Faraday disc

Set disc radius R, signed magnetic field B and signed angular velocity ω, then calculate. The voltage convention is U = V(rim) − V(center). Move the radius probe or enter an exact percentage from 0 to 100%; the initial probe is 50%. Examples keep that percentage. Clear result keeps the inputs. The schematic shows an open external circuit, two brushes and their relative polarity; it is not an animation.

The disc is viewed from +z. Positive B points out of the page (⊙), negative B into it (⊗). Positive ω and positive tangential velocity are counterclockwise. The model assumes a uniform axial magnetic field, an ideal point contact at the center and a rim brush. It describes the stationary open-circuit state of a conducting disc, with current density J = 0.

At radius r, the material's tangential velocity is vθ = ωr. The magnetic force per positive charge points radially with signed value ωBr; the electric field balances it: E + v×B = 0, so E_r = −ωBr. Positive E_r means outward. Thus V(r) − V(0) = ½ωBr² and U = ½ωBR². The center is the zero-potential reference; the contact signs indicate which terminal has the higher potential, not an absolute potential. Reversing B or ω reverses U. Reversing both preserves U. If either is zero, U and E_r are zero and no polarity is shown.

The profile plots the potential in volts against radius in meters. The selected radius is marked in both views. For R = 0.15 m, B = 0.5 T and ω = 50 rad/s, U = 0.28125 V. At half-radius, r = 0.075 m, the potential is 0.0703125 V, E_r = −1.875 V/m and vθ = 3.75 m/s. Doubling R to 0.3 m gives U = 1.125 V. Rotation rate in revolutions per minute is ω·30/π, with the same sign as ω.

Allowed inputs are R = 0.05–0.4 m, B = −2–2 T and ω = −200–200 rad/s; zero B and zero ω are valid. The model does not calculate load current, power, resistance, startup transients or mechanical loading. All calculations stay in your browser.

Source: Markus Zahn, MIT OCW, Electromagnetic Field Theory, Chapter 6, §6-3-3, printed pages 420–421, equations 14–16 and figure 6-15. The source defines its terminal voltage with the center positive, opposite to this tool's explicit rim-minus-center convention. Here the inner contact radius is idealized as zero.