lecture 6.1 : capacitors1 lecture 6.1 : capacitors lecture outline: finding field from potential...

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Lecture 6.1 :Capacitors

Lecture Outline:Finding Field from PotentialCapacitance and CapacitorsEnergy Stored in a Capacitor

Dielectrics

Feb. 19, 2013

Textbook Reading:Ch. 29.3 - 29.7

Announcements

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•HW6 due next Mon. (2/25) at 9pm on Mastering Physics.

•Quiz #3 Thu. (Feb. 21) in class.

•Review session for Exam #2 is scheduled for Wed., Feb. 27, from 7-9pm in Stolkin.

Last Lecture...

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V ≡ Uq+sources

q

∆V = −� f

i

�E · d�s

∆U = −� f

i

�F · d�s

�E = −∇V

Finding Field from Potential

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Clicker Question #1

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Which set of equipotential surfaces matches this electric field?

Stronger Fields = Closer Equipotentials!

Finding Field from Potential

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Kirchhoff’s Loop Law

Finding Field from Potential

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1. Electric Field inside is zero.

2. Excess charge is on surface.

3. Surface is equipotential.

4. Interior is equipotential

5. Exterior field is perpendicular to surface.

6. Field and surface charge density are largest at sharp corners. (Why?)

Conductor in Equilibrium

Capacitance and Capacitors

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Capacitors come in many shapes/sizes. Common elements in electrical circuits. “Caps”.

Capacitance and Capacitors

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Battery charging up a Capacitor

Capacitance and Capacitors

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E =Q

�0A

∆VC = Ed

Q =�0A

d∆VC

1 farad = 1 F ≡    1 C/V

C ≡ Q

∆VC

Capacitance is a geometric property!

C ≡ Q

∆VC=

�0A

d

Clicker Question #2

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What is the capacitance of these two electrodes?

A. 8 nF.B. 4 nF.C. 2 nF.D. 1 nF.E. Some other value.

Capacitance and Capacitors

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Capacitors in SeriesCapacitors in Parallel

Clicker Question #3

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Which arrangement has larger equivalent capacitance?

A. LeftB. RightC. Both are equal

Parallel => add Series => inverse of sum of inverses

Energy Stored in a Capacitor

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dU = dq ∆V =q dq

C

UC =1

C

� Q

0q dq =

Q2

2C

UC =Q2

2C=

1

2C(∆VC)

2

Incremental change in energy (dU) as charge (dq) is added.

Total Energy Stored

Rewriting in terms of Potential across Capacitor.

Energy Stored in a Capacitor

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A 10000 microFarad capacitor is charged to 36V. What is the energy stored inside this capacitor?

If the capacitor discharges in 10 microseconds, what is the power dissipated?

Energy Stored in a Capacitor

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UC =1

2C(∆VC)

2 =1

2

�0A

d(Ed)2 =

�02(Ad)E2

For a Parallel Plate Capacitor:

The energy of a charged capacitor is stored in the electric field!

Dielectrics

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So far we’ve assumed our parallel-plate capacitors are filled with vacuum (i.e. - nothing). What if we insert

some other material?

Capacitance = ?

Dielectrics

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Capacitor with no dielectric

Dielectric is polarized.

Electric Field is diminished.

Dielectrics

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κ ≡ E0

E

∆VC = Ed =E0

κd =

(∆VC)0κ

C =Q

∆VC=

Q0

(∆VC)0/κ= κ

Q0

(∆VC)0= κC0

Voltage diminishes

Capacitance Increases

Dielectric Constant

Dielectrics

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Reminders

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•Show up for recitations! Problems assigned during recitations are fair game for quizzes and exams.

•Quiz #3 Thursday (Feb. 21)