Two identical resistors of 20 Ω each are connected in parallel. This combination, in turn, is connected in series to a 10 Ω resistor. The equivalent resistance of the combination will be:
20 Ω
Understanding Resistor Combinations and Equivalent Resistance
The question asks us to find the total equivalent resistance of a circuit containing resistors connected in both parallel and series configurations. We have two identical resistors of 20 Ω each connected in parallel, and this parallel combination is then connected in series with a 10 Ω resistor. To solve this, we need to first calculate the equivalent resistance of the parallel part and then add it to the resistance of the series resistor.
Step-by-Step Calculation of Equivalent Resistance
1. Calculate the equivalent resistance of the parallel combination:
When resistors are connected in parallel, the reciprocal of the equivalent resistance (\(R_p\)) is equal to the sum of the reciprocals of the individual resistances. For two resistors \(R_1\) and \(R_2\) in parallel, the formula is:
\( \frac{1}{R_p} = \frac{1}{R_1} + \frac{1}{R_2} \)
Alternatively, for two resistors, we can use the product-sum formula:
\( R_p = \frac{R_1 \times R_2}{R_1 + R_2} \)
In this case, we have two identical resistors with \( R_1 = 20 \, \Omega \) and \( R_2 = 20 \, \Omega \). Using the formula:
\( R_p = \frac{20 \, \Omega \times 20 \, \Omega}{20 \, \Omega + 20 \, \Omega} \)
\( R_p = \frac{400 \, \Omega^2}{40 \, \Omega} \)
\( R_p = 10 \, \Omega \)
So, the equivalent resistance of the parallel combination of the two 20 Ω resistors is 10 Ω.
2. Calculate the equivalent resistance of the series combination:
Now, the parallel combination (which has an equivalent resistance of 10 Ω) is connected in series with a 10 Ω resistor. When resistors are connected in series, the total equivalent resistance (\(R_{eq}\)) is simply the sum of the individual resistances. Let \(R_{series}\) be the resistance of the resistor in series (10 Ω). The formula for resistors in series is:
\( R_{eq} = R_p + R_{series} \)
Substituting the values:
\( R_{eq} = 10 \, \Omega + 10 \, \Omega \)
\( R_{eq} = 20 \, \Omega \)
Therefore, the total equivalent resistance of the combination is 20 Ω.
Summary of Equivalent Resistance Calculation
| Component | Resistance (\(\Omega\)) | Connection Type | Equivalent Resistance Calculation | Result (\(\Omega\)) |
|---|---|---|---|---|
| Two identical resistors | 20 each | Parallel | \( \frac{20 \times 20}{20 + 20} \) | 10 |
| Parallel combination + third resistor | 10 (parallel) + 10 (series) | Series | \( 10 + 10 \) | 20 |
The equivalent resistance of the entire combination is 20 Ω.
Revision Table: Key Concepts for Resistor Combinations
| Concept | Series Connection | Parallel Connection |
|---|---|---|
| Diagram Representation | Resistors connected end-to-end in a single path. | Resistors connected across the same two points, providing multiple paths for current. |
| Equivalent Resistance Formula | \( R_{eq} = R_1 + R_2 + R_3 + \dots \) | \( \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \dots \) |
| Current Flow | Current is the same through each resistor. | Total current divides among the branches. |
| Voltage Drop | Total voltage is the sum of voltage drops across each resistor. | Voltage is the same across each resistor. |
Additional Information: Understanding Electrical Resistance
Electrical resistance is a measure of how much a material opposes the flow of electric current. It is measured in Ohms (Ω). Resistors are components specifically designed to provide a certain amount of resistance in a circuit.
- Ohm's Law: Relates voltage (\(V\)), current (\(I\)), and resistance (\(R\)) with the formula \( V = I \times R \).
- Combining Resistors: Resistors can be combined in series or parallel to achieve a desired total resistance or to distribute voltage and current in a circuit.
- Series Combination Properties: Adding more resistors in series increases the total resistance. The current is the same everywhere in a series circuit.
- Parallel Combination Properties: Adding more resistors in parallel decreases the total resistance. The voltage is the same across all components in parallel. The total current entering a parallel combination splits, with more current flowing through branches with lower resistance.
Understanding how to calculate equivalent resistance for different combinations is fundamental to analyzing and designing electrical circuits.
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