The formula for the energy stored in the magnetic field of an inductor is:
E = (1/2) (inductance) (current)² .
In the present situation:
Energy = (3 kilo-watt-hour) x (1,000 / kilo) x (joule/watt-sec) x (3,600 sec/hr)
= (3 · 1000 · 3,600) (kilo·watt·hr·joule·sec / kilo·watt·sec·hr)
= 1.08 x 10⁷ joules .
Now to find the inductance:
E = (1/2) (inductance) (current)²
(1.08 x 10⁷ joules) = (1/2) (inductance) (300 Amp)²
(2.16 x10⁷ joules) = (inductance) (300 Amp)²
Inductance = (2.16 x10⁷ joules) / (300 Amp)²
= 2.16 x10⁷ / 90,000 Henrys
I get 240 Henrys .
This is a big inductance. Possibly the size of your house.
To get a big inductance, you want to wind the coil
with a huge number of turns of very fine wire, in
a small space.
In this case, however, if you plan on running 300A through
your coil, it’ll have to be wound with a very thick conductor …
like maybe 1/4-inch solid copper wire, or even copper tubing,
You have competing requirements.
There are cheaper, easier, better ways to store 3 kWh of energy.
In fact, a quick back-of-the-napkin calculation says that
3 or 4 car batteries will do the job nicely.