Electrostatics #5Capacitance

Capacitance

 

I. Define capacitance and a capacitor:

Capacitance is defined as the ability of an object to store charge.

A capacitor is an electronic component of circuits.

The simplest capacitor is one made of two ________________ metal sheets, or plates.

parallel

QQ Q the amount of charge stored on the capacitor. The parallel plates always carry equal but opposite charges.

Note: Where there is a charge separation, there is also

an ____________ field. The field exists between the

plates and points from _____ towards ____ .

electric

Also, where there is an electric field, there is also an ___________

___________ . The _____ plate is the higher electric potential and the

_____ plate is the lower electric potential.

electric

potential

low

er e

lect

ric

pot

enti

alh

igher electric p

otential

direction of electric field

For a constant electric field, the change of electric potential is given by:

cosV E x

For the capacitor, this can be simplified to

This is just the magnitude of the electric potential between the two plates.

V Ed

The capacitance is defined through the electric potential _____ and the

charge held on the plates, _____ . The equation for capacitance is:

VQ

Q C V

next slide for definitions…

Q C V

|V| = the electric potential between the two plates. When charge is placed on a capacitor, an initial electric potential must be provided by a battery.

Q = the electric charge stored on the capacitor

C = the capacitance of the device. The greater the capacitance, the more charge stored on the device for a given voltage.

The units of capacitance are given the special name of ____________ and have a symbol of ‘F’. What is the farad equivalent to?

farad

QC

V

QC

V

potentialelectric

charge

V

C

volt

coulomb

J

CC

V

CFfarad

CJ

2

11111

Ex. 1: A parallel plate capacitor is made of circular metal sheets placed 0.100 mm apart and has a capacitance of 1.00 F. If air is used as the insulator between the two metal plates, what is the maximum amount of charge that may be stored on this capacitor? Air ceases to be an insulator when the electric field is larger than

C

N6103

VCQ and EdV

CEdQ

mC

NFQ 366 10100.01031000.1

CQ 41000.3

II. The capacitance of a parallel plate capacitor can be calculated from its dimensions: The area of the overlap of the two sheets or plates and the distance between the plates.

A

A = the area that the two surfaces overlap (one covers the other)

d

d = the distance between the plates (plate separation)

The capacitance, C, is proportional to the ________ and

inversely proportional to the _______________ between the

plates.

area

distance

d

AC

ord

AC o

The constant o is called the electric permittivity of free space, and the value of o is:

m

F

mN

Co

122

212 10854.810854.8

The permittivity constant is related to the coulomb constant, 2

2910988.8C

mNk

The coulomb constant is derived from the force between charges, and the permittivity constant is derived through Gauss’ Law. The actual relation is:

o

k4

1

Ex. 2: A 1.00 F capacitor is constructed with its metal plates set 0.100 mm apart. If the plates are circular in shape, what is the diameter of the plates?

d

AC o

d

ro2

o

Cdr

mF

mF

12

36

10854.8

10100.01000.1

mdiametermr 79.3896.1

Ex. 3 A 1.00 F capacitor is constructed with square metal plates set 1.00 mm apart. What is the length of a side for the metal plates?

d

AC o

d

so2

s = the length of one side of the square area

o

Cds

mFmF

12

3

10854.8

1000.100.1

mileskmms 6.66.101006.1 4

III. Energy stored in a capacitor: Since the parallel plate capacitor has two plates that are oppositely charged, there is energy stored in the electric interaction between the two plates. This energy is stored in the electric field between the two plates. The energy is:

C

QVCVQU

2

22

21

21 U = the electric energy

stored in a capacitor.

Ex. 4: A parallel plate capacitor is made with an air gap of 0.0100 mm and circular plates with a diameter of 3.25 cm. a. What is the capacitance of this capacitor?

d

AC o

d

ro2

m

mmF

C3

2212

100100.0

21025.3

10854.8

F101035.7

b. What is the maximum charge that may be placed on this capacitor? Let E have a value of

VCQ and EdV

CEdQ

mC

NFQ 3610 100100.01031035.7

CQ 81020.2

63 10N

C

c. What is the energy stored in this capacitor?

C

QU

2

2

F

C10

28

1035.72

1020.2

J71031.3

d. What is the energy density between the plates of the capacitor?

volume

energy

Vol

Uu

.

u = the energy density, or energy per unit volume.

hr

Uu

2

mm

J

3

22

7

100100.021025.3

1031.3

38.39m

Ju

Note: Alternate form to energy density! You do not need to memorize this derivation…

Ad

VC

volume

Uu

2

21

Ad

EddA

u

o 2

21

221 Eu o

221 Eu o

2612

21 10310854.8

C

N

m

F3

8.39m

J