Why is the equivalent resistance in parallel less than the smallest resistance, and how do I solve series and parallel sums?
GradeHunt_TeamGradeHunt Team · · Gujarat Board 10th
Std 10 GSEB, Science chapter Electricity. In series I just add the resistances and that is fine. But in parallel the answer always comes smaller than every single resistor and I don't understand why. Also I get confused about which quantity is same in series and which in parallel when finding current.
electricityseries and parallelequivalent resistancegseb std 10
1 answer
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Start with what stays the same in each arrangement. That one fact solves most sums.
Series: the same current flows through every resistor. The potential difference divides.
Rs = R1 + R2 + R3
Parallel: the potential difference across every resistor is the same. The current divides.
1/Rp = 1/R1 + 1/R2 + 1/R3
Why is Rp smaller than the smallest resistor? Each resistor you add in parallel is one more path for charge to flow. More paths means more total current for the same voltage, and R = V/I, so the total resistance drops. Even if only the smallest resistor were connected, it would draw a current of V/Rsmallest. Every other branch adds more current on top of that, so the total current is more than V/Rsmallest, and Rp = V/Itotal must be less than Rsmallest. Think of a toll plaza: opening one more booth never slows the traffic.
Worked example: resistors of 2 ohm, 3 ohm and 6 ohm with a 6 V battery.
In series:
Rs = 2 + 3 + 6 = 11 ohm
I = V/R = 6/11 = 0.55 A (approx.), same through all three
Voltages (V = IR): 2 x 6/11 = 1.09 V, 3 x 6/11 = 1.64 V, 6 x 6/11 = 3.27 V, which add up to 6 V
In parallel:
1/Rp = 1/2 + 1/3 + 1/6 = 3/6 + 2/6 + 1/6 = 1
Rp = 1 ohm, smaller than 2 ohm, the smallest one
Each resistor gets the full 6 V:
I(2 ohm) = 3 A, I(3 ohm) = 2 A, I(6 ohm) = 1 A
Total current = 3 + 2 + 1 = 6 A, and 6 V / 1 ohm = 6 A as well, so the answer checks.
The largest current goes through the smallest resistor in parallel.
Two common slips:
1. Finding 1/Rp and forgetting to take the reciprocal at the end.
2. Using Rp = (R1 x R2)/(R1 + R2) for three resistors. This product-over-sum shortcut works only for two at a time.
This is also why household appliances are connected in parallel: each one gets the full mains voltage and can be switched on or off without affecting the others.
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