PX157 - D3a - inductors

self-inductance

a solenoid with $N$ turns, of length, $l$ , and magnetic flux, $ϕ_{B}$
if $I_{s} (t) ⟹ B (t) ⟹ ϕ_{B} (t) :$ faraday's law:

ϕ_{B} = - N \frac{d ϕ_{B}}{d t} = - \frac{d}{d t} (N ϕ_{B})

$N ϕ_{B}$ is called the flux linkage
- units: $w e b e r (W b)$
$ϕ_{B} \sim B \sim I_{s} ⟹ ϕ_{B} \sim I_{s}$
self-inductance is defined as:

L = \frac{N ϕ_{B}}{I}

- units: $h e n r y (H) = k g m^{2} C^{- 2}$
- compare with $C = \frac{Q}{V}$ , or $\frac{1}{R} = \frac{I}{V}$

for a long solenoid:

\begin{aligned} B & = μ_{0} \frac{N}{l} I \\ ϕ_{B} & = μ_{0} \frac{N}{l} A I \\ L & = \frac{μ_{0} N^{2} A}{l} \end{aligned}

if a ferromagnetic material is added inside the solenoid, $μ_{0} \to μ_{0} μ_{r} :$

L = \frac{μ_{r} μ_{0} N^{2} A}{l}