How does the addition of resistors in parallel affect the total resistance of a circuit?
1 Answer
When resistors are connected in parallel in an electrical circuit, the total resistance of the circuit decreases. In a parallel configuration, each resistor provides an alternative path for the current to flow from the source to the load. This results in an overall reduction of resistance compared to if the resistors were connected in series.
To understand why the total resistance decreases in a parallel configuration, let's consider two resistors, R1 and R2, connected in parallel to a voltage source:
Current Division: In a parallel circuit, the voltage across each resistor is the same as the voltage across the source. Therefore, the current flowing through each resistor is determined by Ohm's law: I = V / R, where I is the current, V is the voltage, and R is the resistance. Since both resistors have the same voltage, their currents are inversely proportional to their resistances. The resistor with the lower resistance will allow more current to pass through it.
Equivalent Resistance: When resistors are in parallel, the total current flowing through the circuit is the sum of the currents flowing through each individual resistor. Mathematically, it can be expressed as:
Total Current (I_total) = I1 + I2 + ... + In
Since current is the total voltage divided by the total resistance, we have:
I_total = V / (R1 + R2 + ... + Rn)
Simplified Equivalent Resistance: The equivalent resistance (R_total) of resistors in parallel can be calculated using the formula:
1 / R_total = 1 / R1 + 1 / R2 + ... + 1 / Rn
For two resistors in parallel, the formula becomes:
1 / R_total = 1 / R1 + 1 / R2
Solving for R_total:
R_total = 1 / (1 / R1 + 1 / R2)
As you can see, the equivalent resistance (R_total) is always smaller than the smallest individual resistance. Therefore, adding resistors in parallel reduces the total resistance in the circuit.
In summary, connecting resistors in parallel results in a lower total resistance due to the current division among the parallel branches, which allows more current to flow through the circuit, leading to an overall decrease in resistance.
To understand why the total resistance decreases in a parallel configuration, let's consider two resistors, R1 and R2, connected in parallel to a voltage source:
Current Division: In a parallel circuit, the voltage across each resistor is the same as the voltage across the source. Therefore, the current flowing through each resistor is determined by Ohm's law: I = V / R, where I is the current, V is the voltage, and R is the resistance. Since both resistors have the same voltage, their currents are inversely proportional to their resistances. The resistor with the lower resistance will allow more current to pass through it.
Equivalent Resistance: When resistors are in parallel, the total current flowing through the circuit is the sum of the currents flowing through each individual resistor. Mathematically, it can be expressed as:
Total Current (I_total) = I1 + I2 + ... + In
Since current is the total voltage divided by the total resistance, we have:
I_total = V / (R1 + R2 + ... + Rn)
Simplified Equivalent Resistance: The equivalent resistance (R_total) of resistors in parallel can be calculated using the formula:
1 / R_total = 1 / R1 + 1 / R2 + ... + 1 / Rn
For two resistors in parallel, the formula becomes:
1 / R_total = 1 / R1 + 1 / R2
Solving for R_total:
R_total = 1 / (1 / R1 + 1 / R2)
As you can see, the equivalent resistance (R_total) is always smaller than the smallest individual resistance. Therefore, adding resistors in parallel reduces the total resistance in the circuit.
In summary, connecting resistors in parallel results in a lower total resistance due to the current division among the parallel branches, which allows more current to flow through the circuit, leading to an overall decrease in resistance.