176 Week 5: Resistance
which we can rearrange as:
E=Iρ
A
(322)
If we integrate both sides a second time in the directiond~lfrom one end of the conductor to the
other in the direction of the current, we get the potential difference:
VR=−
∫
E~·d~l=−EL=−IρL
A=−IR (323)
|VR|is thus the amount the electric potentialdecreasesgoing from one side of the resistorRto the
otherin the direction of the field/current. We will often write the potential without theRsubscript
to simplify the algebra a tiny bit when there is no ambiguity introduced by so doing, and will
similarly usually omit the sign and just remember that the potentialdropsgoing across a resistor in
the direction of the current.
We’ve introduced a new quantityR, called theresistanceof of the conducting material in this
particular geometry:
R=ρL
A
(324)
so that in terms of it:
V( orVR) =IR (325)
This equation is known asOhm’s Lawand we will use it extensively in the weeks to come.
The SI units of the resistance are known asOhms(volts per ampere, obviously) and given the
symbol Ω in most literature. Since a volt is a joule per coulomb, and an ampere is a coulomb per
second,
1 Ohm =Joule−Second
Coulomb^2(326)
Note well that the SI units of capacitance, farads, were coulombssquared per joule, so the SI units
ofRtimesCareseconds, a pure time. This will be important to us by the end of this chapter.
Just from the simple relationR=ρL/Awe can tell many things about the ways resistances
will add in various configurations. If we put two identical resistances one right after another in a
circuit, that’s the same as one resistance twice as long, so we expect resistances in series toadd,
increasing the total resistance. If we put two identical resistances in parallel, that’s the same as one
resistance with twice the area, which willdecreasethe resistance by a factor of two. We therefore
expect that parallel resistance will obey areciprocaladdition rule. We will derive these two results
more carefully below.
Before going on, it is worthwhile to point out theanalogybetween current flowing in a wire
with finite resistance and water flowing in a pipe packed with somethinge.g. sand that similarly
resists the flow of water. The flow of water through a sand-filled pipe is proportional to thepressure
difference across the pipe, so pressure difference is analogous to voltage difference. The current of
water is analagous to the current of charge. The resistance of the pipe is analogous to the resistance
of the sand-filled pipe. A pipe twice as long will let half the water through at the same pressure
difference. A pipe twice as wide will let twice the water through at the same pressure difference.
There is even a “current density” for the water in motion that is theanalogue of the current density
of the charge. Even pipes that arenotfilled with sand have an “Ohm’s Law” of the form ∆P=IR
whereRis the “resistance” of the pipe andIis the volumetric current in the pipe, as we discussed
in the chapter on fluids in the first semester textbook.
This is really a rather compelling analogy, and since students are sometimes more comfortable
visualizing the flow of water in pipes than they are imagining electrons flowing in wires, it is offered
up to help you build up your conceptual understanding of the latterusing your prior knowledge and
experience of the former, where a day doesn’t pass where you don’t “switch on and off” the flow
of water by means of increasing or decreasing the area of a pipe using a tap and where the flow