When three resistors, each having resistance r, are connected in parallel, their resultant resistance is x. If these three resistances are connected in series, the total resistance will be
9x
Understanding Resistors in Series and Parallel
This problem involves understanding how to calculate the equivalent resistance when resistors are connected in two different ways: parallel and series. We are given a situation where three identical resistors, each with resistance \(r\), are first connected in parallel, resulting in a total resistance \(x\). We then need to find the total resistance when these same three resistors are connected in series.
Resistors in Parallel Connection
When resistors are connected in parallel, the reciprocal of the total equivalent resistance is the sum of the reciprocals of the individual resistances. The formula for parallel connection is:
\[ \frac{1}{R_{parallel}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \dots \]
In this problem, we have three identical resistors, each with resistance \(r\), connected in parallel. The given resultant resistance is \(x\). So, we can write:
\[ \frac{1}{x} = \frac{1}{r} + \frac{1}{r} + \frac{1}{r} \]
Combining the terms on the right side:
\[ \frac{1}{x} = \frac{1+1+1}{r} \]
\[ \frac{1}{x} = \frac{3}{r} \]
From this equation, we can find the value of \(r\) in terms of \(x\). By cross-multiplying, we get:
\[ r = 3x \]
This tells us the resistance of a single resistor (\(r\)) is three times the total resistance (\(x\)) when they are connected in parallel.
Resistors in Series Connection
When resistors are connected in series, the total equivalent resistance is simply the sum of the individual resistances. The formula for series connection is:
\[ R_{series} = R_1 + R_2 + R_3 + \dots \]
Now, we consider the same three identical resistors, each with resistance \(r\), connected in series. The total resistance in series (\(R_{series}\)) will be:
\[ R_{series} = r + r + r \]
\[ R_{series} = 3r \]
Calculating Total Resistance in Series in terms of x
We have found that the total resistance in series is \(3r\). From the parallel connection analysis, we found that \(r = 3x\). Now, we substitute this value of \(r\) into the series resistance equation:
\[ R_{series} = 3 \times (3x) \]
\[ R_{series} = 9x \]
Therefore, if the three resistors are connected in series, the total resistance will be \(9x\).
Summary of Calculations
| Connection Type | Individual Resistances | Formula | Resultant Resistance Given | Relationship Derived |
|---|---|---|---|---|
| Parallel | \(r, r, r\) | \(\frac{1}{R_{parallel}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}\) | \(R_{parallel} = x\) | \(\frac{1}{x} = \frac{3}{r} \Rightarrow r = 3x\) |
| Series | \(r, r, r\) | \(R_{series} = R_1 + R_2 + R_3\) | \(R_{series} = ?\) | \(R_{series} = 3r\) |
Using the relationship \(r = 3x\) in the series resistance formula \(R_{series} = 3r\):
\[ R_{series} = 3 \times (3x) = 9x \]
The total resistance when the three resistors are connected in series is \(9x\).
Revision Table: Key Concepts for Resistor Networks
| Concept | Description | Formula (for 3 resistors R1, R2, R3) | Special Case (n identical resistors R) |
|---|---|---|---|
| Series Connection | Resistors are connected end-to-end, forming a single path for current. Total resistance increases. | \(R_{total} = R_1 + R_2 + R_3\) | \(R_{total} = nR\) |
| Parallel Connection | Resistors are connected across the same two points, providing multiple paths for current. Total resistance decreases. | \(\frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}\) | \(\frac{1}{R_{total}} = \frac{n}{R} \Rightarrow R_{total} = \frac{R}{n}\) |
| Equivalent Resistance | A single resistor that can replace a combination of resistors (series or parallel) and have the same effect on the rest of the circuit. | Varies based on connection type. | Varies based on connection type. |
Additional Information: Resistors and Circuits
Resistors are fundamental components in electrical circuits used to oppose the flow of electric current. Their resistance is measured in Ohms (\(\Omega\)).
- Ohm's Law: This law relates voltage (\(V\)), current (\(I\)), and resistance (\(R\)): \(V = IR\). Understanding Ohm's law is crucial for analyzing circuits with resistors.
- Voltage Division (Series): In a series circuit, the total voltage is divided among the resistors in proportion to their resistances.
- Current Division (Parallel): In a parallel circuit, the total current is divided among the branches, with current flowing more easily through paths of lower resistance.
- Applications: Resistors are used for various purposes, such as controlling current flow, creating voltage drops, and terminating transmission lines.
Problems involving series and parallel combinations of resistors are common in physics and electrical engineering. Being able to convert between different configurations and calculate equivalent resistance is a key skill.
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