I'm confused. Shouldn't the currents be time-continuous here?
My intuition is that the current upon an inductor (say, a solenoid) will always be time-continuous, without "sudden changes". But below is a case that seemingly contradicts this point of view.
There are two wires $W_1$ and $W_2$, both intensely wrapped around a common core whose section area is $S$, length is $L$ ($L>>\sqrt S$), and permeability is $\mu_0$ (and it is not ferro-magnetic material !). At first the two switches $S_1$ and $S_2$ are both open. Now, close them both. After a time that is long enough, we know that the currents go through $W_1$ and $W_2$ will respectively stabilize at $$I_1=\frac{\mathscr E_1}{R_1}$$ and $$I_2=\frac{\mathscr E_2}{R_2}$$ So the ratio between $I_1$ and $I_2$ is $$\frac{I_1}{I_2}=\frac{\mathscr E_1 R_2}{\mathscr E_2 R_1}\tag{}$$ Now, let's simultaneously switch $S_1$ and $S_2$ open. 628
Then the current $I_1$ that $W_1$ is carrying will have no choice but to go through $R'_1$ and form a loop, which is just the same with $W_2$ side, the other half of the system. And the systems become time-varying. Let's just specify the moment when we switch $S_1$ and $S_2$ open as $t=0$, and let the currents going through $W_1$ and $W_2$ be denoted by $I_1(t)$ and $I_2(t)$, respectively, and the magnetic flux through the core denoted by $\Phi (t)$.
First, let's take a look at the $W_1$ loop. Because $\Phi (t)$ is time-varying, there will be an magnetically-induced emf (denoted by $\mathscr E^i_1(t)$) whose magnitude will be $$\mathscr E^i_1(t)=Ln_1\frac{d\Phi(t)}{dt}$$ where $n_1$ denotes the number of turns of $W_1$ wrapped around the core per unit length.
However, by applying KVL to the $W_1$ loop we will also get $$\mathscr E^i_1(t)=I_1(t)R'_1$$ Therefore $$I_1(t)=\frac{Ln_1}{R'_1}\frac{d\Phi(t)}{dt}$$ Then, just do the same analysis for $W_2$ loop, and likewise we get $$I_2(t)=\frac{Ln_2}{R'_2}\frac{d\Phi(t)}{dt}$$ So the ratio between them is $$\frac{I_1(t)}{I_2(t)}=\frac{n_1R'_2}{n_2R'_1}$$ which does not depend on $t$. Then, by the time-continuity of currents at $t=0$, we have $$\frac{I_1(0)}{I_2(0)}=\frac{n_1R'_2}{n_2R'_1}$$ But as I've specified, $I_1(0)$ is exactly $I_1$, and $I_2(0)$ exactly $I_2$, therefore $$\frac{I_1}{I_2}=\frac{n_1R'_2}{n_2R'_1}\tag{}$$ Comparing () to (**), we'll find it's not possible. Because $\mathscr E, R, R', n$ are all arbitrary, they don't necessarily make the two ratios fit.
So could you please explain to me where I'm wrong? Best regards!
EDIT I think the reasons are probably of the following:
1).I've made too many ideal hypothesis, such as "intensely wrapped", "$L>>\sqrt S$" etc, which may make our assumptions too far away from the fact.
2).(*) is perhaps wrong (which I haven't proven rigorously but wrote by my intuition), i.e. 628