Chapter 26:Capacitance and Dielectrics

CapacitorsA capacitor is made up of 2 conductors carrying

charges of equal magnitude and opposite sign.

The Capacitance (C) of a capacitor is defined as:

where Q is the charge on either conductor and ∆V is the potential difference between them

The SI unit of conductance is the farad (F). - 1 F = 1 coulomb per volt (C/V)

C Q

V

Parallel-Plate CapacitorsTwo metallic plates of equal area “A” are

separated by a distance “d”. One plate carries a + charge while the other carries a – charge.

The surface density of each plate is

The electric field between the plates is

Q

A

E Q

oA

o

Parallel-Plate CapacitorsSince the field between the plates is uniform

and points directly from one plate to the other, the magnitude of the potential difference between the two equals Ed

Therefore the capacitance is given by:

V Ed Qd

oA

C Q

V

oA

d

Combinations of Capacitors

Capacitors can be combined in electric circuits.

When studying circuits a circuit diagram can be used and specific circuit symbols are given to designate a capacitor, battery, and switch.

Capacitor Symbol

Battery Symbol

Switch Symbol

Various CombinationsParallel Combination

∆V1=∆V2=∆V Q total=Q1+Q2 Equivalent capacitor

Q total = Cequiv(∆V)Cequiv = C1+C2+C3…

Series Combination ∆Vtotal=∆V1+∆V2 Q1=Q2=Q Equivalent capacitance

1/Cequiv = 1/C1 + 1/C2 + 1/C3…

Energy Stored in Charged Capacitor

Electric potential energy is stored in the system

Energy stored in a charged capacitor

The energy in a capacitor is stored in the electric field between the two plates as the capacitor is charged.

The electric field is proportional to the charge on the capacitor

U Q2

2C1

2C(V )2

Capacitors with Dielectrics

A Dielectric is a non-conducting material like rubber or glass.

The Dielectric constant (κ) varies from from one material to the next.

The capacitance of a capacitor containing a dielectric is:

With the addition of the dielectric the charge remains the same, the potential difference decreases, and the capacitance increases.

The capacitance increases by the factor “κ” when the dielectric is filling the space between the plates.

C CO

Electric Dipole in an Electric Field

The Electric dipole moment has a magnitude of p=2aq 2a is the distance between – q and +

The torque on an electric dipole in a uniform electric field is given by:

The potential energy of the system of an electric dipole in a unifrom external electric field is given by:

r T

r p

r

r E

U r p

r E

Atomic Description of Dielectrics

The field in the presence of a dielectric equals

Charged dipoles can be viewed as creating another pair of parallel plates establishing an electric field

So, the net electric field in the dielectric has a magnitude

r E

r E O

r E ind

E EO E ind