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I have been taught that magnetic flux is equal to BA, where B is the magnetic flux density and A is the area. When calculating magnetic flux linkage you simply multiply by the number of coils in the solenoid.

I understand what the area A represents when speaking in terms of say a wire passing through a magnetic field, where the area it sweeps out is A. Or a physical sheet. I understand it as being equivalent to the number of field lines that it cuts. When calculating the area of a solenoid are you using the radius of solenoid to calculate the area like it is a solid disk (like this image implies). Because I don't understand this, as it's not actually cutting any of the field lines that pass through the middle.Image found that supposedly helps explain the concept

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For me the diagram needed to answer the question was difficult to draw and I did not make a video which would have illustrated the answer better because the soap film would have been easier to see.

I have made a complete circuit out of bare copper wire and within the circuit formed a soap film which represents a plane defined by the boundaries of the circuit.

enter image description here

In the left-hand image the two surfaces of the soap film, $S_1$ and $S_2$, are relatively easy to see?
There is no magnetic flux through surface $S_1$ and the magnetic flux through surface $S_2$ is $B \,A$ where $A$ is the area of the surface $S_2$ ie the area of the coil.
The total magnetic flux through the soap film is $BA$.
This is equivalent to the left hand diagram in the OP's question.

The right hand circuit which represents a solenoid with two turns was more difficult to photograph whilst at the same time showing the soap film clearly so I had to pull out the two turns into a helix with a much larger pitch than I really wanted.
This has resulted in the area of the soap film being increased and the magnetic field not being at right angles to the soap film.
What I write below would work for such a circuit but to make things easier assume that the two turns are close together as in a "normal" solenoid of two turns.

Again surface $S_1$ does not contribute to the magnetic flux.
Surface $S_2$ contributes $BA$ to the total magnetic flux passing through the whole surface of the soap film as does surface $S_3$.

The total magnetic flux through the soap film is $BA+BA=2BA$ which is $NBA$ with the number of turns $N=2$ again $A$ being the area of a coil.

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