Points to Remember:
- Series combination: Resistances are connected end-to-end.
- Parallel combination: Resistances are connected between the same two points.
- Equivalent resistance: The single resistance that can replace the combination and produce the same overall effect.
Introduction:
The combination of resistances is a fundamental concept in electrical circuits. Understanding how resistances combine in series and parallel configurations is crucial for analyzing circuit behavior and calculating the overall resistance. This involves applying Ohm’s Law (V=IR), which states that the voltage across a resistor is directly proportional to the current flowing through it, with the resistance being the constant of proportionality. Different combinations lead to different formulas for calculating the equivalent resistance.
Body:
1. Series Combination of Resistances:
In a series combination, resistors are connected end-to-end, forming a single path for current flow. The current flowing through each resistor is the same, but the voltage across each resistor is different and proportional to its resistance.
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Diagram: [A simple diagram showing resistors R1, R2, R3 connected in series with a battery and current flowing through them.]
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Formula for Equivalent Resistance (Req): The equivalent resistance in a series combination is the sum of the individual resistances:
Req = R1 + R2 + R3 + ... + Rn
- Explanation: Since the same current flows through each resistor, the total voltage drop across the combination is the sum of the voltage drops across each individual resistor. Applying Ohm’s Law to the entire combination and to each individual resistor leads to the above formula.
2. Parallel Combination of Resistances:
In a parallel combination, resistors are connected between the same two points. The voltage across each resistor is the same (equal to the source voltage), but the current flowing through each resistor is different and inversely proportional to its resistance.
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Diagram: [A simple diagram showing resistors R1, R2, R3 connected in parallel with a battery and current flowing through each branch.]
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Formula for Equivalent Resistance (Req): The reciprocal of the equivalent resistance in a parallel combination is equal to the sum of the reciprocals of the individual resistances:
1/Req = 1/R1 + 1/R2 + 1/R3 + ... + 1/Rn
This can also be expressed as:
Req = 1 / (1/R1 + 1/R2 + 1/R3 + ... + 1/Rn)
- Explanation: Since the voltage across each resistor is the same, the total current flowing into the combination is the sum of the currents flowing through each individual resistor. Applying Ohm’s Law to the entire combination and to each individual resistor leads to the above formula. The reciprocal relationship reflects the fact that adding more parallel paths reduces the overall resistance.
3. Special Cases:
- Two resistors in parallel: A simplified formula for two resistors in parallel is often used:
Req = (R1 * R2) / (R1 + R2)
- Identical resistors: If ‘n’ identical resistors (each with resistance R) are connected in series,
Req = nR. If ‘n’ identical resistors are connected in parallel,Req = R/n.
Conclusion:
The laws of combination of resistances provide a fundamental framework for analyzing and simplifying complex electrical circuits. Understanding the difference between series and parallel combinations, and the corresponding formulas for calculating equivalent resistance, is crucial for electrical engineering and related fields. These formulas allow for the simplification of complex circuits, making it easier to calculate voltage, current, and power. Accurate calculation of equivalent resistance is essential for ensuring the safe and efficient operation of electrical systems, from simple household circuits to large-scale power grids. Further study into more complex circuit topologies involving combinations of series and parallel connections can build upon this foundational knowledge.