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1. Derive an expression for the current in a circuit with external resistance R when (a) n identical cells of emf E and internal resistance r are connected in series (b) m identical cells are connected in parallel 

Dear student

(a) Let there are n identical cells of emf E and internal resistance r connected in series and R is the external resistance in the circuit.


$$\begin{gathered} In the circuit the net emf=nE \\ and net internal resis\tan ce=nr \\ so total resis\tan ce of the circuit=R+nr \\ thus current in the circuit is i=\frac{nE}{R+nr} \\ If nr<<R then i\approx \frac{nE}{R}=n\times current by one cell. \end{gathered}$$
(b) ​Let there are m identical cells of emf E and internal resistance r connected in parallel and R is the external resistance in the circuit.


$$\begin{gathered} Since all cells are in parallel so the net emf=E \\ all internal resis\tan ces are in parallel so net internal resis\tan ce=\frac{r}{m} \\ total resis\tan ce of the circuit=R+\frac{r}{m} \\ Thus the current in the circuit\Rightarrow i=\frac{E}{R+\frac{r}{m}}=\frac{mE}{r+mR} \\ If \frac{r}{m}<<R then i=\frac{E}{R}=current from one cell \end{gathered}$$