This figure from Halliday's book is perfect to go with my question. I've been trying to understand how exactly indutance works almost the entire morning, I think I need another perspective, please.

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Suppose when S1 is closed, S2 must be open, and vice versa. In the initial situation, S1 is closed for enough time so that the current reach its steady state $i=\frac{V_s}{R}$ (I'll just use $V_s$ for simplicity for the source voltage). Then, switch S1 goes open and immediately S2 closes, so the circuit becomes RL with no voltage source. As soon as switch S2 closes, the current changes from $\frac{V_s}{R}$ to $0$ so the magnetic flux decreases, then an EMF is created to oppose the change in magnetic flux, the current will have the same direction as the source current and the magnitude will be about the same, so the magnitude of the current doesnt change as $t\rightarrow n^+$, $n$ being the instant switch S2 closes. But the change in magnetic flux occurred from $t=n$ to $t=n+0.00000000000001$ (lets put it like that). And when $t=n+0.002$, for example, why the induced current still changes if the magnetic flux is no longer changing? It's a stupid question but why it doesn't the current stays for some time and then suddenly drops to zero. I know about the exponential function on the current, but I would like a real perspective on how this works involving the magnetic flux and current relation. Appreciate any help.


As soon as switch S2 closes, the current changes from $\frac{Vs}{R}$ to 0

If, by current, you're referring to the inductor current, then this is where your error is.

The instant after the two switches toggle (S1 opens and S2 closes), the inductor current is unchanged from the steady state value; inductor current is continuous, i.e., no sudden jumps from one value to another. This is because the voltage across the inductor is proportional to the time derivative of the current through:

$$v_L = L \frac{di_L}{dt} $$

An instantaneous change in inductor current implies an infinite voltage across.

However, the voltage across the inductor can change instantaneously and so, the instant after the toggle, the voltage across the inductor changes from zero to $-V_s$ so that the resistor voltage remains unchanged (this must be the case since the resistor current is unchanged)

But, if there is a voltage across the inductor, the current through must be changing and since the voltage across is negative, the current through must be decreasing.

Writing the ordinary differential equation for the inductor current after the toggle and solving reveals that the inductor current must decay exponentially with time constant $\tau = L/R$


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