1/30/07 184 Lecture 13 1

PHY 184PHY 184PHY 184PHY 184

Spring 2007Lecture 13

Title: Capacitors

1/30/07 184 Lecture 13 2

NotesNotesNotesNotes

Homework Set 3 is done! Homework Set 4 is open and Set 5 opens Thursday

morning Midterm 1 will take place in class Thursday, February 8. One 8.5 x 11 inch equation sheet (front and back) is

allowed. The exam will cover Chapters 16 - 19 Homework Sets 1 - 4

1/30/07 184 Lecture 13 3

ReviewReviewReviewReview

The capacitance of a spherical capacitor is

• r1 is the radius of the inner sphere

• r2 is the radius of the outer sphere

The capacitance of an isolated spherical conductor is

• R is the radius of the sphere

C 40

r1r2r2 r1

C 40

r1r2r2 r1

C 40RC 40R

1/30/07 184 Lecture 13 4

Review (2)Review (2)Review (2)Review (2)

The equivalent capacitance for n capacitors in parallel is

The equivalent capacitance for n capacitors in series is

Ceq Cii1

n

Ceq Cii1

n

1

Ceq

1

Cii1

n

1

Ceq

1

Cii1

n

1/30/07 184 Lecture 13 5

Example - System of CapacitorsExample - System of CapacitorsExample - System of CapacitorsExample - System of Capacitors

Let’s analyze a system of five capacitors

If each capacitor has a capacitance of 5 nF, what is the capacitance of this system of capacitors?

1/30/07 184 Lecture 13 6

System of Capacitors (2)System of Capacitors (2)System of Capacitors (2)System of Capacitors (2)

We can see that C1 and C2 are in parallel and that C3 is also in parallel with C1 and C2

We can define C123 = C1 + C2 + C3 = 15 nF … and make a new drawing

1/30/07 184 Lecture 13 7

System of Capacitors (3)System of Capacitors (3)System of Capacitors (3)System of Capacitors (3)

We can see that C4 and C123 are in series We can define

… and make a new drawing

1

C1234

1

C123

1

C4

C1234 C123C4

C123 C4

= 3.75 nF

1/30/07 184 Lecture 13 8

System of Capacitors (4)System of Capacitors (4)System of Capacitors (4)System of Capacitors (4)

We can see that C5 and C1234 are in parallel We can define

And make a new drawing

C12345 C1234 C5 C123C4

C123 C4

C5 (C1 C2 C3)C4

C1 C2 C3 C4

C5 = 8.75 nF

1/30/07 184 Lecture 13 9

System of Capacitors (5)System of Capacitors (5)System of Capacitors (5)System of Capacitors (5)

So the equivalent capacitance of our system of capacitors

More than one half of the total capacitance of this arrangement is provided by C5 alone.

This result makes it clear that one has to be careful how one arranges capacitors in circuits.

C12345 (5 5 5)5

5 5 5 5 5

nF 8.75 nF

1/30/07 184 Lecture 13 10

Clicker QuestionClicker QuestionClicker QuestionClicker Question

Find the equivalent capacitance Ceq

A)

B)

C)

1/30/07 184 Lecture 13 11

Clicker QuestionClicker QuestionClicker QuestionClicker Question

Find the equivalent capacitance Ceq

C)

First Step: C1 and C2 are in series

Second Step: C12 and C3 are in parallel

1/30/07 184 Lecture 13 12

A capacitor stores energy.

Field Theory: The energy belongs to the electric field.

1/30/07 184 Lecture 13 13

A battery must do work to charge a capacitor. We can think of this work as changing the electric potential energy

of the capacitor. The differential work dW done by a battery with voltage V to put a

differential charge dq on a capacitor with capacitance C is

The total work required to bring the capacitor to its full charge q is

This work is stored as electric potential energy

Energy Stored in CapacitorsEnergy Stored in CapacitorsEnergy Stored in CapacitorsEnergy Stored in Capacitors

dW Vdq q

Cdq

Wt dW q

Cdq

0

qt

1

2

qt2

C

U 1

2

q2

C

1

2CV 2

1

2qV

1/30/07 184 Lecture 13 14

We define the energy density, u, as the electric potential energy per unit volume

Taking the ideal case of a parallel plate capacitor that has no fringe field, the volume between the plates is the area of each plate times the distance between the plates, Ad

Inserting our formula for the capacitance of a parallel plate capacitor we find

Energy Density in CapacitorsEnergy Density in CapacitorsEnergy Density in CapacitorsEnergy Density in Capacitors

u U

volume

u U

Ad

12CV

2

AdCV 2

2Ad

u

0A

d

V 2

2Ad

1

20

V

d

2

1/30/07 184 Lecture 13 15

Recognizing that V/d is the magnitude of the electric field, E, we obtain an expression for the electric potential energy density for parallel plate capacitor

This result, which we derived for the parallel plate capacitor, is in fact completely general.

This equation holds for all electric fields produced in any way• The formula gives the quantity of electric field energy per

unit volume.

Energy Density in Capacitors (2)Energy Density in Capacitors (2)Energy Density in Capacitors (2)Energy Density in Capacitors (2)

u 1

20E

2u 1

20E

2

1/30/07 184 Lecture 13 16

An isolated conducting sphere whose radius R is 6.85 cm has a charge of q=1.25 nC.

a) How much potential energy is stored in the electric field of the

charged conductor?

Key Idea: An isolated sphere has a capacitance of C=40R (see previous lecture). The energy U stored in a capacitor depends on the charge and the capacitance according to

Example - isolated conducting sphereExample - isolated conducting sphereExample - isolated conducting sphereExample - isolated conducting sphere

… and substituting C=40R gives

1/30/07 184 Lecture 13 17

An isolated conducting sphere whose radius R is 6.85 cm has a charge of q=1.25 nC.

b) What is the field energy density at the surface of the

sphere?

Key Idea: The energy density u depends on the magnitude of the electric field E according to

so we must first find the E field at the surface of the sphere. Recall:

Example - isolated conducting sphereExample - isolated conducting sphereExample - isolated conducting sphereExample - isolated conducting sphere

u 1

20E

2u 1

20E

2

(Why?)

1/30/07 184 Lecture 13 18

Example: ThundercloudExample: ThundercloudExample: ThundercloudExample: Thundercloud Suppose a thundercloud with horizontal dimensions of 2.0 km by

3.0 km hovers over a flat area, at an altitude of 500 m and carries a charge of 160 C.

Question 1:• What is the potential difference between

the cloud and the ground? Question 2:

• Knowing that lightning strikes requireelectric field strengths of approximately2.5 MV/m, are these conditions sufficientfor a lightning strike?

Question 3:• What is the total electrical energy contained in this cloud?

1/30/07 184 Lecture 13 19

Example: Thundercloud (2)Example: Thundercloud (2)Example: Thundercloud (2)Example: Thundercloud (2) Question 1 We can approximate the cloud-ground system as a parallel plate

capacitor whose capacitance is

The charge carried by the cloud is 160 C, which means that the “plate surface” facing the earth has a charge of 80 C

720 million volts

C 0A

d

(8.85·10-12 F/m)(2000 m)(3000 m)

500 m0.11 F

V q

C

80 C

0.11 F7.2 108 V

++++++++++++…++++++++++++ …

1/30/07 184 Lecture 13 20

Example: Thundercloud (3)Example: Thundercloud (3)Example: Thundercloud (3)Example: Thundercloud (3) Question 2 We know the potential difference between the cloud and ground so

we can calculate the electric field

E is lower than 2.5 MV/m, so no lightning cloud to ground• May have lightning to radio tower or tree….

Question 3 The total energy stored in a parallel place capacitor is

• Enough energy to run a 1500 W hair dryer for more than 5000 hours

E V

d

7.2 108 V

500 m1.5 MV/m

U 1

2qV 0.5(80 C)(7.2 108 V) 2.9 1010 J

1/30/07 184 Lecture 13 21

Clicker QuestionClicker QuestionClicker QuestionClicker Question

A 1.0 F capacitor and a 3.0 F capacitor are connected in parallel across a 500 V potential difference V. What is the total energy stored in the capacitors?

A) U=0.5 J B) U=0.27 J C) U=1.5 J D) U=0.02 J

Hint: Use

1/30/07 184 Lecture 13 22

Clicker QuestionClicker QuestionClicker QuestionClicker Question

A 1.0 F capacitor and a 3.0 F capacitor are connected in parallel across a 500 V potential difference V. What is the total energy stored in the capacitors?

A) U=0.5 J

2221

2

eq

V 500F 4502

F 0.4

.CVC

qU

C

U = 0.5 J

1/30/07 184 Lecture 13 23

Capacitors with DielectricsCapacitors with DielectricsCapacitors with DielectricsCapacitors with Dielectrics

We have only discussed capacitors with air or vacuum between the plates.

However, most real-life capacitors have an insulating material, called a dielectric, between the two plates.

The dielectric serves several purposes:• Provides a convenient way to maintain mechanical

separation between the plates.• Provides electrical insulation between the plates.• Allows the capacitor to hold a higher voltage

• Increases the capacitance of the capacitor• Takes advantage of the molecular structure of the dielectric

material

1/30/07 184 Lecture 13 24

Capacitors with Dielectrics (2)Capacitors with Dielectrics (2)Capacitors with Dielectrics (2)Capacitors with Dielectrics (2)

Placing a dielectric between the plates of a capacitor increases the capacitance of the capacitor by a numerical factor called the dielectric constant,

We can express the capacitance of a capacitor with a dielectric with dielectric constant between the plates as

… where Cair is the capacitance of the capacitor without the dielectric

Placing the dielectric between the plates of the capacitor has the effect of lowering the electric field between the plates and allowing more charge to be stored in the capacitor.

C CairC Cair

1/30/07 184 Lecture 13 25

Parallel Plate Capacitor with DielectricParallel Plate Capacitor with DielectricParallel Plate Capacitor with DielectricParallel Plate Capacitor with Dielectric

Placing a dielectric between the plates of a parallel plate capacitor modifies the electric field as

The constant 0 is the electric permittivity of free space

The constant is the electric permittivity of the dielectric material

E Eair

q

0Aq

A

0QuickTime™ and a

TIFF (Uncompressed) decompressorare needed to see this picture.