25 × 8.0) = e^(−2.0) = 0.1353. Multiply by 100.

Module 6 · ultrasound, MRI & PET

Ultrasound, MRI and PET

Z = ρ c · depth = ½ c tZ = acoustic impedance (kg m⁻² s⁻¹) · the ½ because the pulse travels there AND back

Ultrasound uses a piezoelectric transducer (usually lead zirconate titanate), which both generates and detects the pulse. Reflection occurs at a boundary where acoustic impedance changes.
Coupling gel is essential: without it, the huge impedance mismatch between skin and air would reflect essentially 100% of the pulse at the surface.
A-scan: a single line of echo amplitudes against depth. B-scan: many A-scans combined into a 2-D image — this is the familiar fetal scan. No ionising radiation, so it is safe in pregnancy.
MRI: a strong magnetic field aligns hydrogen nuclei; an RF pulse at the resonant (Larmor) frequency flips them; as they relax they re-emit RF, and the relaxation time distinguishes tissue types. Excellent soft-tissue contrast, no ionising radiation — but it is expensive, noisy, claustrophobic and unusable with ferromagnetic implants.
PET: a positron-emitting tracer (such as fluorine-18 in FDG) is injected. Each positron annihilates with an electron, producing two 511 keV gamma photons back to back. Detecting both and timing them locates the annihilation, building a functional image of metabolic activity.

Why 511 keV? That is exactly the rest energy of an electron: mc² = 9.11 × 10⁻³¹ × 9.00 × 10¹⁶ = 8.20 × 10⁻¹⁴ J = 0.511 MeV. PET is the annihilation of Module 6 turned into a diagnostic tool.

Calculate

Your turn — ultrasound depth

7An ultrasound pulse travelling at 1540 m s⁻¹ returns from a boundary after 60 µs. Calculate the depth of the boundary. Give your answer in mm to 3 significant figures.

mm

Hint: The pulse travels there and back, so depth = ½ × c × t = 0.5 × 1540 × 60 × 10⁻⁶ = 0.0462 m. Convert to mm.

Sort it

Which imaging technique?

Tap a statement, then tap the technique it describes.

🦴 X-ray / CT

🔊 Ultrasound

🧲 MRI

Quick check

Why the gel?

?Why must coupling gel be applied to the skin before an ultrasound scan?

Quick check

How PET works

?A PET scanner detects two gamma photons of 511 keV emitted back to back. Where do they come from?

Quick check

Capacitors in series

?Two 4.0 µF capacitors are connected in series. What is the total capacitance?

Match it

Match the quantity to its equation

Tap an item on the left, then its partner on the right.

Quantity

Equation

Recap

The big ideas to know

Capacitors: C = Q/V · W = ½QV = ½CV² · Q = Q₀e^(−t/RC) · τ = RC, 37% left after one time constant

Electric fields: F = kQ₁Q₂/r² (inverse square, attract or repel) · E = V/d in a uniform field

Electromagnetism: F = BIL sin θ · F = BQv sin θ · r = mv/BQ · Φ = BA · ε = −N dΦ/dt (Lenz = energy conservation)

Nuclear: ΔE = Δmc², 1 u = 931.5 MeV; binding energy per nucleon peaks at iron-56; decay N = N₀e^(−λt)

X-ray / CT: I = I₀e^(−µx); bone has high attenuation; CT gives 3-D at the cost of a higher dose

Ultrasound: Z = ρc; coupling gel removes the air impedance mismatch; depth = ½ct; A-scan and B-scan; no ionising radiation

MRI: resonance of hydrogen nuclei, superb soft-tissue contrast · PET: annihilation gives two back-to-back 511 keV photons

That is OCR Module 6 — and with it, all six modules of H556. Press Finish to see your score.

🏆

Mini-lesson complete!

⭐⭐⭐

You've worked through Particles & Medical Physics for OCR A-level Physics A. 🎉

Your stars: 0 / 0

Next: test yourself in the Evaluate stage Confidence Quiz, then lock it in with Verify.

📣 Smashed it? Share your score

Challenge a mate to beat your stars, or show a parent how you got on.

→ Back to all subjects