Capacitors
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Capacitors. A device storing electrical energy. – – – – – – –. + + + + + + +. –q. +q. Capacitor. A potential across connected plates causes charge migration until equilibrium. Charge stored q = C D V C = capacitance Unit = C/V = henry = H. D V. A. C 2. N m 2.

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Capacitors

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Capacitors

Capacitors

A device storing electrical energy


Capacitor

– –

– –

– –

+ +

+ +

+ +

+

–q

+q

Capacitor

A potential across connected plates causes charge migration until equilibrium

Charge stored

q = CDV

C = capacitance

Unit = C/V = henry = H

DV


Parallel plate capacitance

A

C2

N m2

e0 = 8.8510–13

Parallel Plate Capacitance

Plate area A, separation d

d

Capacitance = Ae0/d


Circuit element symbols

+ –

DV

  • Conductor

  • Capacitor

or

  • Resistor

Circuit Element Symbols

  • Potential Source


At equilibrium

DV

+ –

C

+ –

DV

At Equilibrium

  • Capacitor charges to potential DV

  • Capacitor charge Q = CDV


Energy in a capacitor

DQ

slope = 1/C

DV

area = W

Q

Energy in a Capacitor

  • C = Q/DV so DV = Q/C

  • Work to push charge DQ W =DVDQ = (Q/C)DQ


Energy in a capacitor1

DV

Q/C

Q

Energy in a Capacitor

  • Work to charge to Q is area of triangleW = 1/2 Q(Q/C) = 1/2 Q2/C

  • Work to charge to DVW = 1/2 DV (CDV) = 1/2C(DV)2

CDV


Combining capacitors

Series

Parallel

Combining Capacitors

and


Parallel components

Parallel Components

  • All have the same potential difference

  • Capacitances add

  • (conceptually add A’s)


Series capacitors

Series Capacitors

  • All have the same charge separation

  • Reciprocals are additive

  • (conceptually add d’s)


Gauss s law

  • e0 = 8.8510–13

C2

N m2

Gauss’s Law

  • Electric flux through a closed shell is proportional to the charge it encloses.

    FE = Qin/e0


Field around a point charge

R

q

q

1

q

kq

e04pr2

if k =

=

=

e0A

4pe0

4pe0 r2

r2

Field around a Point Charge

Shell Area = 4pr2

FE = q/e0 = EA

+q

E =

=


Field around infinite plate

s

1

sA

FE =

, so E =

e0

2

e0

Field Around Infinite Plate

With uniform charge density s = Q/A

= E(2A)


Infinite plate capacitor

–q

–q

+q

1/2 s/e0

0

0

+q

s/e0

1/2 s/e0

Infinite ||-Plate capacitor

Individually

Together


Parallel plate capacitance1

  • Field E =

=

s

Q

e0

Ae0

Qd

  • Potential DV = Ed =

Ae0

Q Ae0

Ae0

  • Capacitance Q/DV =

=

Qd

d

Parallel Plate Capacitance

  • Plate area A, plate separation d


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