Field Runner

Between wide, closely spaced plates the field is very nearly uniform and equal to the potential difference divided by the separation, E = V/d. The field is defined by the force per unit positive charge, so a particle of charge q feels F⃗ = qE⃗ and accelerates at a⃗ = qE⃗/m. E⃗ points from the positive plate to the negative one, and because the electron is negative it is pushed the other way — toward the positive plate.

The field has no component along the beam, so the horizontal speed never changes and the time inside the plates is t = L/v₀. The transverse motion is uniform acceleration from rest, y = ½at², which is the parabola you can see bending inside the stack. After the plates the particle travels in a straight line, so the drift multiplies the exit angle by the remaining distance.

Halving d doubles E at fixed voltage and so doubles the deflection — but it also halves the corridor to d/2. The share of the total deflection that has to happen between the plates is fixed by geometry alone, so on every run there is a smallest gap that can possibly work. That is the tension the second run is built on.

The bead on the last run is a point charge, so its field is radial and falls as the inverse square of distance, E = kQ/r². Fields superpose as vectors, so its contribution simply adds to the plate term — but it does not scale with V, which is why the last run needs a genuinely different setting rather than a rescaled one.

• e = 1.602 × 10⁻¹⁹ C elementary charge
• m = 9.109 × 10⁻³¹ kg electron rest mass
• k = 8.99 × 10⁹ N·m²/C² Coulomb constant, k = 1/(4πε₀)
• v₀ = 2.0 × 10⁷ m/s fixed gun speed, 0.067 c
• Semi-implicit Euler, 4000 steps across a 320 mm chamber
• Ignored plate-edge fringing, gravity, radiation and relativity