Inductor, Emf and Krichkoff's law

Question

I have been studying about inductors from last two weeks and i'm stucked here, It would be great if someone helps me.

Suppose i have a circuit like one shown abve;

When i close the switch at $t = 0$ then current starts flowing. As the first electron passes through the inductor it creates a small magnetic-field. As the current is changing the inductor will induce an emf in opposite direction. (True)?

Now the confusing part: According to the Kirchhoff's rule:

The directed sum of the electrical potential differences (voltage) around any closed network is zero.

Suppose at t = 0.001 , current has not reached it's max value then the induced emf $\epsilon \neq V$. I mean $V- \varepsilon \neq 0$?

What i'm missing?

Answer 1

The voltage across the inductor is controlled by the rate of change of current, $\frac{dI}{dt }$, not the current $I$, $V_{\rm L}=L \frac{dI}{dt }$.

So $\mathcal E -V_{\rm L}=\mathcal E -L \frac{dI}{dt }=0$ which means that with no resistance in the circuit the current in the circuit increases at the same rate, $\frac{\mathcal E}{L}$, forever.