Week 4: Capacitance
- Conductorsstore chargeand as they do so, theirpotential(difference)increasesrelative to
ground.
- If we arrange two conductors in a symmetric way and doworkto transfer charge from one
to the other (leaving behind an equal charge of the opposite sign) we call the arrangement a
capacitor– a device for storing energy in the electrostatic field.
- The capacitance of the arrangement is defined to be:
C=|∆Q|
|∆V|
(197)
or, the capacitanceisthe amount of charge we can store that creates a potential difference of
one volt between the conductors.Note the absolute value bars– capacitance is given as a
positive quantity.
- The SI units of capacitance are calledfaradswhere:
1F =
1 Coulomb
1 Volt
(198)
A farad is anenormouscapacitance. Typical values for capacitors in devices range from
picofarads to microfarads, although one can actually buy one farad capacitors for special
projects these days. Large capacitors are dangerous! Especially when strung together
to make a large capacitor at high voltage! Anything over a few hundred microfarads at a
potential of 100+ volts or so can be lethal!
You should be able toderivethe following quantities (from Gauss’s Law, integration of potential
difference, dividing into the presumed total charge):
- Parallel plate capacitor:
C=
ǫ 0 A
d
(199)
whereAis its cross sectional area anddis the separation of the plates.
- Cylindrical capacitor:
C=
2 πLǫ 0
ln(b/a)
(200)
whereais the outer radius of the inner conductor,bthe inner radius of the outer conductor,
andLis its length (where we assumeL≫(b−a)).
- Spherical capacitor:
C= 4πǫ 0
ab
(b−a)
(201)
whereais the outer radius of the inner conductor andbthe inner radius of the outer conductor.
131
