Capacitors

Consider two thin, parallel, conducting
plates of cross-sectional area which are separated by
a *small* distance (*i.e.*, ). Suppose that each plate
carries an equal and opposite charge . We expect this charge to
spread evenly over the plates to give an effective sheet charge density
on each plate. Suppose that the upper plate carries a
positive charge and that the lower plate carries a negative charge. According to
Eqs. (624) and (625), the field generated by the upper plate is normal to the plate and
of magnitude

(638) | |||

(639) |

Likewise, the field generated by the lower plate is

(640) | |||

(641) |

Note that we are neglecting any ``leakage'' of the field at the edges of the plates. This is reasonable if the plates are closely spaced. The total field is the sum of the two fields generated by the upper and lower plates. Thus, the net field is normal to the plates, and of magnitude

(642) | |||

(643) |

Since the electric field is uniform, the potential difference between the plates is simply

(644) |

Clearly, a good charge storing device has a high capacitance. Incidentally, capacitance is measured in coulombs per volt, or farads. This is a rather unwieldy unit, since good capacitors typically have capacitances which are only about one millionth of a farad. For a parallel plate capacitor, it is clear that

Note that the capacitance only depends on

Suppose that the charge on each plate is built up gradually by transferring
small amounts of charge from one plate to another. If the
instantaneous charge on the plates is , and an infinitesimal amount of
positive
charge is transferred from the negatively charged plate to the positively
charge plate, then the work done is
,
where is the instantaneous
voltage difference between the plates. Note that the voltage difference is such
that it opposes any increase in the charge on either plate.
The total work done in charging the capacitor
is

(648) |

The energy of
a charged parallel
plate
capacitor is actually stored in the electric field between the plates. This field
is of approximately constant magnitude , and occupies a
region of volume . Thus, given the energy density of an electric
field,
, the energy stored in the
electric field is

The idea, which we discussed earlier, that an electric field exerts a negative
pressure
on conductors immediately suggests that
the two plates in a parallel plate capacitor *attract* one another with a
mutual force

(650) |

It is not necessary to have two oppositely charged conductors
in order to make a capacitor.
Consider an isolated
sphere of radius which
carries a charge . The radial electric field generated outside the sphere is
given by

Thus, the capacitance of the sphere is

(653) |

Suppose that we have two spheres of radii and , respectively, which are
connected by an electric wire. The wire allows charge to move back and forth between
the spheres until they reach the same potential (with respect to infinity).
Let be the charge on the first sphere and the charge on the
second sphere.
Of course, the total charge carried by the two spheres is a conserved
quantity. It follows from Eq. (652) that

(654) | |||

(655) |

Note that if one sphere is much smaller than the other one,

The ratio of the electric fields generated just above the surfaces of the two spheres follows from Eqs. (651) and (656):

If , then the field just above the smaller sphere is far larger than that above the larger sphere. Equation (657) is a simple example of a far more general rule. The electric field above some point on the surface of a conductor is inversely proportional to the local radius of curvature of the surface.

It is clear that if we wish to store significant amounts of charge on a conductor then the surface of the conductor must be made as smooth as possible. Any sharp spikes on the surface will inevitably have comparatively small radii of curvature. Intense local electric fields are generated in these regions. These can easily exceed the critical field for the break-down of air, leading to sparking and the eventual loss of the charge on the conductor. Sparking can also be very destructive because the associated electric currents flow through very localized regions giving rise to intense ohmic heating.

As a final example, consider two co-axial
conducting cylinders of radii and
, where . Suppose that the charge per unit length carried by the
outer and inner cylinders is and , respectively. We can safely
assume that
, by symmetry (adopting
standard cylindrical polar coordinates). Let us apply
Gauss' law to a cylinder of radius , co-axial with the conductors, and
of length . For , we find that

(658) |

(659) |

(660) |

so

(661) |

(662) |