for the circuit shown in Figure 31.9, the potential difference across the two light bulbs is equal to the emf of the battery.
That is highly misleading. Students are going to read it as a statement of principle, which it isn't. Here is an attempt to repair it:
In the DC limit, every voltage is a potential, in accordance with Faraday's law, as introduced incorrectly back in section 29.2 on page 762 and stated correctly in section 29.5 on pages 770-771; for details see item #152. If we _assume_ the voltage is a potential (DC or otherwise), and if we further _assume_ ideal zero-resistance wires then the voltage across the battery terminals is equal to the voltage across each of the light bulbs. If we further _assume_ an ideal battery, the voltage at the battery terminals is equal to the Thévenin equivalent open-circuit voltage (Voc).
Rationale:
The statement in the book does not elucidate the principles. It is at least three assumptions removed from the principles.
As discussed in item #62, «EMF» is not well defined; the most charitable interpretation is that it refers to Voc.
As discussed in item #60, not every voltage is a potential.
As discussed in item #38, the voltage drop along wires is not always negligible.
I realize this is an introductory course, and it is not possible to teach everything at once. A certain amount of corner-cutting is necessary. However:
Given that Faraday's law has already been covered, referring back to it should be a win/win proposition. It reinforces what was learned in chapter 29, and puts chapter 31 on a much firmer, more-principled foundation. The incremental workload for the students (and teachers) should be negligible. This is a classic example of the spiral approach and the building-block approach.
Talking about "potential difference" instead of voltage is a lose/lose proposition; see item #60.
Given that this is supposed to be a principled book, it would be good to elucidate the distinction between the principles and the simplified model.
In principle there is resistance in the wires;
however sometimes it is convenient to construct a simplified model where the wires are overdesigned so that their resistance is negligible; all the resistance is localized in circuit elements such as light bulbs and discrete resistors.
Note that the concept of resistance has already been introduced (page 816). Also note that there are safety issues here, as discussed in item #38.
Talking about battery «EMF» instead of voltage is a lose/lose proposition. If the battery is ideal, then the terminal voltage is identical to the so-called «EMF», and introducing the «EMF» terminology creates extra work and extra confusion, with no possible upside. Conversely, if the battery is not ideal, than the statement on page 817 is flat-out wrong because the terminal voltage is not equal to Voc under such conditions. To say the same thing in slightly different words: If you are going to gloss over the internal resistance of the battery, basic pedagogical principles suggest also avoiding the «EMF» terminology; otherwise there is unexplained and unexplainable complexity, i.e. a distinction without a difference, i.e. terminology with no corresponding concept. This violates the pedagogical principle of concepts first, terminology afterward. See also item #62.
In section 31.3 on page 817 it says:
That is highly misleading. Students are going to read it as a statement of principle, which it isn't. Here is an attempt to repair it:
Rationale:
I realize this is an introductory course, and it is not possible to teach everything at once. A certain amount of corner-cutting is necessary. However: