Magnetic flux ... your definition is not accurate strictly. The concept of flux is a descriptive mathematical concept. ....
You specify a surface at will, and the vector value f of the vector field at any point on this surface (regardless of electric field, magnetic field and gravitational field),
Point multiplication (vector operator, scalar value obtained by multiplying the absolute value of two vectors by the cosine of their included angle)
Area element vector dS of the surface at this point
The above-mentioned point product integration surface ... {surface integral symbol} ∫ f ds
The popular expression of magnetic flux is: you randomly specify a closed curve, how many magnetic fields pass through this curve? .....
Here, "how many magnetic fields have passed through": "magnetic field" refers to the magnetic induction intensity b; "Crossing the past", we need to confirm its uniqueness, so we adopt the definition of "vertical crossing" ... Otherwise, if regional elements with the same area are projected, the magnetic flux will not be unique ... The corresponding mathematical expression is the dot product of b and dS, BdScosθ ... This can be understood as the component Bcosθ * dS of b in the direction perpendicular to the area element, and it can also be understood as. ......
The above is a mathematical/physical expression that explains the definition of flux/magnetic flux.
To understand the law of electromagnetic induction, you must first understand that you can specify any curved surface to study the induced electromotive force, which is not limited to the shape of the coil ... Only, when calculating the induced electromotive force of the loop, it is more convenient to specify the curved surface which coincides with the shape of the coil directly ... Secondly, find out why the electromotive force is related to the magnetic flux passing through the plane, not to your misunderstanding ("the magnetic flux passing through the wire").
Then, you should be clear that the research method of physics is to abstract the actual situation into a mathematically manageable model ... to ensure that the difference between this model and the actual situation is controllable (that is, mathematically convergent) ... and then calculate the ideal situation with physical laws and mathematical models. .........
When calculating electromagnetic induction, we often abstract the conductor as a mathematical line with no cross-sectional area ... because we don't care about the internal situation of the conductor at this time ... What should we do if the cross-sectional area of the conductor can't be ignored? For example, a thick silver wire will generate induced current in an alternating magnetic field, and we intend to study the eddy current inside the wire ... So we can't simply regard the wire as a mathematical straight line, but abstract its shape into a "cylinder" and so on, and then delimit different magnetic flux surfaces ... calculate the induced electromotive force, and then calculate the eddy current generated at different positions inside the wire.