Week 5: Resistance 177
of water out against the resistance of all of the plumbing isn’t increased or decreased by the water
pressure entering your house from the main.
In this anology, acapacitorcan also be visualized as a wide section of pipe containing apiston on
a spring. The piston blocks water flow, but if one applies a pressure difference then water flowsinto
the pipe section, compressing the spring, until the back-force ofthe spring balances the force on the
piston due to the pressure difference. At that point this “capacitor” has stored somewateron one
side and has had an equivalent amount pushedoffthe other side, just like a regular capacitor. Note
well that this suggestscorrectlythat capacitors will dynamically behave likespringsin an electrical
circuit, storing potential energy and charge and releasing it back to the circuit, causing current and
charge tooscillate. Later we’ll discover a quantity and associated electrical device that behaves just
likemassin such an analogous arrangement, and our work will be complete.
For the moment, though, let’s figure out how to add resistances and then study an actual dy-
namical problem: theRCcircuit.
5.3: Resistances in Series and Parallel
battery capacitor resistor wire ground
V C R
Figure 55: Symbols for batteries, capacitors, resistances, wires, and ground.Before proceding any further, we need to add a symbol to our collection of symbols for circuit
elements. We already have a symbol for capacitance, for a voltagesource or battery and for a
“wire”, but now that conducting wires have this new property of resistance, we need to be a bit more
specific. From now on, wires will be assumed to havezero resistancein all circuit diagrams. This
specifically means, sinceVR=IR, that the voltage drop across any ideal wire iszeroindependent
of the current carried by that wire. Obviously, this is not physical, but if the resistance of the wire
is important, it will (and should) be indicated as an explicit “resistor” inseries with the wire in
question that represents the resistance of that particular segment of wire. Resistance itself has the
new symbol indicated above, typically labelled with its resistance valuein Ohms or a suitably indexed
R. Batteries and capacitances are unchanged (although both may have internal, non-ideal resistance
that will similarly be represented by in-line series or parallel resistance symbols when appropriate).
Finally, the ground symbol, indicating a specific potential ofzerofor all wires connected directly to
it, is recapitulated.
We are now ready to draw collections of individual resistors connected in series or in parallel,
and to derive the effective total resistance of these arrangements. These are pictured in figure 56.
Series
Suppose we apply a fixed voltageVabacross the contacts in the upper (a) diagram. This produces
some currentItotin thesingle(serial) line of resistors. Since charge is conserved and there is nowhere
for it to go but through the resistors, this same current passes through each resistor in turn. We
can thus use Ohm’s Law to determine the voltage drop acrosseachresistor in terms of this total