137 Equivalent resistance
Subject:
If several resistors with the resistances R1, R2, R3, … are connected in
series the total or equivalent resistance of the system is
!
R = R1 + R2 + R3 + …
If they are connected in parallel the equivalent resistance is
!
1
1
1
1
=
+
+
+…
R R1 R 2 R 3
Deficiencies:
These rules are part of the physics syllabus, since electricity is taught at
school, i.e. since about 150 years.
Of course, nothing is incorrect. But one might ask several questions: Why
do these rules belong to the compulsory part of the syllabus? Why do they
have the status of standard rules? Why do we dedicate a whole chapter to
them? One might answer: Because they are important.
But if one considers them as sufficiently important to be treated extensively
in the physics class, why not also a good number of similar or analogues
rules: about connecting capacitors and inductors in series or in parallel, or
Hookean springs and mechanical dampers (dash pots), or heat resistors
and fluid resistors? The corresponding mathematical structure of the rules
is the same as that for the electrical resistors. Is heat resistance less
important than electric resistance? Are electric capacitors less important
than electric resistors?
When taking into account that the above-mentioned rules are
consequences of the the junction rule and the loop rule, another
observation can be made.
The loop rule becomes trivial, when the electric tension is introduced as a
difference of two electric potential values. Then, the loop rule is as trivial as
for instance the following statement: If taking the elevator one goes up first
two floors and then three more, one has gone up five floors in total. One
can say: the loop rule is valid because we have to do with a conservative
field. (But it is better not to express it in such an intimidating way.)
The junction rule is a simple consequence of the conservation of the flowing
quantity. It is valid for any flow of a conserved quantity: energy currents,
electric currents, mass currents and momentum currents. It would be a pity
when treated only in the context of electric charge.
Origin:
The rules had been formulated (in a form that is slightly different from ours)
in 1845 by Kirchhoff, i.e. at the beginning of electricity. At that time
everything was new, and they appeared not trivial at all. The reason why
they survived until the day of today may be due to the fact that they had
gotten their own proper name, namely Kirchhoff’s laws (in contrast to the
above-mentioned analogues rules).
Disposal:
There is nothing to say against treating the various ways of wiring electric
resistors as one of many other problems of electricity. However, one would
not give the results the status of rules or laws. And one will treat similar
questions related to other devices, like capacitors and inductors, and with
other currents, like energy, momentum, heat and water currents.
Finally, one of my favorite rules for writing a syllabus: Always when
someone proposes to introduce a new subject into the syllabus (or into the
curriculum or into a text book) look first for competing subjects, i.e. subjects
that are similar due any kind of analogy. Only if you find an argument, to
treat the subject that was first proposed, and not its competitors, the subject
is accepted. This method has proved to be useful in many occasions.