Why does emf of a cell equal to $\frac{F_b d}{q}\;?$

Question

In Concepts of Physics by H.C.Verma, it is written:

Some internal mechanism exerts forces on the charges of the battery material. ... We show the force on a positive charge $q$ as $\vec F_b\; .$ As positive charge accumulates on $A$ & negative charge on $B,$ a potential difference develops and grows between $A$ & $B$. An electric field $\vec E$ is developed in the battery from $A$ to $B$ and exerts a force $\vec F_e= q\vec E$ on a charge $q.$ The direction of this force is opposite to that of $\vec F_b\;.$ In steady-state, the charge accumulation on $A$ and $B$ is such that $F_b= F_e\;.$

[...] The work done by the battery force per unit charge $$\mathcal{\varepsilon}= \frac{F_b d}{q} \; .$$ This quantity is called the emf of the battery.

I'm dubious of this derivation.

The author himself told, as charges were accumulating an electric field was being developed which would oppose the battery-force $\vec F_b.$ Still the work done is the same as if there were no electric-field. $F_b d$ is the work done when there is no other force. Prior to the steady-state, force on a charge was $F_b- F_e(t)$; but the author seemed to use $\vec F_b$ all the time. Isn't it wrong?

Answer 1

What has been calculated here is the work done performed by the battery force $F_b$ on any charge that moves from one terminal to the other. The net work done on any charge that moves through the battery would be the summation of the battery force's work and the work done by the charges accumulated up to that point of time. But that is not what we mean by the $emf$ of a battery. $emf$ of a battery is precisely defined as the work done on a charge by the forces of battery's chemical mechanisms as the charge moves through one terminal of the battery to the other. In the steady state that would be equal and opposite to the work done by the accumulated charges and that is why it is in a steady state. When the battery is joined in a circuit, the accumulated charges start flowing and thus the work done by the accumulated charges changes and becomes almost negligible, but still the battery forces do the same work done and essentially net work is performed on the charge as it moves from one terminal to the other inside a battery.