= Magnetic flux linkage
{title2=$\Lambda=\sum_{k=1}^N\Phi_k$}
= Flux linkage
{synonym}
<Magnetic flux linkage> adds the signed <magnetic fluxes> through the turns of a coil:
$$
\Lambda=\sum_{k=1}^N\Phi_k,\qquad
\Phi_k=\int_{S_k}\mathbf B\cdot d\mathbf S.
$$
Choose each <oriented surface> $S_k$ consistently with positive <electric current> around its turn using the <right-hand rule>. If all $N$ turns link approximately the same <magnetic flux> $\Phi$, then $\Lambda\approx N\Phi$; equality holds when every turn links exactly the same <magnetic flux>. Uniform <magnetic field> perpendicular to identical flat turns of <area> $A$ gives $\Phi=BA$. For a stipulated $100$-turn coil with <magnetic flux> $0.015\,\mathrm{Wb}$ per turn, <magnetic flux linkage> is $1.5\,\mathrm{Wb}$, not $0.015\,\mathrm{Wb}$. The turn count is dimensionless. https://openstax.org/books/university-physics-volume-2/pages/14-2-self-inductance-and-inductors[OpenStax University Physics 2, §14.2: flux through the turns of a coil]{ref}.
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