Short Answer:
To calculate the total resistance in a parallel circuit, use the reciprocal formula: 1/R_total = 1/R₁ + 1/R₂ + 1/R₃ + …, where R₁, R₂, and R₃ are individual resistances. After calculating the sum of reciprocals, take the reciprocal of the result to get the total resistance.
Unlike a series circuit, the total resistance in a parallel circuit is always less than the smallest individual resistor. This is because parallel connections provide multiple paths for current, reducing the overall opposition to current flow.
Detailed Explanation:
Total Resistance in a Parallel Circuit
In a parallel circuit, all components are connected across the same two points, meaning each resistor or component gets the same voltage, but the current is divided among the branches. This type of circuit is widely used in homes, industries, and electronics because it allows devices to operate independently.
When resistors are connected in parallel, they provide multiple paths for current to flow. As a result, more current can pass through the circuit, which effectively reduces the total resistance.
Formula to Calculate Total Resistance
For two or more resistors connected in parallel, the total resistance (R_total) is given by:
1Rtotal=1R1+1R2+1R3+…\frac{1}{R_\text{total}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \dotsRtotal1=R11+R21+R31+…
After finding the reciprocal sum, take the reciprocal of the final answer to get the total resistance:
Rtotal=1(1R1+1R2+1R3+… )R_\text{total} = \frac{1}{\left(\frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \dots\right)}Rtotal=(R11+R21+R31+…)1
Example Calculation
Example:
Three resistors of 6Ω, 3Ω, and 2Ω are connected in parallel.
1Rtotal=16+13+12\frac{1}{R_\text{total}} = \frac{1}{6} + \frac{1}{3} + \frac{1}{2}Rtotal1=61+31+21 =1+2+36=66=1= \frac{1 + 2 + 3}{6} = \frac{6}{6} = 1=61+2+3=66=1 Rtotal=11=1ΩR_\text{total} = \frac{1}{1} = 1ΩRtotal=11=1Ω
So, the total resistance is 1 ohm, which is less than the smallest individual resistor.
Key Points to Remember
- Voltage Across All Resistors is Same
- In a parallel circuit, each resistor gets the full voltage of the power supply.
- Current Splits Between Branches
- The total current is divided among the different resistors based on their resistance.
- Total Resistance is Always Less
- Adding more resistors reduces the total resistance further.
- For Two Resistors Only:
- A shortcut formula is
Rtotal=R1×R2R1+R2R_\text{total} = \frac{R_1 \times R_2}{R_1 + R_2}Rtotal=R1+R2R1×R2
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- This is useful for quick calculations with two parallel resistors.
Application of Parallel Resistance Calculation
- Household Wiring: All appliances and lights are connected in parallel for independent control.
- Battery Packs: Parallel battery arrangements provide more current without increasing voltage.
- Electronic Circuits: Components like LEDs and sensors are often wired in parallel to maintain constant voltage.
- Power Distribution Systems: Reduce load on individual components by sharing current across multiple paths.
Conclusion:
To calculate total resistance in a parallel circuit, add the reciprocals of all resistor values and then take the reciprocal of the sum. This method ensures you find the correct resistance, which will always be less than the smallest resistor in the circuit. Parallel circuits are highly efficient for distributing current and are widely used in practical electrical systems for their reliability and flexibility.