142 Week 4: Capacitance
This makes the problem of finding the total capacitance really easy!Qtot = Q 1 +Q 2 +Q 3 +...
CtotV = C 1 V+C 2 V+C 3 V+...
Ctot = C 1 +C 2 +C 3 +...=∑
iCi (237)where we note that our rule works foranynumber of capacitors in series and write the final rule
accordingly. Capacitors in parallel add!
We can understand these two rules intuitively in the following way. Capacitors in parallel in-
crease the effectiveareawhere charge is stored, and hence just add. Capacitors in series increases
the effectiveseparationof the plates for a given area, and hence reduce the capacitance,adding
reciprocally.
Before moving on, it is important to make one final observation. Capacitors (as we shall see)
behave in electrical circuits the wayspringsbehave in mechanical systems – they store energy and
exert a restoring force on the charges that are stored that isproportional to the charge. Note well
the analogy:
Fx = −ksx (238)V = −1
C
Q (239)
where 1/Cbehaves like a “spring constant” and where the minus sign indicates that the potential
createdopposesthe addition of more charge (we ignore this in the definition ofC, but used it in the
computation ofU). If one computes the effective spring constant ofspringsin parallel or in series,
one obtains very similar results. Springs in parallel add, with a total spring constant equal to the
sum of the spring constants. Springs in series add as reciprocals, where the total spring constant is
less than the smallestconstant of the springs in the series.
Later we will learn that this analogy is nearly exact, after we discover the quantities which behave
like “friction” or “drag forces” in circuits and even discover a quantity that behaves like a “mass”.
In the end we will find ourselves solving an equation that is identical in form to the damped, driven
harmonic oscillator studied last semester, only this equation will yield the currents flowing in the
circuit as a function of time. At that time it will be very fruitful to be thinking “the capacitor is
like a spring” to help us understand what is going on.
4.4: Dielectrics
We have taken some care to study electric dipoles as the most common arrangement of matter
that leads to an electric field, given the generally neutral character of matter. Indeed, all of the
capacitors studied above can be thought of as stylized “dipoles” storing energy by separating charge.
We have also observed that conductors placed in an electric field polarize and create a (mostly
dipolar) arrangement of surface charge that completely cancels the electric field inside. But what
of insulators? They too are made up of neutral atoms and molecules, but lack the “free charges”
that carry current, as the electrons associated with each molecule prefer to stay home instead of
wandering off long distances under the influence of any vagrant electric field.
To understand what a neutral atom does in the presence of an electric field, it will be very useful
to have amodelof an atom. We know that an atom consists of a tiny, massive nucleuswith a charge
+ZewhereZis theatomic numberof the atom. Surrounding this nucleus is a “cloud” ofZelectrons
(for a total charge of−Zeresulting in an electrically neutral atom), bound to the nucleus by the
electrostatic force. We rather expect the neutral atom to be spherically symmetric in its distribution
of charge so that there is little or no electric field outside of the charge cloud.