Capacitors and Capacitance

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Capacitors: Store Electric EnergyCapacitor: two isolated conductors with equal and opposite charges Q and potential difference V between them.

C Q

VUnits: Coulombs/Volt

orFarads

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Parallel Plate Capacitor

Q A

d

Q A

E 0

E ?

E 0

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Parallel Plate CapacitorWhen you put opposite charges on plates, charges move to the inner surfaces of the plates to get as close as possible to charges of the opposite sign

http://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/electrostatics/35-capacitor/35-capacitor320.html

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Calculating E (Gauss’s Law)

0 0 0 A0oS

inq

E dA E QE

A

AGaussGauss

Note: We only “consider” a single sheet!

Doesn’t the other sheet matter?

Parallel Plate Capacitor

top

bottom

V E dS Ed A

0

Q d Q 0 A

V d

C

C depends only on geometric factors A and dP07 -6

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Spherical CapacitorTwo concentric spherical shells of radii a and b

What is E?

Gauss’s Law � E is not 0 only for a < r < b, where it looks like a point charge:

E Q

rˆ

4r

20

Capacitance of Earth

For an isolated spherical conductor of radius a:

C 40a

8.851012 F m a 6.4 106 m

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0

C 7 104 F 0.7mF

A Farad is REALLY BIG! We usually use pF (10-12) or nF (10-

9)

4p0 is roughly 1010 F/m

1 Farad Capacitor

How much charge?

Q C V 1F 1 2 V 1 2 C

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PRS Question: Changing C Dimensions

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Energy Stored in Capacitor

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Energy To Charge Capacitor

-q

1. Capacitor starts uncharged.2. Carry +dq from bottom to top.

Now top has charge q = +dq, bottom -dq

3. Repeat4. Finish when top has charge q =

+Q, bottom -Q

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+q

Work Done Charging Capacitor

At some point top plate has +q, bottom has –qPotential difference is V = q / CWork done lifting another dq is dW = dq V

+q

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-q

Work Done Charging Capacitor

So work done to move dq is:

dW dq V dq q

1 q dqC C

Total energy to charge to q = Q:

q

dq0

1 Q

W dW C+q

-q1Q2

C 2

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Energy Stored in Capacitor

C Q V

Since

2Q 2 1

2C 2

1

2C V Q V U

Where is the energy stored???

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Energy Stored in Capacitor

Energy stored in the E field!

C o A and V EddParallel-plate capacitor:

U 1 CV 2

2 1 o A Ed 2 o E

2 d 2

2

( Ad )

u

(volume)E

E 2

u E field energy density o

2E

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1 Farad Capacitor - EnergyHow much energy?

1 1F 1 2 V

2

2U 1 C V2

2 7 2 J

Compare to a small capacitor charged to 3kV:

1 1 0 0 µ F 3 k V

2

1 1 1 0 4 F 3 1 03 V 2 4 5 0 J

U 1 C | V|22 2

2

PRS Question: Changing C Dimensions

Energy Stored

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