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; neglecting dispersed fluxes, coil linkage flux $\Psi$ is thus $N$ 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.

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.

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; neglecting dispersed fluxes, coil linkage flux $\Psi$ is thus $N$ 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

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

with $N$ number of windings, $\ell$ coil length, $\frac{N}{\ell}$ number of winding density, $I$ the current in the inductor.

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; neglecting dispersed fluxes, coil linkage flux $\Psi$ is thus $N$ 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.