inductor
Asuuming the following circuit has been turned on for a long time:
I need to find the current Il=current that goes through the inductor.
I know that the indoctor acts as a short circuit, but i still can't figure how to calculate the current.
I was thinking that all the current goes through the inductor, but that still doesn't make sense and doesn't help me solve.
Any tips/help would be really appreciated.
The resistor is not a short circuit at any point, only the inductor. What you may have meant is that the inductor being a short circuit will effectively take the resistor out of the circuit after the switch has been closed for a long time (16Ω in parallel with 0Ω = 0Ω, so regardless of the resistor value it has no effect on the circuit during this time)
The formula for the voltage across an ideal inductor is:
\$ V(t) = L \dfrac{dI(t)}{dt} \$
This means that unless the current is changing (dI), there is 0V across the inductor, i.e:
\$ V(t) = L\dfrac{0}{dt} = 0V \$
Equivalent circuit for the switch closed steady state:
... because that's literally the definition of an inductor!
$$v(t) = L \frac{\delta i(t)}{\delta t}$$
This means that for there to be a discontinuity in i(t), its derivative would have to be infinite, and so would the voltage across it. Since the voltage can only assume finite values, i(t) must be a continuous function. You can safely assume that the inductor current just after the switch opens is equal to the value just before it opens. That becomes the starting point for solving the behavior of the circuit from that point on.
Best Answer
So the 3, 6 and 12 ohms resistors are in parallel (asuming the inductor is 0 ohms) - this means \${12 \over 7} ohms\$. Add the 4 ohms resistor and you got \$ {12 \over 7} + 4 = {40 \over 7} ohms\$. The current through this is \$60 \cdot {7 \over 40} = 10.5 A\$. Using this we calculate the voltage across the three resistors in parallel:
\$ {12 \over 7} \cdot 10.5 = 18 V\$. Then divide this by the 3 ohms resistor: \${18 V \over 3 ohms} = 6 A\$.
simulate this circuit – Schematic created using CircuitLab