CapacitanCapacitancece

and and DielectricDielectric

ssAP Physics CAP Physics CMontwood High SchoolMontwood High School

R. CasaoR. Casao

DefinitionsDefinitionsVoltageVoltage—potential difference between two points in —potential difference between two points in space (or a circuit).space (or a circuit).CapacitorCapacitor—device to store energy as potential energy in —device to store energy as potential energy in an E field.an E field.CapacitanceCapacitance—the charge on the plates of a capacitor —the charge on the plates of a capacitor divided by the potential difference of the plates: C = q/Vdivided by the potential difference of the plates: C = q/VFaradFarad—unit of capacitance, 1F = 1 C/V. This is a very —unit of capacitance, 1F = 1 C/V. This is a very large unit of capacitance, in practice we use large unit of capacitance, in practice we use F (10F (10-6-6) or ) or pF (10pF (10-12-12).).Electric circuitElectric circuit—a path through which charge can flow.—a path through which charge can flow.BatteryBattery—device maintaining a potential difference V —device maintaining a potential difference V between its terminals by means of an internal between its terminals by means of an internal electrochemical reaction.electrochemical reaction. TerminalsTerminals—points at which charge can enter or leave a —points at which charge can enter or leave a battery.battery.

Definition of CapacitanceDefinition of CapacitanceA capacitor consists of A capacitor consists of two uncharged two uncharged conductors connected conductors connected across the terminals of a across the terminals of a voltage source (battery).voltage source (battery).The battery will move The battery will move charge from one charge from one conductor to another conductor to another until the voltage until the voltage produced by the charge produced by the charge buildup is equal to the buildup is equal to the battery voltage.battery voltage.

Definition of CapacitanceDefinition of CapacitanceThe charge on one conductor will be charged to +Q and the other conductor will be charged to -Q. An electric field E is established between the charged conductors.A capacitor is a device whose purpose is to store electrical energy which can then be released in a controlled manner during a short period of time.

When the switch is closed, the battery establishes an electric field in the wire that causes electrons to move from the left plate into the wire and into the right plate from the wire. As a result, a separation of charge exists on the plates, which represents an increase in electric potential energy of the system of the circuit. This energy in the system has been transformed from chemical energy in the battery.

Definition of CapacitanceDefinition of CapacitanceCapacitors are used when a sudden release of energy is needed. Capacitance is the ratio of the charge Q on one conductor of the capacitor to the potential difference between the conductors, Vab.

Capacitance is always a positive quantity.The unit for capacitance is the Farad (F). 1 F = 1 C/V; 1 F = 1 x 10-6 F.

VQC

Capacitance of a Lone Capacitance of a Lone Spherical ConductorSpherical Conductor

The capacitance of a The capacitance of a device depends on the device depends on the geometric arrangement of geometric arrangement of the conductors.the conductors.For a spherical conductor For a spherical conductor of radius R and charge +q of radius R and charge +q (the conductor with the –q (the conductor with the –q charge can be considered charge can be considered as a hollow conducting as a hollow conducting sphere of infinite radius):sphere of infinite radius):– The voltage of the

sphere is

RQkV

Capacitance of a Spherical Capacitance of a Spherical ConductorConductor

The capacitance is given by:The capacitance is given by:

o is the permittivity of the insulating material (called the dielectric) between the plates.

Rεπ4kR

RQk

QVQC o

2

212

o mNC10x854.8ε

Parallel Plate CapacitorParallel Plate CapacitorThe charge on each plate is distributed uniformly over an area A;

Use Gauss's Law to calculate the electric field E between the plates of the charged capacitor:

AQσ

AεQ

εσE

oo

A

A

+Q

-Q

d

Parallel Plate CapacitorParallel Plate CapacitorThe potential difference between the plates is:The potential difference between the plates is:

Substitute into the capacitance equation to Substitute into the capacitance equation to determine the capacitance for a parallel plate determine the capacitance for a parallel plate capacitor:capacitor:

AεdQV

oab

dAε

AεdQ

QVQC o

o

Parallel Plate CapacitorParallel Plate CapacitorThe capacitance of a parallel plate capacitor is The capacitance of a parallel plate capacitor is directly proportional to the area of the plates and directly proportional to the area of the plates and inversely proportional to the distance between inversely proportional to the distance between the plates. the plates. When air or a vacuum is the dielectric material When air or a vacuum is the dielectric material between the plates:between the plates:

When another material is present as the dielectric When another material is present as the dielectric material between the plates, the equation material between the plates, the equation includes a dielectric constant k:includes a dielectric constant k:

dAεC o

dAεkC o

Cylindrical CapacitorsCylindrical CapacitorsA cylindrical capacitor consists of a cylindrical conductor of radius a and length l surrounded by a coaxial cylindrical shell of radius b.The electric field E is perpendicular to the axis of the cylinders and is confined to the region between the cylinders.

Cylindrical CapacitorsCylindrical CapacitorsDetermine the voltage Determine the voltage between the two between the two cylinders:cylinders:

The electric field for a The electric field for a cylinder is given by:cylinder is given by:

b

aab drEVV

rλk2E

Cylindrical CapacitorsCylindrical Capacitors

ablnλk2VV

ablnalnbln

alnblnrlndrr1

drr1λk2VV

drrλk2VV

ab

ba

b

a

b

aab

b

aab

Cylindrical CapacitorsCylindrical CapacitorsThe voltage V is The voltage V is negative based on the negative based on the direction of the direction of the integration. integration. Capacitance is a Capacitance is a positive quantity, so positive quantity, so use the magnitude of use the magnitude of the voltage. You can the voltage. You can also reverse the also reverse the direction of the direction of the integration to Vintegration to Vaa – V – Vbb.. Substitute into the Substitute into the capacitance equation:capacitance equation:

ablnk2

LCablnL

Qk2QC

ablnλk2

QVQC

Cylindrical CapacitorsCylindrical CapacitorsAn example of this type of capacitor is a An example of this type of capacitor is a coaxial cable.coaxial cable.The coaxial cable consists of two The coaxial cable consists of two cylindrical conductors of radii a and b cylindrical conductors of radii a and b separated by an insulator. The cable separated by an insulator. The cable carries currents in opposite directions in carries currents in opposite directions in the inner and outer conductors.the inner and outer conductors.The geometry is useful for shielding the The geometry is useful for shielding the electrical signal from external influences.electrical signal from external influences.

Cylindrical Capacitor w/ Gauss’s LawCylindrical Capacitor w/ Gauss’s Law

ba

enclosedE

o

QE dA

+Q on center conducting cylinder

-Q on outer conducting cylinder

o

QE 2 rL

o

QE2 rL

L

b b

bao o oa a

Q Q dr Q bV dr ln2 rL 2 L r 2 L a

Cylindrical Capacitor w/ Gauss’s LawCylindrical Capacitor w/ Gauss’s Law

abQ CV

ba

L

o

ab

o

2 LQ QCQ b bV ln ln

2 L a a

L

Spherical CapacitorsSpherical CapacitorsThe spheres have uniform The spheres have uniform charge density and the electric charge density and the electric field is uniformly distributed field is uniformly distributed between the inner sphere of between the inner sphere of radius a and the outer sphere of radius a and the outer sphere of radius b.radius b.The electric field E between the The electric field E between the two spheres is given by:two spheres is given by:

2rQkE

Spherical CapacitorsSpherical CapacitorsDetermine the voltage between Determine the voltage between the two spheres:the two spheres:

b

aab drEVV

baabQkVV

bab

baaQka

1b1QkVV

a1

b1

a1

b1

r1drr

1

drr1QkVV

drrQkVV

ab

ab

b

a

b

a 2

b

a 2ab

b

a 2ab

Spherical CapacitorsSpherical CapacitorsThe voltage V is The voltage V is negative based on the negative based on the direction of the direction of the integration. integration. Capacitance is a Capacitance is a positive quantity, so use positive quantity, so use the magnitude of the the magnitude of the voltage. You can also voltage. You can also reverse the direction of reverse the direction of the integration to the integration to VVaa – V – Vbb.. Substitute into the Substitute into the capacitance equation:capacitance equation:

abkbaC

baabQk

QVQC

What is a lightning discharge ?What is a lightning discharge ?•Friction forces between the air molecules within a cloud result in positively charged molecules moving to the lower surface, and the negative charges moving to the upper surface .•The lower surface induces a high concentration of negative charges in the earth beneath.•The resultant electric field is very strong, containing large amounts of charge and energy.•If the electric field is greater than than breakdown strength of air, a lightning discharge occurs, in which air molecules are ripped apart, forming a conducting path between the cloud and the earth.