part one · the sensor
The most celebrated instrument in applied physics detects its signal with a coil of wire. Not a detector array, not a photon counter — a tuned loop, of the same species as the one in a crystal radio, picking up a voltage of order nanovolts by Faraday induction. Everything strange about MRI follows from that.
Four consequences, and each one is a thing the usual explanation gets wrong.
The coil and the patient sit inside each other's near field. At 1.5 T protons precess at about 64 MHz, so the free-space wavelength is metres — hundreds of times the size of any coil in the bore. Nothing radiates and nothing propagates. The coupling is plain magnetic induction, the same effect that makes a dynamo work, and the "radio waves go in, radio waves come out" story survives in textbooks despite thirty years of correction from the person who worked out how the detection actually functions.
Two of those numbers deserve to stop you. The polarisation is a few parts per million: at body temperature the thermal population difference between the two proton spin states is about 4.9 × 10⁻⁶ at 1.5 T. Essentially the entire sample cancels itself out, and the image is built from the residue. And the signal goes as B₀² — once because the magnetisation is proportional to that polarisation, and again because Faraday's law rewards a faster-changing flux. That second factor is a property of the sensor, not of the physics. Hold on to it; it is how §4 ends.
Working out what a coil can detect looks like it ought to be hard. It is not, because of a result of Hoult and Richards': a coil's receive sensitivity at a point is the field it would produce at that point per unit current. Transmit and receive are the same map. So the reception problem collapses into a Biot–Savart integral — which is what the module below is doing, live, as you drag the sliders.
And that map has a hole in it. Only the field component transverse to B₀ couples to precessing magnetisation, so a coil whose field at the sample points along the bore axis is deaf there — not weak, deaf, by sixteen orders of magnitude. Slide the coil around the patient and watch it happen.
The optimum in that last readout is a = √2 · z, which falls out of differentiating the on-axis field of a loop. It is a signal-only optimum, and real coils are smaller than it, because a bigger loop also sees more of the patient — and in a clinical scanner the dominant noise source is the patient, thermal currents in conductive tissue rather than anything in the electronics. That is why coils are made to fit closely, why the bench figure of merit is the ratio of unloaded to loaded Q, and why there is a ceiling on SNR that no hardware can climb past.
A 90° pulse tips the magnetisation into the transverse plane and it precesses, inducing the EMF. Left alone, the signal fades — but two different things are happening at once, and only one of them is a loss. Spins in slightly different fields precess at slightly different rates and fan out; that is reversible. Meanwhile true T₂ relaxation destroys phase coherence for good.
Hahn's trick separates them: flip everything 180° and the fanning-out runs backwards, re-converging into an echo. Turn the field spread up as far as you like — the echo comes back at exactly e−2τ/T₂, unchanged. The free decay is a lie about how much signal you have.
Put the two field dependencies together and induction-detected signal scales as B₀². That is the entire economic argument for superconducting magnets, cryogens, quench pipes, and a machine you must build a room around.
But the second factor of B₀ belongs to the sensor. A SQUID's sensitivity does not depend on frequency at all; nor does an optically pumped atomic magnetometer's. Detect with one of those and the collapse in the graph below simply does not happen — which is how a scanner can work at 0.055 T in an unshielded room, or at 80 mT from a wall socket. The magnet is not the physics. The magnet is a consequence of choosing a coil.
Every number on this page is computed in the browser from constants and closed-form
physics, not quoted. The solver is checked against analytic solutions on each build:
cd sci/engine-rs && cargo test --release (17 known-answer tests), and
cargo run --release --bin verify prints each result beside the closed form it
reproduces. CODATA 2018 constants throughout.
engine: rust → wasm · 35 kB · no dependencies · 17 tests · sci