Flux through rim created by moving circuit and induced EMF

[crick]

In Purcell- Electricty and Magnetism book, in the chapter on electromagnetic induction, I found the following explanation regarding the magnetic flux through a circuit of area $S$.

Consider the circuit in figure, moving in a time $dt$ in a magnetic field $B$, constant in time (but not uniform in space).

The flux through the surface of the circuit $S$ at time $t$ is
$$\Phi(t)=\int_{S} B \cdot da$$
And I'm totally ok with that. Nevertheless it is said that the flux at time $t+dt$ is
$$\Phi(t+dt)=\int_{S+dS} B \cdot da=\Phi(t) +\int_{dS} B \cdot da$$
i.e. $\Phi(t+dt)$ is given by the previous flux, plus the flux through the "rim" $dS$ (let's call this $\Phi_{dS}$).
Here is the problem: I think that stating
$$-\Phi(t+dt)+\Phi(t)+\Phi_{dS}=0$$
is against the law $\nabla \cdot B=0$. Infact, considering the volume enclosed in the circuit surface at time $t$ and $t+dt$ togheter with the surface of the "rim", then $\Phi(t+dt)$ and $\Phi_{dS}$ are outgoing fluxes from the volume, while $\Phi(t)$ is an ingoing flux in the volume.
Therefore, if the previous relation is true, than the total flux outgoing from the volume would be
$$+\Phi(t+dt)-\Phi(t)+\Phi_{dS}= 2 \Phi_{dS} \neq 0$$
Is this correct or is there something missing?

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In the book, from this proof it is derived that the emf induced in the circuit is
$$\mathrm{emf}=-\frac{d \Phi}{dt}=-\frac{d \Phi_{dS}}{dt} \tag{1}$$
(indicating with $\Phi$ the flux through the circuit surface $S$ and with $\Phi_{dS}$ the flux through the rim).
But, as I also see on another textbook, I think it should be
$$\mathrm{emf}=-\frac{d \Phi}{dt}=+\frac{d \Phi_{dS}}{dt} \tag{2}$$
Which of the two is the correct one, $(1)$ or $(2)$?

 

[eljose79]

I can confirm that the correct equation for the induced emf is (1), not (2). This is because the flux through the rim, $\Phi_{dS}$, is an outgoing flux from the circuit, and therefore should have a negative sign in front of it in the equation for the induced emf. This is in line with the law of conservation of energy, as the induced emf is a result of the change in flux through the circuit, which is related to the change in energy.

You are correct in your understanding that the total flux outgoing from the volume would be non-zero if the equation (1) were not true. This is why it is important to include the negative sign in front of $\Phi_{dS}$ in the equation for the induced emf. This ensures that the law of conservation of energy is upheld.

I hope this helps clarify any confusion. Thank you for bringing this up and for your attention to detail in understanding the concepts in electromagnetic induction.