As we have seen, the voltage ( V ) drops that occur across resistors ( R ) in parallel circuits are equal in magnitude to the voltage of the source. In addition to this, the currents ( I ) within parallel circuits split apart ( and later recombine ) at nodes. The magnitude of the current that travels down each circuit branch after leaving a node is inversely proportional to the resistance value of the resistor ( R ) along that branch:
We must now establish a conceptual and mathematically pleasing rationale for determining current values along any given branch of a circuit. Recall that the voltage drops across resistors in parallel are equal in value to the source voltage:
Vs = V1 = V2 = Vn
Furthermore, the current across each of the two resistors shown have magnitudes that are determined by ohm’s law:
( Vs / R1 ) = I1
( Vs / R2 ) = I2
It is equally true that the source voltage is equal to the magnitude of the total current times the net resistance of the circuit:
Vs = ItRt
We are now at liberty to substitute the latter value above into the previous derivations of current along each circuit branch:
( ItRt / R1 ) = I1
( ItRt / R2 ) = I2
By rearranging the terms above, we can derive a general expression that enables us to solve current values of parallel circuits that contain “ x “ number of single, isolated resistors along each branch:
Ix = ( ItRt / Rx ) = ( It )( Rt / Rx )
Thus, the larger the resistance value along a parallel circuit branch, the smaller the current, and vice versa. What about cases where more than a single resistor resides along any given branch in question? As we will see, multiple resistors along branches of parallel circuits must have their values added together to create a composite or sum value. We will have plenty of practice to acquire such skills in our study of combined series-parallel circuits.
RENAISSANCE PHYSICS 