What Happens To Resistance When Connected In Parallel?

When resistors are combined in parallel, the effect is perhaps less obvious than for the series case: rather than adding resistance, we are in fact decreasing the overall resistance of the combination by providing alternative paths for the current.

This is so because in the parallel case the individual charge is not required to travel through every element, only one branch, so that the presence of the parallel elements “alleviates” the current flow through each branch, and thereby makes it easier for the charge to traverse.

It is convenient here to consider resistors in terms of the inverse property, conductance. Thus, we think of the resistor added in parallel not as posing a further obstacle, but rather as providing an additional conducting option: after all, as far as the current is concerned, any resistor is still better than no path at all. Accordingly, the total resistance of a parallel combination will always be less than any of the individual resistances.

Using conductance (G = 1/R), the algebraic rule for combining any number of resistive elements in parallel is simply that the conductance of the parallel combination equals the sum of the individual conductances.

For example, suppose a 10-ohm and a 2.5-ohm resistor are connected in parallel, as in figure below.

We know already that their combined (parallel) resistance must be less than 2.5 ohm. To do the math, it is convenient to first write each in terms of conductance: 0.1 mho and 0.4 mho. The combined conductance is then simply the sum of the two, 0.5 mho. Expressed in terms of resistance, this result equals 2 ohm. In equation form, we would write for resistors in parallel:

1/Rt = 1/R1 + 1/R2 + ....

Note that the voltage drop across any number of elements in parallel is the same. This can easily be seen because all the elements share the same terminals: the points where they connect to the rest of the circuit are, in electrical terms, the same.

While elements connected in parallel thus have a common voltage drop across them, the current flowing through the various elements or branches will typically differ. Intuitively, we might guess that more current will flow through a branch with a lower resistance, and less current through one with a higher resistance.

This can be shown rigorously by applying Ohm’s law for each of the parallel resistances: If V is the voltage drop common to all the parallel resistances, and R1 is the individual resistance of one branch, then the current I1 through this branch is given by V/R1. Thus, the amount of current through each branch is inversely proportional to its resistance.

To summarize, there is a tidy correspondence between the series and parallel cases: In a series connection, the current through the various elements is the same, but the voltage drops across them vary (proportional to their resistance); in a parallel connection, the voltage drop across the various elements is the same, but the currents through them vary (inversely proportional to their resistance).

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