Let's say we have a DC source connected to an inductor. The orientation of the magnetic field through the inductor depends on which direction the coil was wrapped (right hand rule). But why do we not care about it's polarity when we use inductor in the circuit? I guess we do care about polarity in terms of mutual inductance, but what about when only one inductor is used in the circuit?
You might care. For example, if you have a filter choke that has a permanent magnet biasing it.
Or if there is another inductor magnetically coupled to it, then the relative polarity matters.
Suppose you have two identical inductors, except one has a right-handed winding, and the other has a left-handed winding. Ramp up the current through the right-handed inductor. This produces a changing magnet field which induces a voltage in the inductor. That's self-inductance. Now do the same with the left-handed inductor. The handedness reversal reverses the relationship between the current and the magnetic field, but it also reverses the relationship between the magnetic field and the induced voltage. With the double reversal, you get an identical induced voltage, so the self-inductance is the same.
Actually we do care about Inductors, especially with DC circuits. The polarity depends on the direction of the current flowing through it. If a current is moving from point A to point B (say), the Inductor stores the current in the coil inturn producing magentic field. Now if the device is turned off, the inductor that had the current stored in it must dissipate. But because of the reverse EMF (according to your right hand thumb rule) the current travels form point B to A. This current can spike up and destroy your circuitry. That's why you have bypass diodes connected across node A and B such that it creates a loop. Eventually the current dissipates with all its might saving the circuitry behind it. Hope this helped.