Explain the series combination of resistors and derive the formula of equivalent resistance.

Explanation

Two or more than two resistors are said to be connected in series, if they are joined end to end and the same (i.e., total) current flows through each one of them when a potential difference is applied across the combination.

In figure (a), three resistors with resistance `R_1, R_2 and R_3` are connected in series across the points A and B. Here, current (1) flowing through each of the resistors `R_1 R_2 and R_3` is the same, but the total potential difference (p.d.) of the battery V is divided according to the resistances between the two ends of the respective resistors.
If the potential difference (p.d.) across `R_1, R_2 and R_3" are " V_1, V_2 and V_3` respectively, then `V = V_1 + V_2 + V_3`
Now, if one resistor with resistance `R_s`, instead of these three resistors with resistances `R_1, R_2 and R_3` is connected in the circuit in such a way that the current flowing through the circuit remains the same as I, then `R_s` is the resistance of the series combination. It is also called equivalent resistance of the combination (figure (b)).
Now, applying Ohm.s law, `V = IR_s ... ...(12.9)`
From equation (12.8) and (12.9).
`IR_S =V_1 + V_2 + V_3 ... ...(12.10)`
On applying Ohm.s law to the three resistors separately, we have
`V_1 = IR_1`
`V_2 = IR_2`
`V_3 = IR_3`
`:. IR_s = IR_1 + IR_2 + IR_3`
`:. R_s = R_1 + R_2 + R_3` ... ...(12.11)
Thus, the equivalent resistance `R_s` of the series combination is equal to the sum of the individual resistances and is thus greater than any of the individual resistances.
[Note: If n resistors with resistances `R_1, R_2 ...R_n` are connected in series, `R_s = R_1 + R_2 +...+ R_n]`