Electrical – Calculate voltage, current, and power for each element

For part (b), the circuit to the left of the switch is disconnected so you can disregard it for any \$t>0\$.

The key insight for (b) is that the circuit is once again in DC steady state (all transients decay as \$t \rightarrow \infty\$).

Thus, the voltage across the inductor is \$0V\$, i.e., it can be replaced with an ideal wire.

Now, this leaves a simple current divider circuit so the inductor current is given by current division.

I annotated your circuit a bit. In red the loop current \$I\$, while in green currents and voltage references for each resistor. Please note that I followed no particular rule to do the latter.

First of all, when solving a circuit you need to choose a reference for the voltage. I did so on bottom left corner, just under the \$50V\$ generator.

Now since this circuit has only one loop you would normally need only one equation, plus one additional equation for each controlled source: that makes two for this problem.

Starting from the reference: $$ 50 - 2I + 2V_x -3I-10I-10-5I=0 $$ While the additional equation is: $$ V_x = -10I $$ I am sure you can solve it from here.

What about all the green currents then? Whell I've drawn them to prove that how you choose them does not matter at all. For each resistor you can write its voltage and current with respect to the green reference: $$ I_2=-I\qquad V_2=-2I\\ I_3=I\qquad V_3=-3I\\ I_{10}=-I\qquad V_{10}=-10I\\ I_5=-I\qquad V_5 = 5I $$

easy as pie. Please note that usually (as in please do so) for passive components you use a reference for which a positive IV product means dissipated power, i.e. currents enters at the plus terminal, while for generators and such you do the opposite. That's because that's what usually happens: generators source current from the positive terminal, passives drain currents in the positive terminal... But that's just a convention: look at your 10V generator. If we choose a reference with the same direction as I then it has the same references a passive would have... It is actually dissipating power as a matter of fact.