*By Evalf, and other Nutils contributors*

officialelectro-magnetism

Computes the magnetic field induced by a DC or AC current in one or several toroidal conductors. This problem is modeled with the quasi-static magnetic vector potential with Lorenz gauge:

```
∇_j(∇_j(A_i)) = -μ0 J_i
```

where `A`

is the magnetic vector potential, `J`

the current density and `μ0`

the magnetic permeability. The magnetic field `B`

is then given by the curl
of the magnetic vector potential. The current density is the sum of an
external current `Jext`

and the current induced by the magnetic field,
`Jind`

. The external current is given by

```
Jext_i = (I / π rwire²) cos(ω t) eθ_i
```

inside the conductor and zero everywhere else, where `ω = 2 π f`

. The induced
current follows from Faraday's law of induction and Ohm's law:

```
Jind_i = -σ ∂_t(A_i)
```

where `σ`

is the conductivity, which is non-zero only inside the conductor.

We can solve the temporal component of `A`

by letting `A_i = Re(Â_i exp(j ω t))`

. This problem in terms of `Â`

is:

```
∇_j(∇_j(Â_i) = -μ0 Ĵ_i
```

with

```
Ĵext_i = (I / π rwire²) eθ_i
```

and

```
Ĵind_i = -j ω σ Â_i
```