Chapter : 1. Electricity

Derivation of mathematical expression of parallel combination

Derivation of mathematical expression of parallel combination :
Let, V be the potential difference across the two common points A and B. Then, from Ohm’s law
Current passing through R1, I1 = V/R1 ...(i)
Current passing through R2, I2 = V/R2 ...(ii)
Current passing through R3, I3 = V/R3 ...(iii)
If R is the equivalent resistance, then from Ohm’s law, the total current flowing through the circuit is given by,
I = V/R ...(iv)
and I = I1 + I2 + I3 ...(v)
Substituting the values of I, I1, I2 and I3 in Equation (v),
= ...(vi)
Cancelling common V term, one gets

The equivalent resistance of a parallel combination of resistance is less than each of all the individual resistances.
Important results about parallel combination :
(i) Total current through the circuit is equal to the sum of the currents flowing through it.
(ii) In a parallel combination of resistors the voltage (or potential difference) across each resistor is the same and is equal to the applied voltage i.e.
i.e. V1 = V2 = V3 = V :
(iii) (iii) Current flowing through each resistor is inversely proportional to its resistances, thus higher the resistance of a resistors, lower will be the current flowing through it.

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