Cells in Series and in Parallel

When cells are connected one after another, with the positive terminal of one connected to the negative terminal of the next, the cells are said to be connected in series. 

Main result

For two cells connected in series in the same sense:

These are the standard results derived from the source material. 

Interpretation

• In a series combination, emfs add up. 
• Internal resistances also add up. 
• This arrangement is useful when a larger potential difference is needed.

Stepwise derivation idea

• The current through both cells is the same. 
• The potential drop across the combination is given by the sum of the potential drops across the individual cells. 
• By comparing the result with a single equivalent cell, the expressions for equivalent emf and internal resistance are obtained. 

If one cell is reversed

If one of the two cells is connected in the opposite direction, the equivalent emf decreases. The source material gives the result:

and

So, reversing one cell changes the net emf but does not change the total internal resistance. 

For n cells in series

For n cells connected in the same sense:

These relations are directly stated in the source content for the general case.

When like terminals of the cells are connected together, the cells are in parallel. The source material derives the equivalent emf and equivalent internal resistance for this arrangement. 

Main result for two cells

The source also expresses the second relation in reciprocal form. 

Interpretation

• In parallel combination, the effective internal resistance becomes smaller than either individual internal resistance. 
• This combination is useful when the circuit needs a better current supply with lower effective internal resistance.
• Parallel grouping is especially useful when cells are similar and a stable source is desired.

Derivation idea

• Current splits into branches, so the source uses the relation I = I₁ + I₂. 
• The terminal potential difference across both branches is the same. 
• On comparing the combined system with one equivalent cell, the above formulas are obtained. 

For n cells in parallel

The source extends the result to n cells in parallel using the general reciprocal relation for internal resistance and the weighted relation for equivalent emf. 

• In series: ε_eq​ adds, r_eq adds. 

• If one cell is reversed in series, the net emf is the difference of emfs, but internal resistances still add. 

• In parallel, effective internal resistance decreases. 

• For two cells in parallel:
  ε_eq = \[\frac{\varepsilon_1r_2+\varepsilon_2r_1}{r_1+r_2}\], r_eq = \[\frac{r_1r_2}{r_1+r_2}\]​​.