CapacitanceCapacitanceOctober 6, 2012October 6, 2012

Maximum Charge on a Maximum Charge on a ConductorConductor

Earth

Battery

Conductor- - - - ---

- -- - - - -e-e-

A A batterybattery establishes a difference of establishes a difference of potential that can pump electrons potential that can pump electrons ee-- from a from a

groundground (earth) to a conductor (earth) to a conductor

There is a limit to the amount of charge that a conductor can hold

without leaking to the air. There is a certain capacity for holding charge.

There is a limit to the amount of charge that a conductor can hold

without leaking to the air. There is a certain capacity for holding charge.

CapacitanceCapacitanceThe capacitance C of a conductor is

defined as the ratio of the charge Q on the conductor to the potential V

produced.

The capacitance C of a conductor is defined as the ratio of the charge Q on

the conductor to the potential V produced.

Earth

Battery

Conductor- - - - ---

- -- - - - -e-e-

Capacitance:

Q, V

Capacitance in FaradsCapacitance in FaradsOne One farad (F)farad (F) is the capacitance is the capacitance CC of a of a conductor that holds one coulomb of conductor that holds one coulomb of

charge for each volt of potential.charge for each volt of potential.

Example:Example: When 40 When 40 μμC of charge are placed C of charge are placed on a conductor, the potential is 8 V. What is on a conductor, the potential is 8 V. What is

the capacitance?the capacitance?

Capacitance of Capacitance of Spherical ConductorSpherical ConductorAt surface of At surface of

sphere:sphere:

+Q

r

E and V at surface.

Capacitance, C

Example 1:Example 1: What is the What is the capacitance of a metal sphere of capacitance of a metal sphere of

radius 8 cm?radius 8 cm?

r = 0.08 m

Capacitance, C

+Qr

Note: The capacitance depends only on physical para- meters (the radius r) and is

not determined by either charge or potential. This is true for all capacitors.

Note: The capacitance depends only on physical para- meters (the radius r) and is

not determined by either charge or potential. This is true for all capacitors.

Example 1 (Cont.):Example 1 (Cont.): What charge Q What charge Q is needed to give a potential of 400 V?is needed to give a potential of 400 V?

r = 0.08 m

Capacitance, C

+Q

r

Note: The farad (F) and the coulomb (C) are extremely large units for static

electricity. The SI prefixes micro μ, nano n, and pico p are often used.

Note: The farad (F) and the coulomb (C) are extremely large units for static

electricity. The SI prefixes micro μ, nano n, and pico p are often used.

Parallel Plate Parallel Plate CapacitanceCapacitance

d

Area A+Q

-Q

Example 3.Example 3. The plates of a parallel The plates of a parallel plate capacitor have an area of 0.4 mplate capacitor have an area of 0.4 m22 and are 3 mm apart in air. What is the and are 3 mm apart in air. What is the

capacitance?capacitance?

3 mmd

A

0.4 m2

Dielectric MaterialsDielectric MaterialsMost capacitors have a Most capacitors have a dielectric dielectric

materialmaterial between their plates to provide between their plates to provide greater greater dielectric strengthdielectric strength and less and less probability for electrical discharge.probability for electrical discharge.

The separation of dielectric charge allows The separation of dielectric charge allows more charge to be placed on the plates—more charge to be placed on the plates—

greater capacitancegreater capacitance C > C C > Coo..

++++++

------

AirAir

CCoo

EEoo

++++++

------

- - ++- - ++- - ++C > C > CCoo

E < E < EEoo+

+++++

------

- + - - + - ++- + - - + - ++- + - - + - ++

DielectricDielectric

reduced reduced EE

Dielectric Constant, KDielectric Constant, KThe The dielectric constant Kdielectric constant K for a material is for a material is the ratio of the capacitance the ratio of the capacitance CC with this with this

material as compared with the material as compared with the capacitance capacitance CCoo in a vacuum. in a vacuum.

Dielectric constant: K =

1 for Air

Dielectric constant: K =

1 for Air

Energy of Charged Energy of Charged CapacitorCapacitor

The The potential energypotential energy UU of a charged of a charged capacitor is equal to the work (capacitor is equal to the work (qVqV) ) required to charge the capacitor.required to charge the capacitor.

If we consider the average If we consider the average potential difference from 0 to Vpotential difference from 0 to Vf f

to be to be V/2V/2::

Work = Q(V/2) = ½QV

Work = Q(V/2) = ½QV

Only, great theory but how does one use it?

Capacitors in CircuitsLet’s start with the most basic circuit.

Typically, we are solving for the charge on the capacitor and potential difference across the plates

At times, we will also be asked about the stored energy.

Series Circuits

The key here is - no choice

Let’s find Q and ∆V for each capacitor.

Series Circuits

The key here is - no choice

Let’s find Q and ∆V for each capacitor.

Parallel Circuits

The key here is - choice

Let’s find Q and ∆V for each capacitor.

Complex CircuitsThe key here is - make it look either like series or parallel

Let’s find Q and ∆V for each capacitor.

Complex CircuitsThe key here is - make it look either like series or parallel

Let’s find Q and ∆V for each capacitor.