# Tesla / Atmospheric electricity

Source review: 2026-10-07. This module concerns Nikola Tesla, not Tesla, Inc.

The runnable `atmospheric_electricity` experiment is an original, bounded,
modern equivalent-circuit model of a uniform fair-weather atmospheric column.
It connects potential gradient, finite air conductivity, a capacitor, a load
and leakage, with a reproducible energy account. Its 3D scene illustrates the
model. It is not an observation, a calibrated collector design, a reproduction
of a Tesla apparatus, or a simulation of every Tesla invention.

## Historical starting points

These are selected primary documents, not a complete Tesla bibliography.
Patent publication establishes what was disclosed; it does not independently
validate a physical explanation or establish practical performance.

| Document | Published | Relevance and boundary |
| --- | --- | --- |
| [US593138A — Electrical Transformer](https://patents.google.com/patent/US593138A/en) | 1897-11-02 | Historical transformer design. The atmospheric RC recipe does not reproduce its geometry or electromagnetic behavior. |
| [US645576A — System of transmission of electrical energy](https://patents.google.com/patent/US645576A/en) | 1900-03-20 | Historical transmission proposal. It supplies context, not a validated global transmission or energy-generation model. |
| [US685957A — Apparatus for the utilization of radiant energy](https://patents.google.com/patent/US685957A/en) | 1901-11-05 | Describes irradiation of a conducting body connected to a capacitor and use of stored charge. Its examples include ultraviolet and X-ray sources. |
| [US685958A — Method of utilizing radiant energy](https://patents.google.com/patent/US685958A/en) | 1901-11-05 | Related method patent for charging and subsequently using the capacitor. It is not evidence of unlimited energy from fair-weather air. |

The 1901 patents motivate the charge-storage question. They are not the source
of the fair-weather conductivity model below. The present recipe has no
photoemission, radiation transport, X-ray source, resonant transformer, spark
discharge, antenna propagation, lightning or global Earth–ionosphere solver.

## Modern empirical context

The global atmospheric electric circuit supports a small conduction current
through fair-weather air. Conductivity is finite, varies with place and
conditions, and requires an explicit current/energy source in an equivalent
circuit. Positive potential gradient `PG = dφ/dz` means potential increases
upward; the vertical electric field is `E_z = −PG`. In this module `J = σ·PG`
is the **positive downward magnitude** of current density, not its signed
upward component. `σ` is the total bipolar conductivity.

| Primary or official source | Evidence used here |
| --- | --- |
| [NASA-CR-180023, *The Earth's electrical environment* (1986)](https://ntrs.nasa.gov/citations/19870008839), [report PDF](https://ntrs.nasa.gov/api/citations/19870008839/downloads/19870008839.pdf) | Describes the maintained global circuit, typical surface fair-weather fields of about 100–300 V/m, and near-surface conductivity of order 10⁻¹⁴ S/m. These are representative scales, not universal constants. |
| [Harrison & Nicoll (2008), DOI 10.1016/j.atmosres.2008.01.008](https://doi.org/10.1016/j.atmosres.2008.01.008), [author-institution record](https://researchportal.bath.ac.uk/en/publications/air-earth-current-density-measurements-at-lerwick-implications-fo/) | Lerwick measurements at 15 UT during 1978–1984 span 0.5–9.5 pA/m², with a median of 2.5 pA/m². This is a particular site's historical series, not a guaranteed collection rate. |
| [Harrison & Ambaum (2008), DOI 10.1098/rspa.2008.0009](https://doi.org/10.1098/rspa.2008.0009), [author-hosted paper](https://www.met.reading.ac.uk/~sws97mha/Publications/RSPA20080009.pdf) | Uses `J = σE` in a one-dimensional atmospheric model. The Reading observations include 722 fair-weather hours in September–October 2006 with median current density 1.8 pA/m² and range 0.2–4 pA/m². |
| [Harrison & Bennett (2007), DOI 10.5194/adgeo-13-17-2007](https://adgeo.copernicus.org/articles/13/17/2007/) | Shows how columnar resistance depends on conductivity and separates boundary-layer and upper-tropospheric contributions; this explains why a uniform short column is only a simplification. |

The recipe's default `PG = 100 V/m`, `σ = 2 × 10⁻¹⁴ S/m` produces
`J = 2 × 10⁻¹² A/m² = 2 pA/m²`. This is an illustrative choice consistent
with the scale of those observations. It is not a fit to their raw data.
The runtime does not fetch live atmospheric measurements.

## Model, quantities and units

Treat a uniform column of length `h` and cross-sectional area `A` as an ohmic
source resistance. An imposed ideal source maintains its background potential
difference. The source resistance then supplies a capacitor with parallel
load and leakage resistors. This is a defined equivalent circuit; real
electrodes distort the field and their collection geometry is not solved.

| Quantity | Symbol | Unit | Illustrative default |
| --- | --- | --- | --- |
| Potential gradient magnitude | `PG` | V/m | 100 |
| Air conductivity | `σ` | S/m | 2 × 10⁻¹⁴ (20 fS/m) |
| Effective column length | `h` | m | 1 |
| Effective column cross-section | `A` | m² | 1 |
| Storage capacitance | `C` | F | 2 × 10⁻¹¹ (20 pF) |
| Load resistance | `R_L` | Ω | 5 × 10¹³ (50 TΩ) |
| Leakage resistance | `R_K` | Ω | 10¹⁵ (1000 TΩ) |
| Initial capacitor voltage | `V(0)` | V | 0 |
| Model duration | `T` | s | 3000 |

Machine input bounds are explicit:

| Field | Allowed interval in SI units |
| --- | --- |
| `potential_gradient_v_m` | 0–300 |
| `conductivity_s_m` | Exactly 0, or 10⁻¹⁶–10⁻¹³ |
| `height_m` | 0.1–10 |
| `area_m2` | 0.1–10 |
| `capacitance_f` | 10⁻¹²–10⁻⁹ |
| `load_resistance_ohm` | 10¹¹–10¹⁶ |
| `leakage_resistance_ohm` | 10¹²–10¹⁷ |
| `initial_voltage_v` | 0–3000 |
| `duration_s` | 1–36000 |
| `samples` | Integer 21–201; default 121 |

Zero field and zero conductivity are useful mathematical controls, not
representative normal fair-weather conditions. A nonzero initial voltage
represents energy placed in the capacitor before the run. Durations and
parameter ranges bound computation; they do not certify the model across all
those conditions. The recipe registry is the executable schema authority.
The separate zero-conductivity case disconnects the source; the nonzero lower
bound keeps source resistance finite and representable in exported JSON.

The source relations are:

```text
V_s = PG · h                               [V]
G_a = σ · A / h = 1 / R_air               [S]
J = σ · PG                                [A/m²]
I_short = G_a · V_s = J · A                [A]
G_L = 1 / R_L;  G_K = 1 / R_K             [S]
G = G_a + G_L + G_K                        [S]

C · dV/dt = G_a · (V_s − V) − G_L · V − G_K · V
τ = C / G                                 [s]
V_infinity = G_a · V_s / G                 [V]
V(t) = V_infinity + (V(0) − V_infinity) · exp(−t / τ)  [V]
I_source(t) = G_a · (V_s − V(t))           [A]
P_load(t) = G_L · V(t)²                    [W]
U_cap(t) = C · V(t)² / 2                   [J]
```

This exact exponential solution avoids a timestep instability. For zero
initial voltage it reduces to `V_infinity · (1 − exp(−t / τ))`, with
`0 ≤ V(t) ≤ V_infinity ≤ V_s`. More generally voltage stays between its
initial and steady values. A precharged capacitor can discharge, including
back into the ideal reservoir when `V > V_s`; this is a mathematical boundary
condition, not a prediction of a real atmospheric collector's reverse-current
behavior. With `σ = 0`, source conductance and source current are zero;
`R_air` is represented as JSON `null` rather than infinity.
Making an initially uncharged capacitor larger changes the transient; it
does not create energy or increase steady-state available power.

For this *defined source*, an ideal optimally matched load with no leakage has
the maximum-power bound `P_max = V_s² · G_a / 4`. It is not a general bound on
all atmospheric processes or a performance prediction for a physical device.
Leakage lowers the useful steady load power. A precharged capacitor may
temporarily deliver more than this steady-source bound, using its initial
stored energy.

With the defaults, calculations give `R_air = 50 TΩ`, `I_short = 2 pA`,
`τ ≈ 487.804878 s`, and `V_infinity ≈ 48.780488 V`. At 3000 s,
`V ≈ 48.676416 V` and stored energy is approximately `23.693934 nJ`.
Steady load power is approximately `47.590720 pW`; the no-leak matched-source
bound is `50 pW`. These are calculated circuit results, not measured output.

## Energy account and verification

The maintained source can supply or absorb energy. The exported account must
include its signed work, any initial capacitor energy and the loss in the
atmospheric source resistance, not only final capacitor energy and useful
load power:

```text
W_source = ∫ V_s · G_a · (V_s − V(t)) dt
W_air    = ∫ G_a · (V_s − V(t))² dt
W_load   = ∫ G_L · V(t)² dt
W_leak   = ∫ G_K · V(t)² dt

W_source = [U_cap(T) − U_cap(0)] + W_air + W_load + W_leak
```

All integrals run from 0 to `T`. Analytic expressions follow by integrating
the exponential solution. Small-time expressions require numerically stable
evaluation, for example `expm1` and a short-argument series; directly
subtracting nearly equal exponentials can lose precision.

Verification concerns the frozen mathematical model: input bounds, finite
outputs, the differential equation, the steady-state voltage bound, and the
energy identity within declared numerical tolerance. Meaningful controls
include zero field and zero conductivity with an initially uncharged
capacitor, a precharged discharge control, finite leakage, altered load and
capacitance, and rejection of changed evidence. Passing a verifier means the
reported numbers satisfy this model; it does not validate a Tesla patent,
prove a new source of energy or establish atmospheric collection efficiency.

## Reuse and extension

The recipe is original project code under the repository's software license.
Source links and bibliographic descriptions do not relicense third-party
papers or datasets. No full copyrighted paper, trained model weights or live
measurement series is incorporated by this document.

Extensions should declare a new model/version and its assumptions rather than
silently change existing evidence. Useful separate research steps include
height-dependent conductivity, measured time-varying potential gradient,
calibrated electrode geometry, and an independently specified resonance model.
Each needs units, bounded input schema, primary references, independent
controls and explicit data provenance. Use the repository's author-experiment
manifest and runtime extension guides to add such work. An external runtime
listing does not grant access to users' machines, install arbitrary server
code or constitute scientific review.