= Self-inductance
{title2=$\Lambda=LI,\quad\mathcal E_{\rm ind}=-L\dot I$}
<Self-inductance> is <inductance> associated with a coil's own <electric current>, rather than <magnetic flux> supplied by another coil. In the linear model, fixed geometry and magnetic response give constant $L$ in $\Lambda=LI$. With the turn orientations defined in <magnetic flux linkage>, <Faraday's law of induction> gives the self-induced <electromotive force>
$$
\mathcal E_{\rm ind}=-\frac{d\Lambda}{dt}=-L\frac{dI}{dt}.
$$
The final equality requires constant $L$; if the geometry changes, use $-d\Lambda/dt$. The minus sign expresses <Lenz's law>: the induced <electromotive force> opposes the change in <electric current>. It does not always oppose the <electric current> itself. When positive <electric current> decreases, the induced <electromotive force> acts in its direction. https://openstax.org/books/university-physics-volume-2/pages/14-2-self-inductance-and-inductors[OpenStax University Physics 2, §14.2: self-induced emf and its direction]{ref}.
For terminal sign bookkeeping, label the ideal <inductor>'s terminals $a,b$, choose positive <electric current> entering $a$ and leaving $b$, and define $v_L$ as the terminal <electric potential difference> drop from $a$ to $b$ in the lumped <electrical circuit> model. This passive convention gives
$$
v_L=+L\frac{dI}{dt}=-\mathcal E_{\rm ind}.
$$
Thus a coil with <self-inductance> $0.5\,\mathrm H$ and increasing <electric current> $dI/dt=12\,\mathrm{A/s}$ has terminal drop $v_L=6\,\mathrm V$ and induced <electromotive force> $\mathcal E_{\rm ind}=-6\,\mathrm V$. These signs describe the same process with different reference conventions. The <RL circuit> uses the terminal-drop convention throughout.
A finite terminal <electric potential difference> cannot make the <electric current> of an ideal <inductor> with constant $L>0$ jump. Integrating its <differential equation> gives
$$
I(t_2)-I(t_1)=\frac1L\int_{t_1}^{t_2}v_L(t)\,dt.
$$
If $v_L$ stays bounded, the <integral> tends to zero as the interval shrinks, ensuring <continuity> of <electric current>. A step in terminal <electric potential difference> changes $dI/dt$ immediately, while <electric current> remains continuous. At constant <electric current>, the self-induced <electromotive force> is zero, although the <inductor>'s <magnetic energy> can remain nonzero.
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