# Resistance of a System of Resistors

In various electrical circuits, we used to connect more than one resistance in various combinations. Generally there are two methods of connecting the resistors together. The first method is Series combination, in which resistors are connected end to end. The second method is Parallel combination, in which resistors are connected together between some common points.**Resistances in Series**

_{1}, R

_{2}and R

_{3}connected in series. Then a battery is connected as shown, a current flows in the circuit; the current is the same in all the resistances, since any charge flowing into the first resistance must next flow through the second, and so on. There is no other alternative path for the charge to flow. Thus in a series circuit, the current is the same at all points. Let V be the voltage of the battery. Therefore, the potential difference between the points A and D in the circuit is V volts. Let I be the current flowing in the circuit.

Since the current flowing through each resistance is the same (=I) and since the resistances R

_{1}, R

_{2}and R

_{3}are different, Ohmâ€™s law (V= IR) tells us that the potential difference across the ends of each resistance will be different.

If V

_{1}, V

_{2}and V

_{3}are the potential differences across R

_{1}, R

_{2}and R

_{3}respectively, then

V= V

_{1 }+ V

_{2 }+ V

_{3 }

Now from Ohmâ€™s law, V

_{1 }= IR

_{1}, V

_{2}= IR

_{2}and V

_{3 }= IR

_{3}. Let V be the voltage of the battery. Therefore,

V= I (R

_{1 }+ R

_{2 }+ R

_{3})

Let R be the resistance of the combination. The value of R must be such that the same current I flows through it when the same potential difference V is applied across it, i.e

V= IR

Equating the above two equations

R= R

_{1}+ R

_{2}+ R

_{3 }

For n resistances in series, we have

R = R

_{1}+ R

_{2 }+ R

_{3}+ ... R

_{n}

Thus when resistances are connected in series, the total resistance is equal to the sum of the individual resistances.

**Resistances in Parallel**

_{1}, R

_{2}and R

_{3}connected in parallel is shown in the given figure. The current I leaving the battery at its positive terminal divides at the junction into three parts I

_{1}, I

_{2}and I

_{3}, which recombine at other junction to give finally the same total current returning to the battery. It is clear that,

I = I

_{1}+ I

_{2 }+ I

_{3}

_{}

It is evident from the above figure; the potential difference across each resistance is V, the voltage of the battery. Since the potential difference across each resistance is the same and since each resistance has a different value, Ohmâ€™s law tells us that the current in each resistance should be different and must be as follows:

Using this value of I in the above equation we have

For n resistances connected in parallel, we have

Thus when a number of resistances are connected in parallel, the reciprocal of the resistance of the combination is equal to the sum of the reciprocals of the individual resistances.