Current Electricity Class 12 Physics - Cells in series and in parallel
Series combination of cells:
Two cells are said to be in series combination when one end of cells are joined together, other end in either cell is free.
Let ε1 and ε2 be emf and r1 and r2, be internal resistance of both cells connected in series respectively. If VA, VB, Vc are potential difference at point A, B and C then potential difference between A and B is
VAB= VA - VB = ε1-Ir1
Similarly, between point BC, VBC = VB- VC = ε2 - Ir2
Then, potential difference between A and C will be
VAC = VAB + VBC = (ε1-Ir1 ) + ( ε2 - Ir2) = (ε1+ ε2) - I(r1+r2) .
If we assume a single cell between A and C. then εeq and req is equivalent emf and equivalent resistance, then potential difference VAC = εeq - Ireq. .
Hence, we get, εeq = ε1+ ε2 and req = r1+r2.
If we connect negative electrode of first to negative electrode of second then,
εeq = ε1- ε2
From n number of cells we can say that,
The equivalent emf of series combination for n cells is sum of their individual emf
The equivalent internal resistance for n cells is sum of their individual internal resistance.
Parallel combination of cells: consider cells connected in parallel combination.
In this case, current I1 and I2 are leaving positive electrode, also, outgoing current at point B1 is same as incoming current at point B2. hence, current I in this parallel combination is
I = I1+ I2.
let potential at point B1 and B2 be VB1 and VB2 respectively. Potential difference between terminal B1 and B2 for first cell is,
V = VB1 - VB2 = ε1 - I1r1 .
For second cell, is connected to point B1 and B2 same as that of first cell, so potential difference,
V= VB1 - VB2 = ε2 - I2r2
By combining the above equations,
I = I1+ I2 = [(ε1-V)/ r1] +[(ε2 -V )/r2] = (ε1/ r1 + ε2/r2) - V(1/r1 + 1/r2 ) .
From this equation V obtained will be,
V= (ε1r2 + ε2r1)/ (r1+r2) - I(r1r2 / (r1+r2))
If combination is replaced by single cell then internal resistance will be req and emf will be εeq.
V= εeq - Ireq
By comparing above two equations, we get,
εeq = (ε1r2 + ε2r1)/ (r1+r2) and req= r1r2 / (r1+r2)
These equations can also, be represented as,
εeq / req = (ε1 / r1) + (ε2 / r2 ) and 1/ req = 1/r1 + 1/r2
Above expression is true when positive terminals of cells are joined together, and negative terminals are joined together. But these expression also prove to be true when negative terminal of first is connected to positive terminal of second.
If n number of cells are connected in parallel, then ,
1/req = 1/r1+......+ 1/rn
εeq / req = ε1 / r1 +.......+ εn/rn
