Combination of Resistors
There are two ways of combining two or more two resistors; series and parallel combinations.
a. Series Combination
When a number of resistors are connected end to end as shown in the figure they are said to be in series. In this combination, the same current flows through them.
Let R1, R2, and R3 are three connected in series. 'I' is the current flowing through them. V1, V2, and V3 are the potential differences across R1, R2, and R3 respectively. V is the potential difference across the whole circuit which is equal to the sum of potential differences across individual resistors.
Therefore,
\(V=V_1+V_2+V_3\) ----------- (i)
But V1 = IR1, V2 = IR2, V3 = IR3 ------ (ii)
But from relations (i) and (ii), we have,
\(V=I(R_1+R_2+R_3)\)
Or, \(\frac VI=R_1+R_2+R_3\)
Or, \(R=R_1+R_2+R_3\) ------- (iii)
Where, \(R=\frac VI\), is called effective resistance of the system. So, in a series combination of the resistors, the effective resistance is equal to the sum of individual resistances.
i.e., R = R1 + R2 + R3 + ... + Rn for n number of resistance in which I is equal in each resistor.
b. Parallel Combination of Resistors
When one end of the number of resistors are connected to a common point and other ends are connected to another common point as shown in the figure, it is called parallel combination. In this combination, the potential difference across each resistor is the same.
Let, R1, R2, and R3 be three resistors connected in parallel combinations. I is the current entering the system at end A and the same current leaves at end B. V is the common potential difference between common points A and B. I1, I2, and I3 be the current through resistors R1, R2, and R3 respectively. Here, total current, 'I' is equal to the sum of the individual current I1, I2, and I3. Therefore,
I = I1 + I2 + I3 -------- (i)
But, \(I_1=\frac{V}{R_1}\), \(I_2=\frac{V}{R_2}\), and \(I_3=\frac{V}{R_3}\) ------- (ii)
From the relation (i) and (ii), we get,
\(I=\frac V{R_1}+\frac V{R_2}+\frac V{R_3}\)
Or, \(I=V\left(\frac1{R_1}+\frac1{R_2}+\frac1{R_3}\right)\)
Or, \(\frac IV=\left(\frac1{R_1}+\frac1{R_2}+\frac1{R_3}\right)\) ------ (iii)
Where, \(R=\frac VI\), is the effective resistance of the system. So, in a parallel combination of the resistors, the reciprocal of effective resistance is equal to the sum of the reciprocal of individual resistances, for n number of resistors.
