The current in a conductor is found to be moving from (-1, 2, -3) m to (-4, 5, -6) m. Given that the

If the magnitude of the magnetic induction is less than that in a vaccum within a solid, so, this material called
a) Ferrimagnetic
b) Ferromagnetic
c) Diamagnetic
d) Paramagnetic

Gauss's law states that ${\int }_S\overrightarrow{B}\cdot d\overrightarrow{S}=0$. But law of induction states that $\xi =-\frac{d\varphi }{dt}$, where $\varphi ={\int }_S\overrightarrow{B}\cdot d\overrightarrow{S}$
So if Gauss's law was to be correct there should be no induction at all, because then ϕ would be zero through every loop.

State Gauss’ Law for Magnetism. What significant thing does it tell us about the universe?

In Maxwell's equations, I understand intuitively how: $\oint B\cdot da=0$ (because there are no monopoles and so equal number of field lines going in and coming out of the surface).
And then using the divergence theorem:
${\int }_V(\mathrm{\nabla }\cdot B)d\tau ={\oint }_{S}B\cdot da$
Then ${\int }_V(\mathrm{\nabla }\cdot B)d\tau$ must be = 0.
But then I'm not sure why I can say: $\mathrm{\nabla }\cdot B=0$ and forget about the integral.
Does it just mean that $\mathrm{\nabla }\cdot B$ must be zero everywhere?

I'm trying to learn the connection between special relativity and magnetism. I know that if I place a positive charge, at rest, next to wire with current, I should not observe any force on it because there is no electric field and there is no magnetic force as my charge is at rest.
But here is what confuses me - the wire contains moving electrons and according to what I learned, the stationary charge should observe a length contraction of those electrons and so the density of them will increase and a negative electric field should be observed.
This is definitely not the case and I wonder if someone can explain to me what is wrong in my analysts.

What is the magnitude of the net force on the first wire in (Figure 1)?
What is the magnitude of the net force on the second wire in (Figure 1)?
What is the magnitude of the net force on the third wire in (Figure 1)?
photo

How many neutral points can be obtained in a given plane perpendicular to the length of the two parallel wires conducting current in the same direction? Explain, Neglect earth's magnetic field of a bar magnet NS