single-file HTML loop / solenoid / toroid coupled current-density + self-field limit toy cooling + cost model

Wheeeee Loop - a superconductor used like a battery!

See how much you can store. Note: you’ll have to provide lots and lots of cooling!
This treats the coil as a magnetic energy store, not a power cable. The stored energy is E = ½ L I². Tight winding can raise inductance and help storage, but tighter winding also raises self-field, and that pushes the superconductor toward its field limit and cuts down the current density it can carry. The model below couples those effects instead of pretending they are independent. It also adds a rough cryostat-area-based cooling model so you can see how ugly the wall-plug power and dollar cost get as the thing gets bigger and stupider.

What the model is doing

It estimates a maximum circulating current by solving a coupled rule instead of separate hard caps:
Jc(B) = Jc0 · max(0, 1 − B/Bc2)1.5
and then finds I such that
I = margin · Jc(Bself) · Aturn
with geometry-dependent self-field. More turns tend to increase L, but they also increase the field that chokes the conductor. Solenoids pack tightly and often store more for a given footprint. Toroids confine the field better. Big loose rings look silly but are easy to picture.

How to read the cooling model

The cooling numbers are explicit ballpark assumptions, not sacred truth. The page estimates a cryostat envelope around the coil, computes its outside area, multiplies that by an editable heat-leak-per-area, then turns that cold load into wall power and dollars using editable cryogenic efficiency and cost assumptions.
cold load ≈ area × heat leak wall power ≈ cold load × W/W annual electric cost = wall power × hours × $/kWh

What the warnings mean

Any slider marked ⚠ not a direct geometry input is a material or model assumption, not a thing you just dial in by making the wire bigger. Any slider marked ⚠ toy cooling assumption is there so you can change the cryogenic fantasy budget and watch the numbers explode or calm down.

Inputs

Text boxes accept scientific notation. Radius, turns, and sizes use logarithmic sliders so you can sweep from microscopic nonsense to solar-system nonsense.
Geometry
Geometry
choose how the conductor is packed
shape
Mean coil radius R
centerline radius of the winding pack
m
Number of turns N
the turns are assumed to be series-connected
turns
Strand diameter d
diameter of one superconducting turn
m
Solenoid pack aspect ratio ℓ/t
only used for the solenoid: winding length divided by radial thickness
ratio
Superconductor / magnetic model
Zero-field engineering current density Jc0 ⚠ not a direct geometry input
entered as A/mm²; the model reduces it as self-field rises
A/mm²
Characteristic field ceiling Bc2 ⚠ not a direct geometry input
the toy current-density law collapses as self-field approaches this
T
Engineering margin ⚠ not a direct geometry input
headroom multiplier applied after the coupled current limit is solved
× limit
Cooling and cost assumptions
Operating temperature preset ⚠ toy cooling assumption
loads default cryogenic assumptions; you can still override them
Room temperature superconductor exists?
default: No ☹️ because reality keeps being rude
wishful thinking
Cryostat growth factor ⚠ toy cooling assumption
extra shell thickness relative to the winding-pack thickness
× pack thickness
Cold heat leak per cryostat area ⚠ toy cooling assumption
rough watts leaking into the cold stage per square meter of outer cryostat surface
W/m²
Wall-plug power per cold watt ⚠ toy cooling assumption
how many watts from the grid to remove 1 watt at the cold stage
W/W
Cooling hardware capital cost ⚠ toy cooling assumption
ballpark dollars per watt of cold-stage cooling capacity
$/W-cold
Electricity price ⚠ toy cooling assumption
used for annual wall-plug energy cost
$/kWh

Outputs

Modeled maximum stored energy
Coil inductance
Maximum circulating current
Cooling wall power
Annual electricity cost
Cooling hardware capex
Cryostat outside area
One AA battery comparison

Model details

Geometry
Built coil outer width × height
Winding pack thickness / radius
Single-turn cross-section
Zero-field current limit margin·Jc0·A
Self-field at max current
Main geometry field contribution
Local strand-surface field
Effective current density at max current
Cold load / wall-plug power
Cooling assumptions

Radius sweep

maximum storage across radius current selected radius

Scale visualizer

Geometry cheat sheet

Big ring / circular bundle: easy to picture, airy, usually weaker inductance for the same footprint.
Solenoid: compact cylinder, often more inductance, but strong main field and lots of stray field.
Toroid: donut coil, field is more self-contained, but geometry and winding pack matter a lot.

Why compacting does not give free magic

The empty-looking space is not wasted. Magnetic energy lives in the field itself. If you crush the same energy into less space, the field usually has to rise, which pushes the superconductor harder and makes cooling, stress, and quench protection nastier.