The physics
Gauss's law links the field to the charge inside a closed surface. For a sheet or a wire, use a can.
flux out of the can = qin / ε0 k = 1 / 4πε0 = 9 × 10⁹ N m²/C²
A big sheet's lines stay parallel. So its field is the same at every distance.
sheet: E = σ / 2ε0 wire: E = λ / 2πε0r = 2kλ / r
Inside a thick rod, only the charge within r counts. Outside it, the rod acts like a wire.
rod: E = ρr / 2ε0 (inside) E = ρR² / 2ε0r (outside)
Inside metal the field is zero. So two big plates share their charge in a fixed way.
outer faces: (Q₁ + Q₂) / 2 each inner faces: ±(Q₁ − Q₂) / 2
A bead circles a wire when the pull equals mv²/r. The wire's pull falls as 1/r, so r cancels.