🚀 go-pugleaf

> Date: Wed, 22 Feb 1995 14:03:14  From: Malcolm Davis <davis@NOD.BMS.COM>
> (Please excuse the lateness of this addition to the discussion, but I
>  was out of town the last two weeks.)
>
> It seems to me that the "myth" of the ideal capacitor is the same
> "myth" as frictionless planes and massless pulleys.

> Electronic circuits are best understood in terms of "idealized" components,
> e.g., connecting wires are resistanceless.  If in your situation these
> idealized components are not reasonable, then you can introduce a component
> that is the appropriate combination of the ideal components.  For example,
> the "leaky" capacitor is an ideal capacitor with a resistor in parallel, or
> the real battery is an ideal voltage source in series with a resistor.
> Thus, even for "real" systems, you need to be able to understand the
> behavior of "ideal" components.  So, I would hope no one would succeed in
> purging us of these myths.
>                                          Malcolm

Many people bring this argument when they first hear that the electrostatic
model of capacitors in series has nothing to do with reality. Here is the
correct theory again; I need it to discuss idealizations.

*************************************************************************
> To simplify assume that capacitors have parallel plates. Consider C1 and C2
> in series with a source of constant voltage, V. The current is i=V/(R1+R2),
> where R1 and R2 are leakage resistances. The voltage on C1 is V1=V*R1/(R1+R2)
> and therefore Q1=C1*V1. Similarly, the voltage on C2 is V2=V*R2/(R1+R2) and
> therefore Q2=C2*V2. Thus the ratio Q1/Q2=(C1*R1)/(C2*R2). Use parallel plate
> formula to show that (C1/C2)=e1/e2, where e1 and e2 stand for dielectric
> constants. Also note R1/R2=r1/r2, where r1 and r2 stand for resistivities
> (because R=r*length/area).
>
> In other words, C1*R1=e1*r1 and C2*R2=e2*r2. The geometrical factors (length
> and area) simply cancel in each CR product. Thus Q1/Q2=(e1*r1)/(e2*r2); the
> ratio of charges depends only on material constants, e and r. That ratio
> is equal to unity when dielectric materials are identical in two capacitors.
> In general, however, Q1 and Q2 are expected to be different. Try a cheap low
> voltage paper capacitor and an expensive high voltage mica capacitor to show
> how different Q1 and Q2 can be.
****************************************************************************

Is there any error in the above?
Please show us how that theory leads to Q1=Q2 when R1 and R2 are very large.

Yes, idealizations are essential in physics and they often lead to practically
correct solutions. For example, we know that the thin lens formula is an
approximation which is very useful when rays are paraxial and when thicknesses
are small in comarison with the radii of curvature. The same is true for
frictionless planes, massless pulleys, linear springs, newtonian kinematics,
wires without resistance, fluids without viscosity, point-like objects, etc.
Learning about nature (and teaching physics) would not be possible without
such models and without methods of successive approximations. We will always
use idealizations to understand reality. Note, however, that "to idealize"
means to ignore factors which are not essential in a given situation. In the
problem of distribution of equilibrated charges on capacitors in series (with
a source of constant voltage) leakage resistors are essential factors.

The above simple theory and the electrostatic theory of capacitors in series
lead to mutually contradicting conclusions. Which "theory" is correct?

         FORTUNATELY, THAT QUESTION CAN BE ANSWERED IN A LABORATORY.

The equality of charges leads to 1/C=1/C1+1/C2 and the validity of the formula
leads to equality of charges. This means that at least two experimental
approaches are possible to prove or to disprove the textbook description;
one is to measure charges, another is to measure capacitances. Get your
students involved in this discussion. How often do we have a situation in
which the establishment (authors of physics textbooks) can possibly be
wrong?
                  Ludwik Kowalski   Kowalski@apollo.montclair.edu.