Basic Circuit Analysis - confused about self polarity

[FrogPad]

This question is bugging me, and I think I need to get it answered before I move on to working on the "harder" problems. I hate being stumped by the little stuff like this. Anywas, I'm including the question that I have as an attachment, it will be easier for all.

Thank you!

 

[berkeman]

I'm not sure this will answer your question, but think of the case where you have a single long winding with a center tap. When you drive the full winding with an AC voltage, you see half that AC voltage at the center tap. Now drive just the first half of the winding with half of the AC voltage, and really nothing has changed, right? You still will measure half of the AC voltage on the second half of the winding, even though you are only driving the first half. That's because of the transformer action. Now think about how the dots would go on the two windings. If you dot the top of the first half winding, then you would dot the start of the second half winding at the center tap. You can also split the center tap, so that you have two separate windings, and the dots stay in the same places.

 

[doodle]

Yes, you have to make it negative. In the counter-clockwise direction (the second diagram), voltage drop across the inductor is $I_1 (j4)$. If you represent all voltage drops in the negative (as what you did with the resistor), then this term should be in the negative too. Also, Vs should be in negative in the second equation as well since, in the counter-clockwise direction, there is a voltage drop across its terminals.