Parallel and series

Similar rules govern electrical components connected in series and in parallel as those known in the mechanical domain.  These rules are derived using the fact that current is a through variable and that voltage is an across variable, along with Kirchoff's Current Law and Kirchoff's Voltage Law.

For N elements connected in series:

Since current is a through variable, the current in each element must be the same:

i = i1 = i2 = i3 = . . . = iN
Since V is a relative voltage, or voltage change across an element, the total voltage change across the series is just a simple sum:
V = V1 + V2 + V3 + . . . + VN
The equation above can also be obtained by applying Kirchoff's Voltage Law to the loop made by the voltmeter and the N elements ( V1 + V2 + V3 + . . . + VN-V=0). Notice that the rule for voltage across elements in series is similar to the velocity across elements in series in the mechanical domain.

For N elements connected in parallel:

Since the current into a junction must equal the current out of a junction (Kirchoff's Current Law):

i = i1 + i2 + i3 + . . . + iN
Since V is the voltage change across the elements, and all elements are connected together at their ends (and thus must share a common potential at their ends), the voltages across all elements must be the same:
V = V1 = V2 = V3 = . . . = VN
The equation above can also be obtained by applying Kirchoff's Voltage Law to the loops around any two elements (eg, loop around element 1 and element 2 (V1-V2=0); or loop around element 1 and the voltmeter (V1-V=0)).

Examples:

What is the effective resistance of two resistors in series?

We know: i1 = i2= i , V1 = i R1, V2 = i R2, V = V1 + V2

Thus: V = i R1 + i R2 = i (R1 + R2)

Thus: Rredbox = R1 + R2

 

What is the effective resistance of two resistors in parallel?

We know: i = i1 + i2, V1 = i R1, V2 = i R2, V = V1 = V2

Thus: i = V/R1 + V/R2 = V*(1/R1 + 1/R2)

Thus: Rredbox = (1/R1 + 1/R2)-1