## Tuesday, February 15, 2011

### DERIVATION OF CELLS IN SERIES AND PARALLEL

jaipreet singh asked: "can you please tell me the DERIVATION OF CELLS IN SERIES AND PARALLEL?"

Answer: I am posting here the simplified treatment to calculate the current in a circuit with combination of cells.

In this derivation it is assumed that all cells have the same EMF and same internal resistance.

CELLS IN SERIES

Consider n identical cells of emf E and internal resistance r connected in series across an external resistor of resistance R.

The total internal resistance = nr (since the internal resistances come in series)

The total resistance in the circuit = nr+R

The total emf = nE (since the emfs add up in series circuit)

Therefore, the current in the circuit;$current =\frac{Total Emf}{Total resistance}$
$I=\frac{nE}{nr+R}$

PARALLEL COMBINATION OF CELLS

Consider m identical cells of emf E and internal resistance connected in parallel across an external resistor of resistance R.

The total emf in circuit = E (Since each cell has the same emf and they are in parallel)

The net internal resistance = r/m (since the cells are in parallel, their resistances are also in parallel. If m identical resistances are in parallel, the effective resistance is r/m)

The total resistance in circuit = R + r/m

Therefore, the current in circuit;$Current = \frac{Total EMF}{Total Resistance}$
$I = \frac{E}{mr+R}$

MIXED COMBINATION OF CELLS

Consider a combination of m rows of n cells each. The emf of each cell is E and the internal resistance of each cell is r. This combination is connected across and external resistance R.

The total EMF = nE

The net internal resistance = nr/m

The total resistance in circuit = R + nr/m

The current in circuit; $Current = \frac{Total EMF}{Total Resistance}$
$I = \frac{nE}{\frac{nr}{m}+R}=\frac{nmE}{nr+mR}$