# Optimal size of a reactor

## Reference reactor

What size should a reactor be? It would be useful to have some kind of typical reactor configuration to use as a yardstick when investigating this, something that can be scaled. Let our reference reactor have a central electrode of length `2 a` and set the maximum size of the orbit to `a`. The central electrode can have diameter `k_i a` where `k_i` is constant, and have a charge of `Q = a \lambda`. This setup preserves the voltage with respect to `a` as the reactor is scaled (see below).

It would also be useful to have a simple orbit that can be used a reference betweeen scalings. The simplest orbits are circular orbits around the equator which are achieved by balancing the tangential velocity and the radius. Setting `r = a` provides a handy reference orbit.

## Electrode charge

What affect does varying the size of the reactor have on the charge of the central electrode?

The electric field of an infinite line charge is given by Guass' Law for lines:

`E = \lambda / (2 \pi r \epsilon_0)`

where `E` is the electric field, `\lambda` the charge per unit length, `r` the radius of the cylinder,
and `\epsilon_0 = 8.85 \times 10^{12}` F/m
(see this page on hyperphysics).

Let `V` be the voltage, then:

`V = \int E dr
= \int \lambda / (2 \pi \epsilon_0 r) dr
= \lambda / (2 \pi \epsilon_0) [ln |r|]_j^i`

Rearranging:

`\lambda = 2 \pi \epsilon_0 V [ln |r|]_j^i`

If `i=1` and `j=0.05` then `[ln |r|]_j^i \approx 2.996`. If we divide `i` and `j` by the same
amount (as would be reasonable when scaling the reactor) then `[ln |r|]_j^i` is constant. So
if `V` is constant throughout the scaling process then `Q \prop a`

## Circular orbit equations

Consider a reactor with a pole of length `2 a`, centred at `z=0`. For simplicity we'll look at circular orbits where `z=0`, although in practice we want all orbital paths to be unchanged, with only linear scaling of the particle positions taking taking place. Note the orbital frequency will be allowed to change during scaling.

The *radial* force towards the pole is:

`F_r = k/(2 a r) ( (z-a)/(sqrt(r^2 + (z-a)^2)) - (z+a)/(sqrt(r^2 + (z+a)^2)) )`

where `k = k_e q Q` and `q` is the charge on a particle. Holding the voltage constant throughout scaling is probably the easiest thing
to do and `V prop lambda` so substituting `Q = a lambda` we get:

`F_r = (k_e q lambda)/(2 r) ( (z-a)/(sqrt(r^2 + (z-a)^2)) - (z+a)/(sqrt(r^2 + (z+a)^2)) )`

When `z=0`:

`F_r = -(k_e q lambda a)/(r sqrt(r^2 + a^2))`

In order to achieve a circular orbit we need `F_r = (m v^2)/r`. Equating the two equations
for force gives us:

`v^2 = - (k_e q lambda a)/(m sqrt(r^2 + a^2))`

The sign here can be ignored. It can be seen there are multiple circular orbits given by different `(r,v)` pairs. Larger radii lead to slower particles, although presumably such particles would have higher potential energy.

We wish the paths followed by particles to all be scaled linearly with respect to `a`.
For our circular prototype orbit let `c a = r` where `c` is a constant. This gives us:

`v^2 = (k_e q lambda)/(m sqrt(c^2 + 1))`

This give us our main result that `v` independent of `a` provided the voltage is held constant.

The constancy of the velocity is very helpful for analysing these systems. As the reactor shrinks nothing changes at the tiny scale of particle encounters - the starting velocities and ending velocities of particles involved during encounters are identical as for large reactors. Consequently the ratio of fusion events to particles lost are independent of `a`.

The frequency of orbits is inversely proportional to `a`, and thus the output of a single reactor is inversely proportional to `a`. The next section, which looks at the capacity of the reactor, confirms this finding, as do computer simulations.

## Collision frequency

The cross-section for a trip through the plasma is:

`\sigma = (\pi b^2 n) / V = (\pi b^2 n) / (k_V a^3)`

where `b` is the furthest approach distance deemed to be a collision, `V = k_V a^3` is the effective volume
of the plasma (the main body of the plasma, excluding outliers),
and `n` is the number of particles in the
plasma. Let `n = 6.242 \times 10^{18} a \lambda k_\lambda` where `k_\lambda` is the plasma charge
proportional to the central electrode charge (negated) and `\lambda` is the charge per length on the
central electrode.

The collision frequency is:

`X = \sigma n v`
` = (\pi b^2 n^2 v) / (k_V a^3)`
` = (\pi b^2 v (6.242 \times 10^{18} \lambda k_\lambda)^2 ) / (k_V a)`

Now `\lambda` is being held constant with respect to `a` and from the section above
`v` (the average velocity and indeed the velocity distribution) is constant with respect to `a`.
Thus:

`X \prop 1/a`

The number of collisions increases inversely with respect to `a`.

If an array of reactors is used then per volume then the output would be proportional to `1/a^4`, i.e. `X prop 1/a^4`.

## Limits to scaling

The increase in cyclotron/synchrotron radiation is the most obvious limiting factor. [TODO calculate]

Note that Bremsstrahlung radiation increases with respect to `a`, as do the number of fusions and particle losses, so this is less of a concern.

Are there any implications to having the high voltage at small distances? E.g. electrodes may be more susceptable to breaking as they are thinner... Any other voltage issues?