Magnetic flux and number of turns

Question

The number of windings is not written in the magnetic flux formulas in high school textbooks, but I come across articles saying that the number of windings affects the magnetic flux formulas. Is this true or not?

Answer 1

You need to focus on the surface across which you compute the magnetic flux: a coil with $N$ windings can be thought as $N$ circular surface.

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Let's call $\Phi$ the flux across a section of the coil, and $\Psi$ the linkage flux across. Neglecting dispersed fluxes, coil linkage flux $\Psi$ is $N$ times the flux through $\Phi$ one of these sections,

$$\Psi = N \, \Phi \ ,$$

being $\Phi = B \, A$, with $A$ the section of the coil, and $B$ the uniform magnetic flux inside the coil, that can be evaluated (using Ampére law in steady conditions) as

$$B \, \ell = \mu \, N \, i \qquad \rightarrow \qquad B = \mu \, \frac{N}{\ell} i$$

with $N$ number of windings, $\ell$ coil length, $\frac{N}{\ell}$ number of winding density, $i$ the current in the inductor, and $\mu$ the permeability of the medium in the core of the coil.

Putting everything together and using Faraday's law on the coil, the voltage across the ends of the coil is

$$v = \dfrac{d \Psi}{dt} = \frac{d}{dt} \left( N \Phi \right) = \frac{d}{dt} \left( N A B \right) = \frac{d}{dt} \left( \mu \frac{A N^2}{\ell} i \right) = L \dfrac{d i}{d t} \ ,$$

being $L$ the inductance, here assumed to be constant.

Answer 2

When talking about coils, the term magnetic flux is somewhat ambiguous without context. For any given surface $S$, bounded by some loop $C$, we can define it as the surface integral $$\Phi = \int\limits_S \vec B\cdot d\vec A.$$ This can be thought qualitatively as the number of magnetic field lines through $S$. The ambiguity arises from which surface $S$ (or loop $C$) one has in mind.

Consider the cross-section $S_1$ of this core somewhere along a coil with $N$ turns. We can call the flux through it $\Phi_1$. If the coil is sufficiently small, or is wound around a magnetic core which is good at confining the magnetic field within, $\Phi_1$ is roughly the same wherever you take the cross-section $S_1$.

Next, we take $C$ to be a circuit loop that runs along the conductor of the coil. The surface $S_2$ bounded by it is very complicated, but it essentially intersects the magnetic core about $N$ times. Therefore, the flux through $S_2$ is $\Phi_2 \approx N\Phi_1$.

If the magnetic field is created by some other source, then $\Phi_1$ doesn't depend on $N$, but $\Phi_2$ is approximately proportional to $N$. In the context of magnetic circuits and coils, $\Phi_1$ is often referred to as the flux (or the flux in the core if there is a magnetic core), and $\Phi_2$ as the flux linked by the coil.