I'm studying magnetic fields and I have a big problem. The Gauss theorem states that the magnetic flux through a closed surface is null since the line of flux enter and exit from it in the same amount. However, the Faraday law talks about the flux linkage with a surface which can be not null in case, for example, the magnetic field is not constant in time. Why is not null as well, since it is the flux through a surface?
[Physics] Gauss Theorem and Faraday’s Law
electromagnetismgauss-law
Related Solutions
In an attempt to be brief: The big thing to remember is that the flux is also proportional to the area (technically, the surface integral of the field over the area). Crudely speaking, the side of the enclosed surface with exiting field lines are further away from the external charge than the side with "entering" field lines, and the surface area increases by $r^2$ (remember the formula for the surface area of a sphere). Gauss's Law is just a mathematically precise way of stating this compensation.
(Image source: http://www.ux1.eiu.edu/~cfadd/1360/24Gauss/Gauss.html)
Gauss' Law says $\iint_S \mathbf{E} \cdot \mathbf{dA} = Q/\epsilon_0$ whereas the total vector area is $\iint_S \mathbf{dA}=\mathbf{0}$ for some closed surface $S$. The total vector area is taking vector sum of all the differential area vectors that are normal to the surface, whereas $\iint_S \mathbf{E} \cdot \mathbf{dA} = Q/\epsilon_0$ is taking the dot product of the electric field and the differential area vector.
Let's use a sphere of radius $R$ centered at the origin as an example. Its clear that $\iint_S \mathbf{dA}=\mathbf{0}$ since, for every point on the sphere, the normal vector of that point cancels out with the point opposite it on the sphere. Therefore, $\iint_S \mathbf{dA}=\mathbf{0}$.
Now suppose there's a charge $Q$ at the origin. The electric field and the normal vector will be parallel at every point on the sphere, so the symmetry of the problem means $\iint_S \mathbf{E} \cdot \mathbf{dA} = EA$ where $E$ and $A$ are the electric field and surface area of the sphere respectively.
Best Answer
Faraday's law talks about the flux piercing through an open surface, while Gauss' law for magnetic fields is for a closed surface.