W9_parallel_resonance.eps

132 Week 4: Capacitance

• Energy stored in a capacitor:

U=^1

2

QV=^1

2

CV^2 =^1

2

Q^2

C

(202)

where the first form is the simplest to understand.

One question that is very important iswhereis all this energy stored in the capacitor? The

“best” answer will be: in the electric field! If we write the energy in terms of the electric field,

we find that theenergy density of the electric fieldis given by:

ηe=^1

2

ǫ 0 E^2 (203)

• Adding capacitors in parallel:
  Ctot=C 1 +C 2 +... (204)


• Adding capacitors in series:
  1
  Ctot

=

1

C 1

+

1

C 2

+... (205)

• Dielectrics areinsulatorsthatpolarizewhen placed in an electric field. This builds up a surface
  charge thatreducesthe electric field inside the material – itdisplacesit from its usual value.
  For “weak fields” this reduced field is:

E~=E~^0

ǫr (206)

whereE~ 0 is the external field,E~is the field inside the dielectric, andǫr≥1 is therelative per-

mittivity(also called thedielectric constantκin many “standard” physics textbooks, although

this usage has been deprecated as being too ambiguous) and is characteristic of the material.

One can consistently describe both conductors and insulators in terms of their dielectric prop-

erties by evaluating theirpermittivity(relative to the vacuum permittivityǫ 0 we’ve used so

far) and using it to compute the electric field inside the material:

ǫ=ǫrǫ 0 (207)

This is theactualpermittivity of the material, and in the general case of a time dependent

applied electric field is a complex-valued function of frequency, leading (eventually) to a con-

sistent description ofresistanceand Ohm’s Law, and todispersionand the rainbow!

• Dielectrics perform three important functions in the engineering ofcapacitors:

a) They physically separate the plates (which, recall, experience a possibly strong force of

attraction).

b) They reduce the field in between the plates, which reduces the potential difference, which

increases the amount of charge one can store per volt – the capacitance. If the material

fillsthe space between the plates you should be able to (easily) show that:

C=ǫrC 0 (208)

whereC 0 is the capacitance without the dielectric.

c) They preventdielectric breakdown, so the physical separation of the platesdcan be much

smaller (and the capacitance much larger) at some design voltage.