Capacitance & DielectricsAP E&M

Introduction

What is a capacitor?

What is a capacitance?

Conductors of equal and opposite charge separated by an insulating material

The ratio of the magnitude of the charge on either conductor to the magnitude of the potential difference between them.

Capacitance:

CQ

VAlways positive

Measured in [C/V] also known as [F] Farads

1 farad is very large. A 1 farad capacitor could store 1 coulomb of

charge per unit of voltage.

A measure of how much electric potential energy or charge that a capacitor can store.

Capacitance

Parallel Plate:

Cylindrical:

Spherical:

CQ

V

CA

d0

CL

2k lnb

a

Cab

k b a

Energy stored in a capacitor

Suppose that the charge on a capacitor is q during an instant while charging. At this instant the potential difference is V=q/C.

What is the work necessary to move charge dq from the –q plate to the q plate (which is at higher potential)?

CQ

V

dW Vdq

dW q

Cdq

W q

C0Q dq

W Q2

2C

Energy stored in a capacitor

The work done in charging the capacitor can be considered as the potential energy

U stored in the capacitor.

W Q2

2C

CQ

V

UCV 2

2C1

2CV2

UQ2

2Q

V

1

2QV

Energy stored in a parallel-plate capacitorWe can consider the energy to be stored in

the electric field of the capacitor.

CQ

V

U1

2CV2

CA

d0

VEd

UA02d

Ed 2

U1

2Ad0E

2

Energy Density

For a parallel-plate capacitor, the electric field occupies a space of volume Ad, so we will define

the energy per unit volume, or Energy Density as:

U1

2Ad0E

2

uE U

Ad1

20E

2

CQ

V

The energy density in any field is proportional to the square of the electric field in the unit volume.

Capacitors with dielectrics

Dielectric: a nonconducting material such as rubber, glass,

or waxed paper.

CQ

V

VV0

CQ0

VQ0

V0 /C0

1

Capacitors with dielectrics

CQ

VIf a dielectric fills the space between

conductors in a capacitor, the capacitance increases by a factor of a dimensionless factor, called κ, the dielectric constant.

Capacitors with dielectrics

CQ

VThe dielectric material is polarized. This reduces the effective electric field between plates, which

also reduces the potential difference between plates. Lower potential difference results in

higher capacitance.

Benefits of dielectrics in capacitors:

The dielectric strength (maximum electric field that a material can be exposed to without

breaking down and behaving as a conductor) of most materials is higher than air.

Increases the capacitance of a capacitor.

Increases the maximum operating voltage of a capacitor.

My provide mechanical support between the conducting plates.

CQ

V