


Capacitors
Capacitors rarely come up on SAT II Physics,
but they do sometimes make an appearance. Because capacitance is
the most complicated thing you need to know about DC circuits, questions
on capacitors will usually reward you simply for knowing what’s
going on. So long as you understand the basic principles at work
here, you’re likely to get a right answer on a question most students
will answer wrong.
A capacitor is a device for storing charge, made up of
two parallel plates with a space between them. The plates have an
equal and opposite charge on them, creating a potential difference
between the plates. A capacitor can be made of conductors of any
shape, but the parallelplate capacitor is the most
common kind. In circuit diagrams, a capacitor is represented by
two equal parallel lines.
For any capacitor, the ratio of the charge to the potential
difference is called the capacitance, C:
For a parallelplate capacitor, C is
directly proportional to the area of the plates, A,
and inversely proportional to the distance between them, d.
That is, if the area of the plates is doubled, the capacitance is
doubled, and if the distance between the plates is doubled, the capacitance
is halved. The proportionality constant between C and A/d is , called the permittivity of free space,
which we encountered in the previous chapter in relation to Coulomb’s
constant. In case you forgot, C^{2} /N
· m^{2}.
The unit of capacitance is the farad (F).
One farad is equal to one coulomb per volt. Most capacitors have
very small capacitances, which are usually given in microfarads,
where 1 µF = 10^{–6} F.
Energy
To move a small amount of negative charge from the positive
plate to the negative plate of a capacitor, an external agent must
do work. This work is the origin of the energy stored by the capacitor.
If the plates have a charge of magnitude q,
the potential difference is . If q = 0,
and work is done to add charge until q = Q,
the total work required is:
This is the energy stored by the capacitor. Manipulating
this equation and the equation for capacitance, , we can derive a number of equivalent forms:
Equivalent Capacitance
Like resistors, capacitors can be arranged in series or
in parallel. The rule for adding capacitance is the reverse of adding
resistance:
Capacitors in series add like resistors in parallel,
and capacitors in parallel add like resistors in series.
For two capacitors in series:
For two capacitors in parallel:
Example

First, we find the equivalent capacitance of and . Since they are in parallel, = + = 8 µF. Then is given by:
Dielectrics
One way to keep the plates of a capacitor apart
is to insert an insulator called a dielectric between
them. A dielectric increases the capacitance. There is an electric
field between the plates of a capacitor. This field polarizes the
molecules in the dielectric; that is, some of the electrons in the
molecules move to the end of the molecule, near the positive plate:
The movement of electrons creates a layer of negative
charge by the positive plate and a layer of positive charge by the
negative plate. This separation of charge, in turn, creates an electric
field in the dielectric that is in the opposite direction of the
original field of the capacitor. This reduces the total electric
field:
The Greek letter is called the dielectric
constant, and it varies from material to material. For all
materials, > 1.
For a parallelplate capacitor, the reduction in E means
that is also reduced by a factor of . Then, since C = Q/ , we find that:
If the potential difference across the capacitor is too
large, then the electric field will be so strong that the electrons
escape from their atoms and move toward the positive plate. This dielectric
breakdown not only discharges the capacitor, but also burns
a hole in the dielectric and ruins the capacitor.
