The energy of a capacitor, booked as a density

At its defaults it draws the energy of a capacitor, booked as a density. The energy stored by a parallel-plate capacitor of 100 square centimetres — 0.0100 square metres — against the separation of its plates, drawn twice. Held at 10 nC the energy rises in proportion to the separation; held at 113 V it falls as the inverse. Both curves are obtained by integrating the energy density ½ε₀E² over the volume between the plates, and each agrees with ½QV to better than a part in 10¹². The two describe the same capacitor at 1.00 mm, where they cross at 565 nJ, and there their slopes are equal and opposite: the attraction between the plates is 565 µN, or 5.647·10⁻⁴ N, whichever quantity is held fixed. That force is Q²/2ε₀A — a property of the field in the gap and of the area it crosses, with no reference to the plates at all.

field-energy is one function in lib/figures/fields.js — charge, current, flux and the lines drawn between them. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page. A figure here is the figure a reader meets in an essay, and if the generator changes, this page changes with it.

The energy stored by a parallel-plate capacitor of 100 square centimetres — 0.0100 square metres — against the separation of its plates, drawn twice. Held at 10 nC the energy rises in proportion to the separation; held at 113 V it falls as the inverse. Both curves are obtained by integrating the energy density ½ε₀E² over the volume between the plates, and each agrees with ½QV to better than a part in 10¹². The two describe the same capacitor at 1.00 mm, where they cross at 565 nJ, and there their slopes are equal and opposite: the attraction between the plates is 565 µN, or 5.647·10⁻⁴ N, whichever quantity is held fixed. That force is Q²/2ε₀A — a property of the field in the gap and of the area it crosses, with no reference to the plates at all.

The electric field between two oppositely charged plates separated by 0.20 of their own width, traced by following the field of 26 discrete charges on each plate rather than drawn as parallel lines. In the middle the lines are straight and evenly spaced; near the ends they bow outward. The field nine-tenths of the way to the edge is 80 per cent of the field at the centre.

Capacitance of a parallel-plate capacitor with plates of 100 square centimetres, plotted against their separation in millimetres. It is an inverse curve: halving the gap doubles the capacitance, and nothing but the geometry and the permittivity of free space enters it.

The electric field between two oppositely charged plates separated by 0.60 of their own width, traced by following the field of 26 discrete charges on each plate rather than drawn as parallel lines. In the middle the lines are straight and evenly spaced; near the ends they bow outward. The field nine-tenths of the way to the edge is 68 per cent of the field at the centre.

A ring carrying 10⁻⁶ C on a freely pivoted disc, with a solenoid on the axis threading 0.002 Wb through it, switched off over 1 second. Nothing is turning at the start and nothing has been touched. The collapsing flux drives a circumferential electric field round the ring, the ring is torqued, and the disc ends up spinning with 3.183·10⁻¹⁰ kg m²/s of angular momentum. Where was it? The two curves are the field's share, ε₀∫r × (E × B), and the matter's share integrated from the torque — computed by different routes and summing to a constant to 4.2e-15 of the total throughout. So the angular momentum was there before the switch was thrown, in a static electric field crossed with a static magnetic one, in a room where nothing whatever was moving. It is qΦ/2π, it does not depend on the radius of the ring or the shape of the solenoid, and it is the plainest demonstration available that the field is not a bookkeeping device for forces between distant charges.

The energy stored by a parallel-plate capacitor of 100 square centimetres — 0.0100 square metres — against the separation of its plates, drawn twice. Held at 10 nC the energy rises in proportion to the separation; held at 113 V it falls as the inverse. Both curves are obtained by integrating the energy density ½ε₀E² over the volume between the plates, and each agrees with ½QV to better than a part in 10¹². The two describe the same capacitor at 1.00 mm, where they cross at 565 nJ, and there their slopes are equal and opposite: the attraction between the plates is 565 µN, or 5.647·10⁻⁴ N, whichever quantity is held fixed. That force is Q²/2ε₀A — a property of the field in the gap and of the area it crosses, with no reference to the plates at all.

physicscheck asserts something about field-energy that could fail — it draws it and measures the result against a value reached some other way.

Across the library: 100 interrogated, 2 exercised only, 1 untouched, of 103. Read out of the gate's source by the gate's own two patterns — and the gate's last claim fails the build if that read disagrees with what it was handed while running.