Week 5: Resistance 179
each resistor, given a common voltageVabacross them:
I 1 =
Vab
R 1(335)
I 2 =
Vab
R 2(336)
I 3 =
Vab
R 3(337)
Now, consider the total currentItotflowing into the arrangement from pointa. Charge is
conserved, so that all of the charge that flows into the first junction connecting the three independent
conducting pathways through the resistorsmust flow out of itand into the three resistors. From
this we conclude that:
Itot=I 1 +I 2 +I 3 =Vab
R 1+Vab
R 2+Vab
R 3=Vab(
1
R 1
+^1
R 2
+^1
R 3
)
(338)
As before in the lower (b) figure we have:Itot=Vab
Rtot(339)
and when we equate these two forms and cancel the commonVabwe get:
1
Rtot=^1
R 1
+^1
R 2
+^1
R 3
(340)
There is nothing special about three resistors, and once again we can easily generalize this
argument toNresistors as:
1
Rtot=1
R 1 +
1
R 2 +...+
1
RN=
∑N
i=11
Ri (341)We conclude that the total resistance of several resistors in series is the simple sum of the
individual resistances, while thereciprocalof the total resistance of serveral resistors in parallel is
the sum of thereciprocalsof the individual resistances. This is the exact opposite of the rulesfor
summing capacitances in seris and parallel.
5.4: Kirchhoff’s Rules and Multiloop Circuits
In the previous sections we used two rules implicitly that we should make explicit so that we can
use them in the more complicated circuits we will study over the next few weeks. In studying series
capacitors and series resistors, we used the idea that we couldaddthe changes in voltage across
objects in a common wire carrying a steady state current (includingno current at all) to find the
voltage changes between any two points in the wire. This is an idea related toenergy conservation.
In studying parallel capacitors and and parallel resistors, we usedthe idea that the total charge
moving around in these circuits must be conserved to track its distribution over time whether or not
it is actually moving.
These two rules (which we will derive and discuss below) are known as Kirchhoff’s Rules^55.(^55) Wikipedia: http://www.wikipedia.org/wiki/Kirchhoff’s Circuit Laws.