Derive capacitance and bound a plate trajectory

Derive the interior field

One ideal infinite sheet has field magnitude sigma/(2epsilon0) on either side. Equal and opposite sheets cancel outside and add inside, giving E = sigma/epsilon0. Real finite plates approximate this uniform result only sufficiently far from edges when separation is small compared with plate dimensions. Integrating across the gap gives delta V = Ed.

Separate geometry from operating state

Using Q = sigma A and C = Q/delta V gives C = epsilon0 A/d for vacuum. For a fixed linear geometry, changing Q and delta V proportionally does not change C. The displayed boundary maintains 6 V, so changing d or A changes the supplied free charge. It would be inconsistent to call that a simultaneous fixed-Q experiment.

Transfer to motion with a collision check

A passive body entering halfway between plates with horizontal speed vx and zero initial vertical velocity has ay = (q/m)E, x = vx t and y = ay t²/2. For q/m = 1 C/kg, vx = 100 m/s and a model length of 0.05 m, compute both the candidate exit time and the time to reach y = d/2. Stop at whichever occurs first. A formula evaluated beyond a plate does not establish an accessible exit state; contact dynamics are not modeled.

Reverse the signed force

Changing the sign of q/m reverses vertical deflection while the source field remains unchanged. Horizontal motion stays uniform only because the ideal field has no horizontal component and back reaction is ignored. This comparison with projectile motion depends on a uniform field; edge fields or a changing source invalidate constant acceleration.

Worked original case

For A = 0.02 m² and d = 2 mm, C ≈ 88.5419 pF, Q ≈ 531.251 pC and E = 3000 N/C. Candidate exit time is 0.0005 s and vertical displacement 0.375 mm, less than the 1 mm half-gap. At d = 1 mm, candidate deflection is 0.75 mm, exceeding the 0.5 mm half-gap, so the body reaches a plate first.

Investigation

Change area at fixed gap and predict why trajectory is unchanged although charge changes. Then change gap, compare the exit and impact conditions, and retain a rejected exit calculation. Explain the geometric limit and why the supplied q/m does not identify a real particle species.

Changed-case practice

Sources and boundaries

Original cases checked against the official framework on 30 October 2026, with OpenStax verification on 31 October 2026 against the current official framework, Unit 10 and OpenStax capacitance. These are virtual supplied models, not instructions to charge, ground, wire or handle real equipment. Public answers are practice; retained plans and self-reports do not establish independent mastery, actual laboratory participation or AP authorization.