Chapter 27: 4-5

Electric Field formulas for several continuous distribution of charge.

We can represent an electric dipole by two opposite charges ±q separated by the small distance s.The dipole moment is defined as the vector

The Electric Field of a Dipole

The dipole-moment magnitude p = qs determines the electric field strength. The SI units of the dipole moment are C m.

The electric field at a point on the axis of a dipole isThe Electric Field of a Dipole

where r is the distance measured from the center of the dipole.The electric field in the plane that bisects and is perpendicular to the dipole is

This field is opposite to the dipole direction, and it is only half the strength of the on-axis field at the same distance.

Problem-Solving Strategy: The electric field of a continuous distribution of charge

Problem-Solving Strategy: The electric field of a continuous distribution of charge

Problem-Solving Strategy: The electric field of a continuous distribution of charge

Example 26.3Derivation on pages 827-828

See appendix A-3 for integral

Derivation on pages 827-828

Electric Field of a infinite line of charge

Electric Field of a line of charge

2L to reduces radical then L

2LrrQ

41E

220rod

Electric Field on Ring

232204

1RzzQE

xring

2/322

0

12/322

01

2/3220

22220

22

22

20

41

41

4141

cos

cos41cos

RzzQE

QRzzEE

QRzzE

Rzz

RzQE

Rz

zrzRzr

rQEE

zring

N

i

N

iizz

iz

iz

ii

i

ii

iiiz

A Disk of Charge

Electric Field of a disk of charge

220

02/322

0

12/322

0

1 12/322

0

12

2)(

,

2

2

4

Rz

zzE

rz

rdrzE

drrNrz

rrzE

rrAQrz

QzEE

zdisk

R

zdisk

N

i i

izdisk

ii

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zizdisk

The on-axis electric field of a charged disk of radius R, centered on the origin with axis parallel to z, and surface charge density η = Q/πR2 is

A Disk of Charge

NOTE: This expression is only valid for z > 0. The field for z < 0 has the same magnitude but points in the opposite direction.NOTE 2: The formula reduces to a plane of charge relationship at R>>z

A Plane of ChargeThe electric field of an infinite plane of charge with surface charge density η is:

For a positively charged plane, with η > 0, the electric field points away from the plane on both sides of the plane.For a negatively charged plane, with η < 0, the electric field points towards the plane on both sides of the plane.

Formula above can be derived from charge disk formula, see page 830.

The Parallel-Plate Capacitor• The figure shows two electrodes, one with charge +Q and the other with –Q placed face-to-face a distance d apart. • This arrangement of two electrodes, charged equally but oppositely, is called a parallel-plate capacitor. • Capacitors play important roles in many electric circuits.

The electric field inside a capacitor is

where A is the surface area of each electrode. Outside the capacitor plates, where E+ and E– have equal magnitudes but opposite directions, the electric field is zero.

The Parallel-Plate Capacitor

Use two infinite size disks to create the relationship for a capacitor. If z <<R, second fraction reduces to 0. Since there are 2 disks, then formula is doubled.

EXAMPLE 27.7 The electric field inside a capacitor

QUESTIONS:

EXAMPLE 27.7 The electric field inside a capacitor

EXAMPLE 27.7 The electric field inside a capacitor

EXAMPLE 27.7 The electric field inside a capacitor

A sphere of charge Q and radius R, be it a uniformly charged sphere or just a spherical shell, has an electric field outside the sphere that is exactly the same as that of a point charge Q located at the center of the sphere:

A Sphere of Charge