Week 5: Resistance 165
consumer. I personally expect that by 2020 battery technology will advance so that the full retail
cost crosses the $0.10/watt-hour threshold (where now it is morelike $0.15), so that a full-day
battery will cost roughly $3000 (with a two day supply or supply for alarger household still quite
affordable). There is no good reason to think that retail costs will not continue to fall beyond this
point as technology improves and manufacturing capacity increases and enables various economies of
scale. Well within the decade, individual houses will be easily and cheaply equippable with “backup
batteries” that can store days’ worth of energy for the entire house that will last for a decade or
more with most of their charge storage capability intact.
Similarly, as of this writing an array of rooftop solar cells capable of recharging these batteries
with the energy received in a single “typical” sunny day in most of the United States costs around
$5000, and this number willalsocontinue to decrease to 2020 and beyond as new technologies emerge
and maufacturing capacity increases.
Electrical energy purchased from a utility company in the United States currently costs an
averageof 12 cents per kilowatt-hour, so a year’s worth of electrical energy for a “typical household”
is around $1300 in 2017. If one invests approximately $10,000 (plus $3000 for installation), one can
alreadygo “off-grid” in most U.S. locations (ones with adequate insolation) and generate very close
to 100% of the electrical energy needed to run a typical household, andbreak evenon the investment
in roughly a decade, for about what a top of the line high efficiency air-conditioner/heat pump for
that same household would cost.
The amortization time required to recover the investment will very likely drop to seven years or
even less within the next few years, making this a no-brain decision for most households – one can
borrow the money required to convert over and pay off the loan in a matter of years for about what
one would pay for a new car and entirely funded by reduced electrical utility bills and enjoy “free”
electrical power for the rest of the useful lifetime of the hardware, estimated at this time to be in
excess of twenty years.
5.2: Resistance and Ohm’s Law
Fine, so now we have a battery. We place a chunk of conducting matter between the poles/terminals
of the battery, and what happens? Well,current flows, that’s what happens! We have created a
situation where a conductor isnotin electrostatic equilibrium, andcharge moves in timethrough
the conductor in response to the force created by the battery,withenergy releasedin the process.
This is actuall fine, and we might even say, it’s abouttimethat we got out of statics (which are
kind of boring, as not much happens, right?) and intodynamics, where things happen. All we need,
then, is to come up with a model for what goes on inside the conductor as the current flows, and we
can start to analyze dynamical electrical systems once again, which has to be more interesting than
just thinking about a charged capacitor sitting around all do doing nothing much but just storing
charge.
A microscopic picture, of course, begins with atoms, each with a heavy nucleus and surrounded
by electrons, arranged in some sort of solid lattice, with some of theelectrons “free” to move within
the lattice. Free to move, however, is not the same thing as non-interacting. Electrons that move
through the lattice interact with the lattice and transfer their momentum to the lattice so that (in
equilibrium) their average velocity is zero. The lattice therefore exerts a kind ofdrag forceon the
electrons that brings them back to equilibrium.
Thesimplestmodel for conduction of electrons through a material that “resists” their motion
via a drag force caused by the collision of the moving electrons with each other and the underlying
atoms in the lattice is one with alinear drag force– one that is proportional to the average velocity
of transport of the electrons through the resistive lattice. If the electrons are being pushed through
the conductor by some constant force, then, they’ll arrive quickly at aterminal velocitythat is