Why is the magnetic flux proportional to the current (Inductance)? I'm trying to understand the reasoning behing this formula, or how to get there, if possible, using Maxwell's equations:
$$\Phi_i = L_i I_i + M_{ij} I_j$$
From the wikipedia page of inductance I got this formula which is basically faraday's law:
$$v(t) = - \frac{d\Phi(t)}{dt}$$
But then they somehow get to this one which is what I don't understand:
$$v(t) = L \frac{d i(t)}{dt}$$
I've tried doing some things with the Maxwell equations but I couldn't really get to anything interesting.
 A: Your first formula is in fact one possible definition of $L_i$ and $M_{ij}$, so it's not surprising that you're having a hard time deriving it.
The reasoning is simple: The magnetic field generated by a circuit is proportional to the current in the circuit. The magnetic field at circuit $i$ is just the sum of the magnetic fields generated by each circuit, including the same circuit $i$, and each of these is proportional to the current of the corresponding circuit. We can write the contribution of circuit $j$ as
$$\mathbf{B}_j(\mathbf{r}) = I_j \tilde{\mathbf{B}}_j(\mathbf{r})$$
where $\tilde{\mathbf{B}}_j$ is some function that depends on the shape of circuit $j$ but not on its current. The flux across circuit $i$ is then
$$\Phi_i = I_i \iint_\text{circuit $i$} \tilde{\mathbf{B}}_i \cdot \mathbf{dS} + \sum_{j\neq i} I_j \iint_\text{circuit $i$} \tilde{\mathbf{B}}_j \cdot \mathbf{dS}.$$
Each of these integrals depends only on the geometry of the circuits but not on their currents, and they're what we call the inductances:
$$\Phi_i = L_i I_i + \sum_{j \neq i} M_{ij} I_j.$$
Your last formula follows if we take just one circuit; in that case we have $\Phi = LI$ and $d\Phi/dt = L\, dI/dt$. Note that the voltage in your formula seems to be missing a sign.
