Re: There's work, and then there's work

["John S. Denker" <jsd@MONMOUTH.COM>]

Ludwik Kowalski wrote:

> The amount of "ohmoc heat" deposited in R, when C is being CHARGED
> from a constant voltage battery, V, does not depend on R; it depends
> on C and V only.

It's a fairly well-known result.

> This nearly paradoxical fact is worth emphasizing, I think.

The result might be surprising if you think in
terms of integrating the power with respect to time.

But at a more fundamental level you can think of
integrating the voltage with respect to charge.
In that view, you have a rectangle of height
V and width Q, so the area is VQ.

The discharge curve runs down the diagonal.
The diagonal is a straight line whenever the
resistor is linear.  Since triangles have half
the area of rectangles, the dissipated energy is
VQ/2.

Also note that by integrating by parts, you can
change the VdQ integral into a QdV integral and
come to the same conclusion.  This is equivalent
to flipping the rectangle 90 degrees, exchanging
its base and altitude.  But the diagonal still
divides the area in half.

Also note that you dissipate VQ/2 if you charge
the capacitor through a resistor from a source
with constant voltage V, so
 -- the total energy drawn from the source is VQ.
 -- during charging VQ/2 is dissipated and the
    other VQ/2 ends up in the capacitor.
 -- during discharging another VQ/2 is dissipated
    and the capacitor ends up where it started.
 -- during the whole charge/discharge cycle a
    full VQ/2 is dissipated.
 -- The foregoing assumes charging from a constant
    voltage source (V) and discharging in to a constant
    voltage (ground).  If you can arrange for
    time-dependent sources and sinks, all bets are
    off.  If you do it right, you can charge and
    discharge your RC circuit with arbitrarily
    little dissipation.  This is sometimes useful;
    see
http://patft.uspto.gov/netacgi/nph-Parser?Sect1=PTO1&d=PALL&p=1&u=/netahtml/srchnum.htm&r=1&f=G&l=50&s1=5,559,463.WKU.&OS=PN/5,559,463&RS=PN/5,559,463
    and references therein.

===============

Similar logic can be used in many other situations.
For instance, consider the lift-to-drag ratio of
an airplane.  It is, to a good approximation,
independent of the weight of the airplane.  The
heavy airplane and the light airplane will stick
to the same glide slope;  one will move down the
slope faster, that's all (assuming no-wind
conditions).

A smaller resistance-value will cause the system
to slide down the diagonal of the VQ rectangle
more quickly.  A larger resistance-value will
follow the same diagonal, just more slowly.