Chapter 11 · Question 7

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

Answer Revealed

Direct Answer:

For resistors in parallel, the potential difference

V

across each resistor is the same. The total current

I

drawn from the source equals the sum of individual currents through each resistor:

I = I_1 + I_2 + I_3

. Using Ohm's law,

I_1 = V/R_1

I_2 = V/R_2

I_3 = V/R_3

. Therefore,

I = V(1/R_1 + 1/R_2 + 1/R_3)

. Comparing with

I = V/R_p

, we get

1/R_p = 1/R_1 + 1/R_2 + 1/R_3

Simple Explanation

In a parallel connection, all resistors are connected to the same two points, so each gets the full battery voltage. The total current splits among the different paths — more current flows through the path with less resistance. Think of it like multiple checkout counters: each counter gets the same 'voltage' (number of staff), but more 'current' (customers) goes through the fastest counter. The total equivalent resistance is always less than the smallest individual resistance.

Exam-Ready Structure

When resistors are connected in parallel, the following analysis applies: 1. Circuit arrangement: Resistors are connected such that one end of each resistor is joined to a common point and the other ends are joined to another common point. The resistors thus provide separate paths for current. 2. Potential difference in parallel: The potential difference

V

across each resistor connected in parallel is the same as the potential difference of the source. This can be verified using a voltmeter connected across each resistor. 3. Current in parallel: The total current

I

drawn from the source is equal to the sum of the currents flowing through the individual branches:

I = I_1 + I_2 + I_3

. An ammeter in the main line measures total current, while ammeters in individual branches measure individual currents. 4. Derivation of equivalent resistance: Using Ohm's law,

I_1 = V / R_1

I_2 = V / R_2

I_3 = V / R_3

. The total current

I = I_1 + I_2 + I_3 = V / R_1 + V / R_2 + V / R_3 = V (1/R_1 + 1/R_2 + 1/R_3)

. If

R_p

is the equivalent resistance of the parallel combination, then

I = V / R_p

. Therefore,

1/R_p = 1/R_1 + 1/R_2 + 1/R_3

. 5. The equivalent resistance of a parallel combination is always less than the smallest individual resistance in the combination. 6. For two resistors

R_1

and

R_2

in parallel, the equivalent resistance

R_p = (R_1 \times R_2) / (R_1 + R_2)

. 7. If

n

identical resistors each of resistance

R

are connected in parallel,

R_p = R / n

Key Points

In parallel: same potential difference (voltage) across each resistor
Total current $I = I_1 + I_2 + I_3$ (sum of individual branch currents)
Equivalent resistance: $1/R_p = 1/R_1 + 1/R_2 + 1/R_3$
Equivalent resistance of parallel combination is less than the smallest individual resistance
For $n$ identical resistors $R$ in parallel: $R_p = R / n$

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