Chapter 11 · Question 6

Derive the equivalent resistance of three resistors R1, R2, R3 connected in series. How does current and potential difference behave in a series combination?

Answer Revealed

Direct Answer:

For resistors in series, the same current

I

flows through each resistor. The total potential difference

V

across the combination equals the sum of individual potential differences:

V = V_1 + V_2 + V_3 = I R_1 + I R_2 + I R_3 = I (R_1 + R_2 + R_3)

. Comparing with

V = I R_s

, the equivalent series resistance is

R_s = R_1 + R_2 + R_3

Simple Explanation

In a series connection, the resistors are connected end-to-end like a chain. The same current has to pass through each one (there is only one path). The voltage splits across them — each resistor takes a share of the total voltage based on its resistance. The total or equivalent resistance is just the sum of all the individual resistances:

R_s = R_1 + R_2 + R_3

Exam-Ready Structure

When resistors are connected in series, the following analysis applies: 1. Circuit arrangement: Resistors are connected end-to-end so that the same current flows through each of them in sequence. There is only a single conducting path. 2. Current in series: The current

I

is the same through every resistor in the series combination. This can be verified experimentally using an ammeter connected at different points in the series circuit. 3. Potential difference in series: The total potential difference

V

across the combination is equal to the sum of the potential differences across each individual resistor:

V = V_1 + V_2 + V_3

. This is verified by connecting a voltmeter across the combination and then across individual resistors. 4. Derivation of equivalent resistance: Using Ohm's law,

V_1 = I R_1

V_2 = I R_2

V_3 = I R_3

. Therefore,

V = V_1 + V_2 + V_3 = I R_1 + I R_2 + I R_3 = I (R_1 + R_2 + R_3)

. If

R_s

is the equivalent resistance of the series combination, then

V = I R_s

. Comparing,

R_s = R_1 + R_2 + R_3

. 5. The equivalent resistance of a series combination is always greater than any individual resistance in the combination. 6. If

n

resistors each of resistance

R

are connected in series, the equivalent resistance is

nR

Key Points

In series: same current flows through each resistor
Total potential difference $V = V_1 + V_2 + V_3$ (sum of individual voltage drops)
Equivalent resistance $R_s = R_1 + R_2 + R_3$
Equivalent resistance of series combination is greater than each individual resistance
For $n$ identical resistors $R$ in series: $R_s = nR$

Derive the equivalent resistance of three resistors R1, R2, R3 connected in series. How does current and potential difference behave in a series combination?