What is called series connection of cell ? Derive equation of equivalent emf of two cell with emf `epsilon_(1) and epsilon_(2)` connected in series.

Explanation

`rArr `When one terminal of two cell connected with each other and another terminal of each cell are kept free then such connection is called series connection of cell.

`rArr` In figure, battery having emf `epsilon_(1)` and internal resistance `r_(1)` is connected between A arid B and battery with emf `epsilon_(1)` and internal resistance `r_(2)` is connected between B and C.
`rArr ` Let equivalent emf between A and C be `E_(eq)` and equivalent resistance of `r_(1) and r_(2)` be `r_(eq)` .
`rArr ` Let potential at point A, B and C be V(A), V(B) and V(C) respectively.
P .d. between positive and negative terminal of frist cell `V_(AB) = V(A) - V(B)` .
`rArr ` p.d. between positive and negative terminal of second cell = V(B) - V(C).
`V(A) - V(B) = epsilon_(1) - Ir_(1) " "` ... (1)
`V(B) - V(C) = epsilon_(2) - Ir_(2) " " `... (2)
`rArr` p.d. between A and C ,
`V_(AC) = V_(AB) + V_(BC)`
= [V(A) - V(B)) + [V(B) - V(C)]
`= epsilon_(1) - Ir_(1) + epsilon_(2) - Ir_(2) `
= ` epsilon_(1) + epsilon_(2) - Ir_(1) - Ir_(2)`
` = epsilon_(1) + epsilon_(2) - I(r_(1) + r_(2)) " " ` ... (3)
For given combination equivalent emf between A and C be `e_(eq)` and internal resistance `r_(eq)` ,
`V_(AC) = V(A) - V(C) = e_(eq) - Ir_(eq) " " `... (4)
Comparing (3) and (4) ,
`e_(eq) = epsilon_(1) + epsilon_(2)`
and `r_(eq) - r_(1) + r_(2)`
If cells are connected in series in opposing condition then,
`V_(BC) = V(B) - V(C) `
= `epsilon_(2) - Ir_(2)`
`therefoe e_(eq) = epsilon_(1) - epsilon_(2) " " (epsilon_(1) gt epsilon_(2))`
`rArr` If .n. cells are connected in series helping condition then,
`epsilon_(eq) = epsilon_(1) + epsilon_(2) + ... epsilon_(n)`
equivalent internal resistarice of combination will be equal to summation of individual internal resistance of each cell
`r_(eq) = r_(1) + r_(2) + ... + r_(n)`
If given cell is connected in opposing condition then emf of given cell will be considered as negative.