According to Gauss's law of magnetism, the total magnetic flux through a closed surface is zero. But during induction, we study that the magnetic field lines passing through a coil change, as does flux given by $\Phi = LI$. But even if they change, the net lines coming in= net lines going out. So, flux should be zero?
[Physics] total magnetic flux through a coil
electric-circuitselectricityelectromagnetic-inductionelectromagnetismmagnetic fields
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The flux through the closed hemisphere is zero, $$\Phi_{\mathrm{hemi}}+\Phi_{\mathrm{disk}} = 0.$$ This allows us to find the flux through the hemisphere knowing the (more easily calculable) flux through the disk, $$\Phi_{\mathrm{hemi}} = -\Phi_{\mathrm{disk}}.$$
Would the ‘net’ flux through the area bound by the coil be zero?( because of opposite directions of magnetic field at its two faces?)
The magnetic field actually is in the same direction on each side of the plane. You can see this in the image below and as explained here. As you can see, there is definitely a magnetic flux through the area bound by the coil itself.
If so, Gauss’ Law states that net flux through a closed surface is zero, but the area bound by the coil is not a closed surface.
Yes, you are right. Gauss's law for magnetic fields tells us $$\oint\mathbf B\cdot\text d\mathbf A=0$$
but this is a surface integral over a closed surface. The area bound by the coil is not a closed surface, so we don't need to worry about this applying here.
So, even though the flux through thiis area is in fact not $0$, I will address a concern you seem to have in linking these two ideas together. You seem to be thinking that a $0$ flux means that the surface integral must have been done over a closed surface. This is not the case. The statement "If the integral is over a closed surface, then the magnetic flux is $0$" is not a biconditional statement. In other words, the statement "If the magnetic flux is $0$ then the integral was done over a closed surface" is a false statement.
Best Answer
Gauss's law of magnetism does not give us a tool to "measure" flux. As you said it just says that the total magnetic flux through a closed surface is zero.
If we take a coil with changing magnetic field and imagine a sphere around it, Gauss law tell's us that the total flux through the surface of the sphere is zero, what helps us in no way.
Field lines are just a tool to represent intensity of the magnetic field with drawing (less) denser lines, and representing the direction the field at a given position. You can't argue with the "amount" of field lines.
But as you stated magnetic flux through a coil is given by $$\Phi = LI$$
So flux is not zero for $I, L \ne 0$.