Three resistors of resistances 11 Ω, 22 Ω and 33 Ω are connected in parallel. Their equivalent resistance is equal to
6 Ω
Understanding Resistors in Parallel
When resistors are connected in parallel, the current divides among them, and the voltage across each resistor is the same. The total or equivalent resistance of a parallel combination is less than the smallest individual resistance in the circuit. This is because connecting resistors in parallel provides more paths for the current to flow, effectively reducing the overall opposition to current.
Formula for Equivalent Resistance in Parallel
The formula used to calculate the equivalent resistance (\(R_{eq}\)) of resistors connected in parallel is given by the reciprocal of the sum of the reciprocals of individual resistances. For three resistors with resistances \(R_1\), \(R_2\), and \(R_3\), the formula is:
Alternatively, for two resistors, a simpler formula exists: \(R_{eq} = \frac{R_1 \times R_2}{R_1 + R_2}\). However, the reciprocal sum method is general for any number of parallel resistors.
Calculating the Equivalent Resistance
In this problem, we are given three resistors with resistances:
- \(R_1 = 11 \, \Omega\)
- \(R_2 = 22 \, \Omega\)
- \(R_3 = 33 \, \Omega\)
We need to find the equivalent resistance (\(R_{eq}\)) when they are connected in parallel. Using the formula:
To add these fractions, we need a common denominator. The least common multiple (LCM) of 11, 22, and 33 is 66.
Now, we add the fractions:
Simplify the fraction on the right side:
To find \(R_{eq}\), we take the reciprocal of both sides:
The equivalent resistance of the three resistors connected in parallel is 6 Ω.
Verification
As expected, the equivalent resistance (6 Ω) is less than the smallest individual resistance (11 Ω), which is a characteristic of parallel resistor connections.
Revision Table: Resistor Combinations
| Combination Type | Equivalent Resistance Formula | Key Characteristic |
|---|---|---|
| Series | \(R_{eq} = R_1 + R_2 + R_3 + \dots\) | Current is the same through each resistor. Equivalent resistance is greater than the largest individual resistance. |
| Parallel | \(\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \dots\) | Voltage is the same across each resistor. Equivalent resistance is less than the smallest individual resistance. |
Additional Information: Applications of Parallel Resistors
Parallel resistor connections are common in many electrical circuits. Some examples include:
- Household Wiring: Appliances in a house are connected in parallel so that each appliance receives the full mains voltage and can be operated independently.
- Circuit Design: Parallel resistors can be used to obtain a specific equivalent resistance value that may not be available as a standard single resistor value.
- Current Division: In parallel circuits, current divides among the branches. The amount of current through each branch depends on the resistance of that branch (lower resistance gets more current).
Understanding both series and parallel connections is fundamental to analyzing and designing electrical circuits.
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