1/25/07184 Lecture 111 PHY 184 Spring 2007 Lecture 11 Title: Capacitors.

1/25/07 184 Lecture 11 1

PHY 184PHY 184PHY 184PHY 184

Spring 2007Lecture 11

Title: Capacitors

1/25/07 184 Lecture 11 2

AnnouncementsAnnouncementsAnnouncementsAnnouncements

Homework Set 3 is due Tuesday, Jan. 30 at 8:00 am.

Homework Set 4 opened this morning. Today we will finish up the electric potential

and start with capacitors.

1/25/07 184 Lecture 11 3

Review - Electric Potential, V(x)Review - Electric Potential, V(x)Review - Electric Potential, V(x)Review - Electric Potential, V(x)

)definition - potential (electric

energy) (potential

ΔU/qΔV

WUUU eif

x

sdExV

)(

zV

yV

xV

EEE zyx ,,,,

► If a charge q moves in an electric field…

► For reference V=0 at infinity…

► E is the gradient of V…

1/25/07 184 Lecture 11 4

Review - Electric Potential (2)Review - Electric Potential (2)Review - Electric Potential (2)Review - Electric Potential (2)

rkQ

rV )(

)()(1

xVxVn

ii

► For a point source Q …

► For many sources …

(superposition)

1/25/07 184 Lecture 11 5

Problem-Solving StrategiesProblem-Solving StrategiesProblem-Solving StrategiesProblem-Solving Strategies

Given a charge distribution, calculate the electric field and the electric potential:

(i) Use Gauss Law to derive the electric field E:

(ii) For the potential (V=0 at infinity) use

For an arrangement of charges, remember the superposition superposition principleprinciple! This holds for the electric force, the electric field, the electric potential and the electric potential energy.

enclosed0 qAdE

x

sdExV

)(

cosabba Note:

1/25/07 184 Lecture 11 6

Capacitors

1/25/07 184 Lecture 11 7

CapacitorsCapacitorsCapacitorsCapacitors

Capacitors are devices that store energy in an electric field.

Capacitors are used in many every-day applications• Heart defibrillators• Camera flash units

Capacitors are an essential part of electronics.• Capacitors can be micro-sized on computer

chips or super-sized for high power circuits such as FM radio transmitters.

1/25/07 184 Lecture 11 8

CapacitanceCapacitanceCapacitanceCapacitance Capacitors come in a variety of sizes

and shapes. Concept: A capacitor consists of two

separated conductors, usually called plates, even if these conductors are not simple planes.

We will define a simple geometry and generalize from there.

We will start with a capacitor consisting of two parallel conducting plates, each with area A separated by a distance d .

We assume that these plates are in vacuum (air is very close to a vacuum).

1/25/07 184 Lecture 11 9

Parallel Plate CapacitorParallel Plate CapacitorParallel Plate CapacitorParallel Plate Capacitor

We charge the capacitor by placing• a charge +q on the top plate• a charge -q on the bottom plate

Because the plates are conductors, the charge will distribute itself evenly over the surface of the conducting plates.

The electric potential, V, is proportional to the amount of charge on the plates.

q q

e.g., using a battery

More precisely, potential difference V(+)-V(-) = V

1/25/07 184 Lecture 11 10

The proportionality constant between the charge q and the electric potential difference V is the capacitance C.

We will call the electric potential difference V the “potential” or the “voltage” across the plates.

The capacitance of a device depends on the area of the plates and the distance between the plates, but does not depend on the voltage across the plates or the charge on the plates.

The capacitance of a device tells us how much charge is required to produce a given voltage across the plates.

Parallel Plate Capacitor (2)Parallel Plate Capacitor (2)Parallel Plate Capacitor (2)Parallel Plate Capacitor (2)

CVq

q q

VqC

1/25/07 184 Lecture 11 11

Clicker QuestionClicker QuestionClicker QuestionClicker Question

What is the NET CHARGE on the charged capacitor?

q q

A: +q+(-q)=0B: |+q|+|-q|=2qC: qD: none of the above

1/25/07 184 Lecture 11 12

Clicker QuestionClicker QuestionClicker QuestionClicker Question

What is the NET CHARGE on the charged capacitor?

A: +q+(-q)=0

Charges are added with their signs.

However, we refer to the charge of a capacitor as being q (the charge of a capacitor is not the net charge!)

q q

1/25/07 184 Lecture 11 13

Definition of CapacitanceDefinition of CapacitanceDefinition of CapacitanceDefinition of Capacitance

The definition of capacitance is

The units of capacitance are coulombs per volt. The unit of capacitance has been given the name

farad (abbreviated F) named after British physicist Michael Faraday (1791 - 1867)

A farad is a very large capacitance• Typically we deal with F (10-6 F), nF (10-9 F),

or pF (10-12 F)

Vq

CVq

C

V 1

C 1 F 1

1/25/07 184 Lecture 11 14

Charging/Discharging a CapacitorCharging/Discharging a CapacitorCharging/Discharging a CapacitorCharging/Discharging a Capacitor

We can charge a capacitor by connecting the capacitor to a battery or to a DC power supply.

A battery or DC power supply is designed to supply charge at a given voltage.

When we connect a capacitor to a battery, charge flows from the battery until the capacitor is fully charged.

If we then disconnect the battery or power supply, the capacitor will retain its charge and voltage.

A real-life capacitor will leak charge slowly, but here we will assume ideal capacitors that hold their charge and voltage indefinitely.

1/25/07 184 Lecture 11 15

Charging/Discharging a Capacitor (2)Charging/Discharging a Capacitor (2)Charging/Discharging a Capacitor (2)Charging/Discharging a Capacitor (2)

Illustrate the charging processing using a circuit diagram.

Lines represent conductorsThe battery or power supply is represented byThe capacitor is represented by the symbol

This circuit has a switch (ab)(open) When the switch is between positions a and b, the circuit is open (not connected).(pos a) When the switch is in position a, the battery is connected across the capacitor. Fully charged, q = CV.(pos b) When the switch is in position b, the two plates of the capacitor are connected. Electrons will move around the circuit – a current will flow -- and the capacitor will discharge.

1/25/07 184 Lecture 11 16

Demo: Big SparkDemo: Big SparkDemo: Big SparkDemo: Big Spark

Energy stored in this particular capacitor: 90 J

This is equivalent to the kinetic energy of a mass of 1 kg moving at a velocity of 13.4 m/s!

m/s 4.13kg 1

J 9022

221

mE

v

mvE

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Parallel plate capacitor –

-- a simple ideal model

1/25/07 184 Lecture 11 18

Parallel Plate CapacitorParallel Plate CapacitorParallel Plate CapacitorParallel Plate Capacitor

Consider two parallel conducting plates separated by a distance d

This arrangement is called a parallel plate capacitor. The upper plate has +q and the lower plate has –q. The electric field between the plates points from the positively

charged plate to the negatively charged plate. We will assume ideal parallel plate capacitors in which the electric

field is constant between the plates and zero elsewhere. Real-life capacitors have fringe field near the edges.

1/25/07 184 Lecture 11 19


enclosed0 qAdE

We can calculate the electric field between the plates using Gauss’ Law

We take a Gaussian surface shown by the red dashed line

Flux (through bottom surface of the top plate only!) = EAEnclosed charge = q

Aq

E0

1/25/07 184 Lecture 11 20


Now we calculate the electric potential across the plates of the capacitor in terms of the electric field.

We define the electric potential across the capacitor to be V and we carry out the integral in the direction of the blue arrow.

dEdsE

sdEVVVf

iif

)180cos(

V = Ed

Note: V = V(+) V()

1/25/07 184 Lecture 11 21

Remember the definition of capacitance…

… so the capacitance of a parallel plate capacitor is


Vq

C

dA

dA

q

qEdq

C 0

0

Variables: A is the area of each plate d is the distance between the platesNote that the capacitance depends only on the geometrical factors and not on the amount of charge or the voltage across the capacitor.

dA

C 0

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Demo: Parallel Plate CapacitorDemo: Parallel Plate CapacitorDemo: Parallel Plate CapacitorDemo: Parallel Plate Capacitor

The voltage of a charged capacitor as function of the distance between the plates

For constant charge, the voltage is proportional to the distance between the plates

Aqd

Cq

V0

1/25/07 184 Lecture 11 23

Example - Capacitance of a Parallel Plate CapacitorExample - Capacitance of a Parallel Plate CapacitorExample - Capacitance of a Parallel Plate CapacitorExample - Capacitance of a Parallel Plate Capacitor

We have a parallel plate capacitor constructed of two parallel plates, each with area 625 cm2 separated by a distance of 1.00 mm.

What is the capacitance of this parallel plate capacitor?

Result: A parallel plate capacitor constructed out of square conducting plates 25 cm x 25 cm separated by 1 mm produces a capacitor with a capacitance of about 0.5 nF.

F 1053.5

m 0.001

m 0.0625 F/m108.85

10

2-120

dA

C

C = 0.553 nF

1/25/07 184 Lecture 11 24

Example 2 - Capacitance of a Parallel Plate Example 2 - Capacitance of a Parallel Plate CapacitorCapacitor

Example 2 - Capacitance of a Parallel Plate Example 2 - Capacitance of a Parallel Plate CapacitorCapacitor

We have a parallel plate capacitor constructed of two parallel plates separated by a distance of 1.00 mm.

What area is required to produce a capacitance of 1.00 F?

Result: A parallel plate capacitor constructed out of square conducting plates 10.6 km x 10.6 km (6 miles x 6 miles) separated by 1 mm produces a capacitor with a capacitance of 1 F.

28

12-0

m 1013.1

F/m 108.85

m 0.001 F 1

Cd

A

1/25/07 184 Lecture 11 25

Example - Capacitance, Charge and Electrons …Example - Capacitance, Charge and Electrons …Example - Capacitance, Charge and Electrons …Example - Capacitance, Charge and Electrons …

A storage capacitor on a random access memory (RAM) chip has a capacitance of 55 nF. If the capacitor is charged to 5.3 V, how many excess electrons are on the negative plate?

Idea: We can find the number of excess electrons on the negative plate if we know the total charge q on the plate. Then, the number of electrons n=q/e, where e is the electron charge in coulomb.

Second idea: The charge q of the plate is related to the voltage V to which the capacitor is charged: q=CV.

1/25/07184 Lecture 111 PHY 184 Spring 2007 Lecture 11 Title: Capacitors.

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