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Question
Explain the equivalent resistance of a parallel resistor network.
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Solution
Resistors are in parallel when they are connected across the same potential difference as shown in the following figure.
Three resistors in parallel
In this case, the total current I that leaves the battery in split into three separate paths. Let I1, I2 and I3 be the current through the resistors R1, R2 and R3 respectively. Due to the conservation of charge, total current in the circuit I is equal to sum of the currents through each of the three resistors.
I = I1 + I2 + I3 ……. (1)
Since the voltage across each resistor is the same, applying Ohm’s law to each resistor, we have
`"I"_1 = "V"/"R"_1, "I"_2 = "V"/"R"_2, "I"_1 = "V"/"R"_3`
Substituting these values in equation (1), we get
I = `"V"/"R"_1 + "V"/"R"_2 + "V"/"R"_3`
= V`[1/"R"_1 + 1/"R"_2 + 1/"R"_3]`
I = `"V"/"R"_"p"`
`1/"R"_"p" = 1/"R"_1 + 1/"R"_2 + 1/"R"_3`
Here RP is the equivalent resistance of the parallel combination of the resistors. Thus, when a number of resistors are connected in parallel, the sum of the reciprocal of the values of resistance of the individual resistor is equal to the reciprocal of the effective resistance of the combination as shown in the following figure.
Equivalent resistance (RP) has the same current
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