The physics
A long solenoid with n turns per metre makes a uniform field inside it. Outside, the field is almost zero.
B = μ0nI L = μ0N²A / l u = B² / 2μ0
The coil's energy ½LI² sits in its field. Each cubic metre holds u = B²/2μ0.
A changing B makes an electric field that swirls round the axis. Inside the coil, E grows with r. Outside, it falls as 1/r.
E · 2πr = (rate of change of flux through the circle)
Two coils share flux. M is the flux in one coil for each ampere in the other. The emf in one coil is M times the other current's rate.
M = μ0nNA L = L₁ + L₂ ± 2M ε₁ = −(L₁ di₁/dt + M di₂/dt)
μ0 = 4π × 10⁻⁷ T m/A