What is meant by the series connection of cells? Derive the equation for the equivalent electromotive force $(emf)$ of two cells with $emf$ $\varepsilon_1$ and $\varepsilon_2$ connected in series.

(N/A) When one terminal of one cell is connected to the opposite terminal of another cell,and the remaining two terminals are left free,such a connection is called a series connection of cells.
In the figure,a battery having $emf$ $\varepsilon_1$ and internal resistance $r_1$ is connected between points $A$ and $B$,and a battery with $emf$ $\varepsilon_2$ and internal resistance $r_2$ is connected between points $B$ and $C$.
Let the equivalent $emf$ between $A$ and $C$ be $\varepsilon_{eq}$ and the equivalent internal resistance be $r_{eq}$.
Let the potentials at points $A, B,$ and $C$ be $V(A), V(B),$ and $V(C)$ respectively.
The potential difference $(p.d.)$ across the first cell is $V_{AB} = V(A) - V(B) = \varepsilon_1 - I r_1$ ... $(1)$
The potential difference across the second cell is $V_{BC} = V(B) - V(C) = \varepsilon_2 - I r_2$ ... $(2)$
The total potential difference between $A$ and $C$ is:
$V_{AC} = V_{AB} + V_{BC} = (V(A) - V(B)) + (V(B) - V(C))$
$V_{AC} = (\varepsilon_1 - I r_1) + (\varepsilon_2 - I r_2) = (\varepsilon_1 + \varepsilon_2) - I(r_1 + r_2)$ ... $(3)$
For the equivalent combination,the potential difference is:
$V_{AC} = \varepsilon_{eq} - I r_{eq}$ ... $(4)$
Comparing equations $(3)$ and $(4)$,we get:
$\varepsilon_{eq} = \varepsilon_1 + \varepsilon_2$
$r_{eq} = r_1 + r_2$
If the cells are connected in series in an opposing condition (i.e.,positive terminal to positive terminal),the equivalent $emf$ becomes $\varepsilon_{eq} = \varepsilon_1 - \varepsilon_2$ (assuming $\varepsilon_1 > \varepsilon_2$).