Parallel Circuits

When a number of resistors are conducted in such a way that one end of each of them is joined to a common point and the other ends being joined to another common point, as shown in then resistors are said to be connected in parallel and such circuits are known as parallel circuits. In these circuits current is divided into as many paths as the number of resistances.

Let the resistors R₁ and R₂ be connected in parallel, as shown in and the potential difference of V volts be applied across the circuit.

Since potential difference across each resistor is same and equal to potential difference applied across the circuit.

 According to Ohm’s law

Current in resistor R₁,   I₁ = V
                                        R₁

Current in resistor R₂,  I₂ = V
                                        R₂

Current in resistor R₃,  I₃ = V
                                       R₃

Adding expressions (i), (ii) and (iii) we get

                            I₁ + I₂ + I₃ = V + V + V
                                              R₁   R₂   R₃
                            I₁ + I₂ + I₃ = V (  1 +  1 + 1 )
                                                   R₁   R₂   R₃

and since I₁ + I₂ + I₃ = I, the total current flowing through the circuit

So             I = I₁ + I₂ + I₃ = V (  1 +  1 + 1 )
                                             R₁   R₂   R₃
or               I  =   1 +  1 + 1
                 V     R₁   R₂   R₃

and since   I   =  1 where R is the equivalent resistance of the whole circuit
                V      R

                I  = 1 +  1 + 1
               R    R₁   R₂   R₃

Thus when a number of resistors are connected in parallel, the reciprocal of the equivalent resistance is given by the arithmetic sum of the reciprocals of their individual resistances.

In general if n resistors of resistance R1 , R2, R3.........Rn are connected in parallel, then equivalent resistance R of the circuit is given by

              I  = 1 +  1 + 1 +................+ 1
             R    R₁   R₂   R₃                         Rn

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