The following exercise has brought to my mind something that confuses me:
Here, I don't see why there isn't positive charge residing on the inner surface of the conductor at $r=a$.
The way I see it, at first the conductors were neutrally charged. They were then given additional positive charge. As the outer shell is positively charged, it will exert a force on the positively charged inner shell, such that this added positive charge will be pushed to the inner surface of the conductor and the negative charge (which existed when the inner conductor was electrically neutral and hasn't gone anywhere despite being outnumbered by the positive charge) moves to the outer surface of the conductor, attracted to the outer shell.
Why is this not happening here?