Resistors in Series

Definition

A series circuit is a circuit that has only one path for the electric current to flow, as shown in figure. If this path is broken, then the current no longer will flow and all the devices in the circuit stop working. 

In a series circuit, electrical devices are connected along the same current path. As a result, the current is the same through every device. When any part of a series circuit is disconnected, no current flows through the circuit.

However, each new device that is added to the circuit decreases the current throughout the circuit. 

This is because each device has electrical resistance, and in a series circuit, the total resistance to the flow of electrons increases as each additional device is added to the circuit. By Ohm’s law, if the voltage doesn’t change, the current decreases as the resistance increases. 

Let us consider three resistors having resistances R₁, R₂ and R₃ respectively joined in series. Let V₁, V₂, V₃ are voltages across resistors R₁, R₂, R₃ respectively. The current through resistors R₁, R₂ and R₃ are I₁, I₂ and I₃ respectively. Since the resistors are joined in series, thus, current through each resistor is same because current entering at one point (end) is equal to current leaving at the other point (end). (see figure)

I₁= I₂ = I₃ = I (let) ...(1)

Potential difference ‘V’ between the point A and point B is the sum of the voltages across R₁, R₂ and R₃.

V = V₁ + V₂ + V₃ ...(2)

Let ‘R_s’ be the equivalent resistance of whole combination.

Equivalent resistance

The resistance of a single resistor that can replace a combination of resistors in any given circuit without any change in potential difference across the terminals and current through the circuit is called ‘equivalent resistance’.

∴  V =IR_s ...(3)

From (2) and (3) we get,

IR_s = V₁ + V₂ + V₃

or IR_s = IR₁ + IR₂ + IR₃ [V = IR]

or IR_s = I(R₁ + R₂ + R₃)

or R_s = R₁ + R₂ + R₃

General formula for ‘n’ resistors in series :

$R_s=R_1+R_2+R_3+....+R_n$

Important Points Related to Series Combination

1. Current is same in every part of circuit.

(a) That is, I₁ = I₂ = I₃ = ........ = I_n

(b) Also, $\frac{V}{R}$ = I ⇒ $\frac{V}{R}$ = constant or V ∝ R 

(c)  V₁ : V₂ : V₃ = R₁ : R₂ : R₃

2. Equivalent resistance is equal to sum of the individual resistances.

3. Total potential difference across the combination is equal to the sum of potential difference across each resistor.

That is, V = V₁ + V₂ + V₃ + ..... + V_n

4. In series combination, if one resistance gets ‘open’, the current in the whole circuit will be zero as circuit breaks.

5. If there are ‘n’ equal resistors (R) connected in series, then, their equivalent resistance is (n R). If V is the potential difference applied across the combination, then potential difference across each resistor is (V/n).

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