Question

if two resistances R1 and R2 are connected in series and parallel, derive an expression for resultant resistance in each​

Answer 1

Series:

Two resistors of resistance R1 and R2 are connected in series. Let I be the current through the circuit. The current through each resistor is also I. The two resistors joined in series is replaced by an equivalent single resistor of resistance R such that the potential difference V across it, and the current I through the circuit remains same.



As , V = IR , V1 = IR1 , V2 = IR2

IR = IR1 + IR2

IR = I (R1 + R2)
R= R1 + R2

Parallel:

It is observed that the total current I, is equal to the sum of the separate currents through each branch of the combination.

I + I1 + I2 + I3

Let Rp be the equivalent resistance of the parallel combination of resistors. By applying Ohm's law to the parallel combination of resistors, we have

I = V/Rp

On applying Ohm's law to each resistor, we have

I1 = V/R1 ; I2 = V/R2; and I3 + V/R3

From eqns. we have

V/Rp = V/R1 + V/R2 + V/R3

or 1/Rp = 1/R1 + 1/R2 + 1/R3




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